Integrand size = 54, antiderivative size = 72 \[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-2^{3/4} \arctan \left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{2} x}\right )+2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b+a x^3}}\right ) \]
-4/3*(a*x^3+b)^(3/4)/x^3-2^(3/4)*arctan(1/2*(a*x^3+b)^(1/4)*2^(3/4)/x)+2^( 3/4)*arctanh(2^(1/4)*x/(a*x^3+b)^(1/4))
Time = 1.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-2^{3/4} \arctan \left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{2} x}\right )+2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b+a x^3}}\right ) \]
(-4*(b + a*x^3)^(3/4))/(3*x^3) - 2^(3/4)*ArcTan[(b + a*x^3)^(1/4)/(2^(1/4) *x)] + 2^(3/4)*ArcTanh[(2^(1/4)*x)/(b + a*x^3)^(1/4)]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a x^3+4 b\right ) \left (-a x^3-b+x^4\right )}{x^4 \sqrt [4]{a x^3+b} \left (-a x^3-b+2 x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {a}{x \sqrt [4]{a x^3+b}}+\frac {4 b}{x^4 \sqrt [4]{a x^3+b}}+\frac {a x^3+4 b}{\sqrt [4]{a x^3+b} \left (a x^3+b-2 x^4\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 b \int \frac {1}{\sqrt [4]{a x^3+b} \left (-2 x^4+a x^3+b\right )}dx+a \int \frac {x^3}{\sqrt [4]{a x^3+b} \left (-2 x^4+a x^3+b\right )}dx-\frac {4 \left (a x^3+b\right )^{3/4}}{3 x^3}\) |
3.10.50.3.1 Defintions of rubi rules used
\[\int \frac {\left (a \,x^{3}+4 b \right ) \left (-a \,x^{3}+x^{4}-b \right )}{x^{4} \left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (-a \,x^{3}+2 x^{4}-b \right )}d x\]
Timed out. \[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=\text {Timed out} \]
integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b), x, algorithm="fricas")
Timed out. \[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=\int { \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b), x, algorithm="maxima")
\[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=\int { \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
integrate((a*x^3+4*b)*(-a*x^3+x^4-b)/x^4/(a*x^3+b)^(1/4)/(-a*x^3+2*x^4-b), x, algorithm="giac")
Timed out. \[ \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx=\int \frac {\left (a\,x^3+4\,b\right )\,\left (-x^4+a\,x^3+b\right )}{x^4\,{\left (a\,x^3+b\right )}^{1/4}\,\left (-2\,x^4+a\,x^3+b\right )} \,d x \]