Integrand size = 56, antiderivative size = 95 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}+\frac {\arctan \left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \]
2/3*(a*x^6-b)^(3/4)/x^3+1/2*arctan(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2 ^(3/4)+1/2*arctanh(2^(1/4)*x*(a*x^6-b)^(3/4)/(-a*x^6+b))*2^(3/4)
Time = 8.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\frac {2 \left (-b+a x^6\right )^{3/4}}{3 x^3}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-b+a x^6}}\right )}{\sqrt [4]{2}} \]
(2*(-b + a*x^6)^(3/4))/(3*x^3) - ArcTan[(2^(1/4)*x)/(-b + a*x^6)^(1/4)]/2^ (1/4) - ArcTanh[(2^(1/4)*x)/(-b + a*x^6)^(1/4)]/2^(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.
\(\displaystyle \int \frac {\left (a x^6+2 b\right ) \left (a x^6-b-x^4\right )}{x^4 \sqrt [4]{a x^6-b} \left (a x^6-b-2 x^4\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\sqrt [4]{a x^6-b}}+\frac {-3 b-2 x^4}{\left (-a x^6+b+2 x^4\right ) \sqrt [4]{a x^6-b}}+\frac {2 b}{x^4 \sqrt [4]{a x^6-b}}+\frac {a x^2}{\sqrt [4]{a x^6-b}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 b \int \frac {1}{\left (-a x^6+2 x^4+b\right ) \sqrt [4]{a x^6-b}}dx+2 \int \frac {x^4}{\sqrt [4]{a x^6-b} \left (a x^6-2 x^4-b\right )}dx+\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},\frac {a x^6}{b}\right )}{\sqrt [4]{a x^6-b}}+\frac {2 \left (a x^6-b\right )^{3/4}}{3 x^3}\) |
3.14.23.3.1 Defintions of rubi rules used
Time = 1.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {6 \arctan \left (\frac {\left (a \,x^{6}-b \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}} x^{3}-3 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (a \,x^{6}-b \right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} x^{3}+8 \left (a \,x^{6}-b \right )^{\frac {3}{4}}}{12 x^{3}}\) | \(99\) |
1/12*(6*arctan(1/2*(a*x^6-b)^(1/4)/x*2^(3/4))*2^(3/4)*x^3-3*ln((-2^(1/4)*x -(a*x^6-b)^(1/4))/(2^(1/4)*x-(a*x^6-b)^(1/4)))*2^(3/4)*x^3+8*(a*x^6-b)^(3/ 4))/x^3
Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\text {Timed out} \]
integrate((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
integrate((a*x^6+2*b)*(a*x^6-x^4-b)/x^4/(a*x^6-b)^(1/4)/(a*x^6-2*x^4-b),x, algorithm="maxima")
\[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} - x^{4} - b\right )} {\left (a x^{6} + 2 \, b\right )}}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\left (a x^{6} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (2 b+a x^6\right ) \left (-b-x^4+a x^6\right )}{x^4 \sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx=\int \frac {\left (a\,x^6+2\,b\right )\,\left (-a\,x^6+x^4+b\right )}{x^4\,{\left (a\,x^6-b\right )}^{1/4}\,\left (-a\,x^6+2\,x^4+b\right )} \,d x \]