Integrand size = 51, antiderivative size = 141 \[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\frac {2 \sqrt [4]{b+a x^6}}{x}+\sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )-\sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^6}}\right ) \]
2*(a*x^6+b)^(1/4)/x+2^(1/2)*c^(1/4)*arctan(2^(1/2)*c^(1/4)*x*(a*x^6+b)^(1/ 4)/(-c^(1/2)*x^2+(a*x^6+b)^(1/2)))-2^(1/2)*c^(1/4)*arctanh((1/2*c^(1/4)*x^ 2*2^(1/2)+1/2*(a*x^6+b)^(1/2)*2^(1/2)/c^(1/4))/x/(a*x^6+b)^(1/4))
Time = 8.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\frac {2 \sqrt [4]{b+a x^6}}{x}+\sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}{-\sqrt {c} x^2+\sqrt {b+a x^6}}\right )-\sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {b+a x^6}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b+a x^6}}\right ) \]
(2*(b + a*x^6)^(1/4))/x + Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a *x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^6])] - Sqrt[2]*c^(1/4)*ArcTanh [(Sqrt[c]*x^2 + Sqrt[b + a*x^6])/(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(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.
| \(\displaystyle \int \frac {\left (a x^6-2 b\right ) \left (a x^6+b-c x^4\right )}{x^2 \left (a x^6+b\right )^{3/4} \left (a x^6+b+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 c^2}{a \left (a x^6+b\right )^{3/4}}+\frac {2 c \left (3 a b x^2-b c-c^2 x^4\right )}{a \left (a x^6+b\right )^{3/4} \left (a x^6+b+c x^4\right )}-\frac {2 c x^2}{\left (a x^6+b\right )^{3/4}}+\frac {a x^4}{\left (a x^6+b\right )^{3/4}}-\frac {2 b}{x^2 \left (a x^6+b\right )^{3/4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c^3 \int \frac {x^4}{\left (a x^6+b\right )^{3/4} \left (a x^6+c x^4+b\right )}dx}{a}-\frac {2 b c^2 \int \frac {1}{\left (a x^6+b\right )^{3/4} \left (a x^6+c x^4+b\right )}dx}{a}+6 b c \int \frac {x^2}{\left (a x^6+b\right )^{3/4} \left (a x^6+c x^4+b\right )}dx-\frac {4 \sqrt {b} c \left (\frac {a x^6}{b}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} x^3}{\sqrt {b}}\right ),2\right )}{3 \sqrt {a} \left (a x^6+b\right )^{3/4}}+\frac {2 c^2 x \left (\frac {a x^6}{b}+1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},-\frac {a x^6}{b}\right )}{a \left (a x^6+b\right )^{3/4}}+\frac {2 b \left (\frac {a x^6}{b}+1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{4},\frac {5}{6},-\frac {a x^6}{b}\right )}{x \left (a x^6+b\right )^{3/4}}+\frac {a x^5 \left (\frac {a x^6}{b}+1\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{6},\frac {11}{6},-\frac {a x^6}{b}\right )}{5 \left (a x^6+b\right )^{3/4}}\) |
3.21.3.3.1 Defintions of rubi rules used
Time = 0.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(\frac {-\ln \left (\frac {\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{6}+b}}{\sqrt {a \,x^{6}+b}-\left (a \,x^{6}+b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}\right ) c^{\frac {1}{4}} \sqrt {2}\, x -2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {1}{4}} \sqrt {2}\, x -2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{6}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {1}{4}} \sqrt {2}\, x +4 \left (a \,x^{6}+b \right )^{\frac {1}{4}}}{2 x}\) | \(173\) |
1/2*(-ln(((a*x^6+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2+(a*x^6+b)^(1/2))/( (a*x^6+b)^(1/2)-(a*x^6+b)^(1/4)*x*c^(1/4)*2^(1/2)+c^(1/2)*x^2))*c^(1/4)*2^ (1/2)*x-2*arctan((2^(1/2)*(a*x^6+b)^(1/4)+c^(1/4)*x)/c^(1/4)/x)*c^(1/4)*2^ (1/2)*x-2*arctan((2^(1/2)*(a*x^6+b)^(1/4)-c^(1/4)*x)/c^(1/4)/x)*c^(1/4)*2^ (1/2)*x+4*(a*x^6+b)^(1/4))/x
Timed out. \[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\text {Timed out} \]
integrate((a*x^6-2*b)*(a*x^6-c*x^4+b)/x^2/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b), x, algorithm="fricas")
Timed out. \[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - c x^{4} + b\right )} {\left (a x^{6} - 2 \, b\right )}}{{\left (a x^{6} + c x^{4} + b\right )} {\left (a x^{6} + b\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
integrate((a*x^6-2*b)*(a*x^6-c*x^4+b)/x^2/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b), x, algorithm="maxima")
\[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - c x^{4} + b\right )} {\left (a x^{6} - 2 \, b\right )}}{{\left (a x^{6} + c x^{4} + b\right )} {\left (a x^{6} + b\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
integrate((a*x^6-2*b)*(a*x^6-c*x^4+b)/x^2/(a*x^6+b)^(3/4)/(a*x^6+c*x^4+b), x, algorithm="giac")
Timed out. \[ \int \frac {\left (-2 b+a x^6\right ) \left (b-c x^4+a x^6\right )}{x^2 \left (b+a x^6\right )^{3/4} \left (b+c x^4+a x^6\right )} \, dx=\int -\frac {\left (2\,b-a\,x^6\right )\,\left (a\,x^6-c\,x^4+b\right )}{x^2\,{\left (a\,x^6+b\right )}^{3/4}\,\left (a\,x^6+c\,x^4+b\right )} \,d x \]