Why aren't these two integration methods yielding the same answer? I'm trying to solve this (not homework, if it matters), and both u-substitution and integration by parts are both yielding two different answers. Where am I going wrong?
Equation: $$\int \frac{(4x^3)}{(x^4+7)}dx$$
u-substitution answer: $$=\ln\big|(x^4+7)\big|+C$$
integration by parts answer: 
$$=\int4x^3*(x^4+7)^{-1}dx$$
$$=4x^3*\ln\big|x^4+7\big|-\int 12x^2*(x^4+7)^{-1}dx$$
$$=4x^3*\ln\big|x^4+7\big|-(12x^2*ln\big|x^4+7\big|-\int 24x*(x^4+7)^{-1}dx)$$
$$=4x^3*\ln\big|x^4+7\big|-(12x^2*ln\big|x^4+7\big|-24x*ln\big|x^4+7\big|-\int 24(x^4+7)^{-1}dx)$$
$$= 4x^3*\ln\big|x^4+7\big|-(12x^2*\ln\big|x^4+7\big|-(24x\ln\big|x^4+7\big|-24\ln\big|x^4+7\big|))$$
$$=(4x^3-12x^2+24x-24)(\ln\big|x^4+7\big|)$$

I know you are asking about a specific math problem, however, I ran into the same issue with finding the integral of 
$$ \int x(x-2)^5 dx.$$
The two answers I received were 
$$\frac{(x-2)^7}{7}+\frac{x-2^6}{3}+C$$
by using the Chain Rule and 
$$\frac{x (x-2)^6}{6}-\frac{(x-2)^7}{42} +C$$ 
using the Product Rule (integration by parts or tabular method). At first I thought they were different answers but I expanded both answers in my TI-89 (could be done by hand but will take a minute). Then I factored each expanded answer and received 
$$\frac{(x-2)^6 (3x+1)}{21}+C$$ 
for both answers. So through that route I figured out both yield the same answer but in different algebraic form. 
