Integrand size = 42, antiderivative size = 107 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\frac {x \left (-b+a x^2+c x^4\right )^{3/4}}{2 c \left (b-a x^2\right )}-\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \]
[Out]
\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\frac {-\frac {2 \sqrt [4]{c} x \left (-b+a x^2+c x^4\right )^{3/4}}{b-a x^2}-\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(-\frac {-4 \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {3}{4}} x \,c^{\frac {1}{4}}+\left (\ln \left (\frac {-c^{\frac {1}{4}} x -\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )\right ) \left (a \,x^{2}-b \right )}{c^{\frac {5}{4}} \left (8 a \,x^{2}-8 b \right )}\) | \(131\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.23 \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\frac {{\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - {\left (a c x^{2} - b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (-\frac {c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x - {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + {\left (-i \, a c x^{2} + i \, b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {i \, c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + {\left (i \, a c x^{2} - i \, b c\right )} \frac {1}{c^{5}}^{\frac {1}{4}} \log \left (\frac {-i \, c^{4} \frac {1}{c^{5}}^{\frac {3}{4}} x + {\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (c x^{4} + a x^{2} - b\right )}^{\frac {3}{4}} x}{8 \, {\left (a c x^{2} - b c\right )}} \]
[In]
[Out]
\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int \frac {x^{4} \left (a x^{2} - 2 b\right )}{\left (a x^{2} - b\right )^{2} \sqrt [4]{a x^{2} - b + c x^{4}}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int { \frac {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=\int { \frac {{\left (a x^{2} - 2 \, b\right )} x^{4}}{{\left (c x^{4} + a x^{2} - b\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx=-\int \frac {x^4\,\left (2\,b-a\,x^2\right )}{{\left (b-a\,x^2\right )}^2\,{\left (c\,x^4+a\,x^2-b\right )}^{1/4}} \,d x \]
[In]
[Out]