Integrand size = 51, antiderivative size = 123 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]
4/3*(a*x^3-b)^(3/4)/x^3-2*2^(1/2)*arctan((-1/2*2^(1/2)*x^2+1/2*(a*x^3-b)^( 1/2)*2^(1/2))/x/(a*x^3-b)^(1/4))+2*2^(1/2)*arctanh(2^(1/2)*x*(a*x^3-b)^(1/ 4)/(x^2+(a*x^3-b)^(1/2)))
Time = 1.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b+a x^3}}{\sqrt {2} x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]
(4*(-b + a*x^3)^(3/4))/(3*x^3) - 2*Sqrt[2]*ArcTan[(-x^2 + Sqrt[-b + a*x^3] )/(Sqrt[2]*x*(-b + a*x^3)^(1/4))] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-b + a*x ^3)^(1/4))/(x^2 + Sqrt[-b + a*x^3])]
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.
\(\displaystyle \int \frac {\left (a x^3-4 b\right ) \left (-a x^3+b+x^4\right )}{x^4 \sqrt [4]{a x^3-b} \left (a x^3-b+x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {a}{x \sqrt [4]{a x^3-b}}+\frac {4 b}{x^4 \sqrt [4]{a x^3-b}}+\frac {2 \left (4 b-a x^3\right )}{\sqrt [4]{a x^3-b} \left (-a x^3+b-x^4\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 b \int \frac {1}{\sqrt [4]{a x^3-b} \left (-x^4-a x^3+b\right )}dx+2 a \int \frac {x^3}{\sqrt [4]{a x^3-b} \left (x^4+a x^3-b\right )}dx+\frac {4 \left (a x^3-b\right )^{3/4}}{3 x^3}\) |
3.19.13.3.1 Defintions of rubi rules used
\[\int \frac {\left (a \,x^{3}-4 b \right ) \left (-a \,x^{3}+x^{4}+b \right )}{x^{4} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (a \,x^{3}+x^{4}-b \right )}d x\]
Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int { -\frac {{\left (a x^{3} - x^{4} - b\right )} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} + x^{4} - b\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
\[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int { -\frac {{\left (a x^{3} - x^{4} - b\right )} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} + x^{4} - b\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int -\frac {\left (4\,b-a\,x^3\right )\,\left (x^4-a\,x^3+b\right )}{x^4\,{\left (a\,x^3-b\right )}^{1/4}\,\left (x^4+a\,x^3-b\right )} \,d x \]