How Do You Simplify Complex Root Integrals?

[transgalactic]

i tried this:
$$\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}=\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}*\frac{1+\sqrt{x+1}}{1+\sqrt{x+1}}*\frac{1-\sqrt[3]{x+1}+(x+1)^{\frac{3}{2}}}{1-\sqrt[3]{x+1}+(x+1)^{\frac{3}{2}}}$$
but when i got read of 2 roots i got another two roots which are more complicated
??

 

[tiny-tim]

$$\int \frac{1-\sqrt{x+1}}{1+\sqrt[3]{x+1}}$$

Hi transgalactic!

Hint: substitute!

 

[transgalactic]

i tried t=(x+1)^1/3
but it creates anoted roots in the dt
??

 

[tiny-tim]

i tried t=(x+1)^1/3
but it creates anoted roots in the dt
??

uhh?

dx = … ?

anyway, (x+1)^1/6 might be easier.

 

[transgalactic]

but i don't have members of 1/6 power
??

 

[Mark44]

Sure you do. a^(1/2) = a^(1/6)^3, and b^(1/3) = b^(1/6)^2.