Evaluating Integral with Mathematica: A & B Cases

[ChrisVer]

Could someone please help me evaluate the integral with mathematica

$\int \frac{dx}{(a(1+x^{-1})+b(1+x^{2})-1)^{1/2}}$

For better in your code the integral must be:
1/sqrt[a(1+(1/x))+b(1+x^(2))-1]

For $a≤1$ and for cases:

A)$0<b<1$
B)$b>1$

I am sorry,but I haven't been able to receive mathematica yet... *sad face*
Deep thanks in regard

*not to be misunderstood that I'm asking to find everything ready I even know the codes I'd use in such a case:
Expand[Assuming[0<a<1 && b>1, Integrate[1/sqrt[a(1+(1/x))+b(1+x^(2))-1]]]]
Expand[Assuming[0<a<1 && 0<b<1, Integrate[1/sqrt[a(1+(1/x))+b(1+x^(2))-1]]]]
(if there would be an error I'd try to remove the expand)...I just still don't have the software at hand

 

[Ray Vickson]

Could someone please help me evaluate the integral with mathematica

$\int \frac{dx}{(a(1+x^{-1})+b(1+x^{2})-1)^{1/2}}$

For better in your code the integral must be:
1/sqrt[a(1+(1/x))+b(1+x^(2))-1]

For $a≤1$ and for cases:

A)$0<b<1$
B)$b>1$

I am sorry,but I haven't been able to receive mathematica yet... *sad face*
Deep thanks in regard

*not to be misunderstood that I'm asking to find everything ready I even know the codes I'd use in such a case:
Expand[Assuming[0<a<1 && b>1, Integrate[1/sqrt[a(1+(1/x))+b(1+x^(2))-1]]]]
Expand[Assuming[0<a<1 && 0<b<1, Integrate[1/sqrt[a(1+(1/x))+b(1+x^(2))-1]]]]
(if there would be an error I'd try to remove the expand)...I just still don't have the software at hand

I don't understand your statement that you " haven't been able to receive mathematica yet...". Does that mean that you have placed an order to buy Mathematica but it has not arrived yet, or what?

Anyway, I don't have access to Mathematica, so I did it in Maple instead. The results are exceedingly complicated, involving Elliptic functions of complex arguments, etc. Here is the code and result for 0 < b < 1:
> lprint(f); <---I call your function 'f'
1/(a*(1+1/x)+b*(1+x^2)-1)^(1/2)
J1:=int(f,x) assuming a<1,b>0,b<1: <---output suppressed by ending in ':'
lprint(J1);-4*(EllipticF(6^(1/2)*((3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)/(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b))^(1/2),((-3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+3*12^(1/3)*b*a+3*12^(1/3)*b^2-3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(-((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+12^(1/3)*b*a+12^(1/3)*b^2-12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b))^(1/2))-EllipticPi(6^(1/2)*((3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)/(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b))^(1/2),(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b),((-3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+3*12^(1/3)*b*a+3*12^(1/3)*b^2-3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(-((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+12^(1/3)*b*a+12^(1/3)*b^2-12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b))^(1/2)))/((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)*(-(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b)*(12*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)+12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-b*12^(2/3)*a-12^(2/3)*b^2+12^(2/3)*b+I*3^(1/2)*12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(2/3)*b*a+I*3^(1/2)*12^(2/3)*b^2-I*3^(1/2)*12^(2/3)*b)/(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b))^(1/2)*((((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b)*(12*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)+12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-b*12^(2/3)*a-12^(2/3)*b^2+12^(2/3)*b-I*3^(1/2)*12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-I*3^(1/2)*12^(2/3)*b*a-I*3^(1/2)*12^(2/3)*b^2+I*3^(1/2)*12^(2/3)*b)/(-((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+12^(1/3)*b*a+12^(1/3)*b^2-12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b))^(1/2)*(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b)^2*((3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)/(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b))^(1/2)*(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*x*((a*x+a+b*x+b*x^3-x)/x)^(1/2)/(x*(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+b*12^(2/3)*a+12^(2/3)*b^2-12^(2/3)*b)*(12*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)+12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-b*12^(2/3)*a-12^(2/3)*b^2+12^(2/3)*b-I*3^(1/2)*12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-I*3^(1/2)*12^(2/3)*b*a-I*3^(1/2)*12^(2/3)*b^2+I*3^(1/2)*12^(2/3)*b)*(12*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)+12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-b*12^(2/3)*a-12^(2/3)*b^2+12^(2/3)*b+I*3^(1/2)*12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(2/3)*b*a+I*3^(1/2)*12^(2/3)*b^2-I*3^(1/2)*12^(2/3)*b)/(-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2)))^(1/2)/(3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(x*(a*x+a+b*x+b*x^3-x))^(1/2)

Something similar is obtained for the case b > 1.

Note: the command 'lprint' gives AASCII output suitable for inclusion as text. The on-screen output looks much better, but still needs 9 pages to display.

 

[ChrisVer]

Could your program then, instead, find solution to:
$a(x+1)+b(x^3+x)-x=0$
a(x+1)+b(x^3+x)-x=0
for the same domains of a,b?

(Also for the mathematica, I'm having it offered by my univ, but unfortunately I learned today that I have to send a mail first to them in order to be able to download it.)

 

[Mark44]

Questions about integrals should be posted in the Calculus & Beyond section.

 

[Ray Vickson]

Could your program then, instead, find solution to:
$a(x+1)+b(x^3+x)-x=0$
a(x+1)+b(x^3+x)-x=0
for the same domains of a,b?

(Also for the mathematica, I'm having it offered by my univ, but unfortunately I learned today that I have to send a mail first to them in order to be able to download it.)

Yes, Maple can solve that equation---it just uses standard formulas for the solutions of a cubic equation, that you can find easily on-line. You can solve the equation yourself using Wolfram Alpha, which is like Mathematica lite and is freely available on the web. PF rules forbid me from writing the answer here.

 

[SammyS]

I don't understand your statement that you " haven't been able to receive mathematica yet...". Does that mean that you have placed an order to buy Mathematica but it has not arrived yet, or what?

Anyway, I don't have access to Mathematica, so I did it in Maple instead. The results are exceedingly complicated, involving Elliptic functions of complex arguments, etc. Here is the code and result for 0 < b < 1:
> lprint(f); <---I call your function 'f'
1/(a*(1+1/x)+b*(1+x^2)-1)^(1/2)
J1:=int(f,x) assuming a<1,b>0,b<1: <---output suppressed by ending in ':'
lprint(J1);-4*(EllipticF(6^(1/2)*((3*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-3*12^(1/3)*b*a-3*12^(1/3)*b^2+3*12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)/(((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)-12^(1/3)*b*a-12^(1/3)*b^2+12^(1/3)*b+I*3^(1/2)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(2/3)+I*3^(1/2)*12^(1/3)*b*a+I*3^(1/2)*12^(1/3)*b^2-I*3^(1/2)*12^(1/3)*b)/(6*x*b*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-4)/b)^(1/2))*b^2)^(1/3)-12^(1/3)*((-9*a+3^(1/2)*((4*a^3+39*a^2*b-12*a^2+12*a*b^2-24*a*b+12*a+4*b^3-12*b^2+12*b-

...

Hey Ray:

I think there's a mistake in line 57, the 31's chara...

Oh! Nevermind.

I had the wrong eyeglasses on.

 

[webdevelopment]

tell me about your post details.

Dear,
Would you tell me about your thread details. I want to clear that. Thank you for your post.

 

[ChrisVer]

Nevermind I'm getting weird results... For example I was expecting the cubic expression I gave above not to have positive solutions in this domains.
However $a=0.25=b$ have solution at $1$ and $0.618$...