Integrand size = 43, antiderivative size = 96 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{2^{3/4} c^{3/4}} \]
-1/2*arctan(2^(1/4)*c^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(1/4)/c^(3/4)+ 1/2*arctanh(2^(1/4)*c^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(1/4)/c^(3/4)
Time = 8.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=-\frac {-\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+a x^6}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+a x^6}}\right )}{2^{3/4} c^{3/4}} \]
-((-ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + a*x^6)^(1/4)] + ArcTanh[(2^(1/4)*c^(1 /4)*x)/(-b + a*x^6)^(1/4)])/(2^(3/4)*c^(3/4)))
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.
\(\displaystyle \int \frac {x^2 \left (a x^6+2 b\right )}{\left (a x^6-b\right )^{3/4} \left (a x^6-b-2 c x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 a b x^2+2 b c+4 c^2 x^4}{a \left (a x^6-b\right )^{3/4} \left (a x^6-b-2 c x^4\right )}+\frac {2 c}{a \left (a x^6-b\right )^{3/4}}+\frac {x^2}{\left (a x^6-b\right )^{3/4}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 c^2 \int \frac {x^4}{\left (-a x^6+2 c x^4+b\right ) \left (a x^6-b\right )^{3/4}}dx}{a}-\frac {2 b c \int \frac {1}{\left (-a x^6+2 c x^4+b\right ) \left (a x^6-b\right )^{3/4}}dx}{a}+3 b \int \frac {x^2}{\left (a x^6-b\right )^{3/4} \left (a x^6-2 c x^4-b\right )}dx+\frac {\sqrt {\frac {a x^6}{\left (\sqrt {a x^6-b}+\sqrt {b}\right )^2}} \left (\sqrt {a x^6-b}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^6-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a \sqrt [4]{b} x^3}+\frac {2 c x \left (1-\frac {a x^6}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {3}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{a \left (a x^6-b\right )^{3/4}}\) |
3.14.36.3.1 Defintions of rubi rules used
Time = 3.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {1}{4}} c^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} c^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,c^{\frac {1}{4}}}\right )\right )}{4 c^{\frac {3}{4}}}\) | \(84\) |
-1/4*2^(1/4)/c^(3/4)*(ln((-2^(1/4)*c^(1/4)*x-(a*x^6-b)^(1/4))/(2^(1/4)*c^( 1/4)*x-(a*x^6-b)^(1/4)))+2*arctan(1/2*(a*x^6-b)^(1/4)/x*2^(3/4)/c^(1/4)))
Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]
\[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + 2 \, b\right )} x^{2}}{{\left (a x^{6} - 2 \, c x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {3}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (2 b+a x^6\right )}{\left (-b+a x^6\right )^{3/4} \left (-b-2 c x^4+a x^6\right )} \, dx=\int -\frac {x^2\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{3/4}\,\left (-a\,x^6+2\,c\,x^4+b\right )} \,d x \]