Integrand size = 39, antiderivative size = 62 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \left (3 a+x+3 a x^4\right ) \sqrt {x+x^5}}{3 \left (1+x^4\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \]
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\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {x^{3/2} \left (-1+3 x^4\right )}{\left (1+x^4\right )^{5/2} \left (a-x+a x^4\right )} \, dx}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \left (-1+3 x^8\right )}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (\frac {3 x^4}{a \left (1+x^8\right )^{5/2}}+\frac {x^4 \left (-4 a+3 x^2\right )}{a \left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \left (-4 a+3 x^2\right )}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ & = \frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 a x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}+\frac {3 x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}}+\frac {\left (7 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {x+x^5}} \\ & = \frac {7 x^3}{8 a \sqrt {x+x^5}}+\frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (7 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{8 a \sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ & = \frac {7 x^3}{8 a \sqrt {x+x^5}}+\frac {x^3}{2 a \left (1+x^4\right ) \sqrt {x+x^5}}-\frac {7 x^3 \sqrt {1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{8},\frac {13}{8},-x^4\right )}{40 a \sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{5/2} \left (a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ \end{align*}
Time = 22.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5} \left (x+3 a \left (1+x^4\right )\right )}{3 \left (1+x^4\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right ) \]
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Time = 1.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(\frac {-6 a^{\frac {3}{2}} \sqrt {x \left (x^{4}+1\right )}\, \left (x^{4}+1\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}\, \sqrt {a}}{x}\right )+6 x \left (a \,x^{4}+a +\frac {1}{3} x \right )}{\sqrt {x \left (x^{4}+1\right )}\, \left (3 x^{4}+3\right )}\) | \(70\) |
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Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.73 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {3 \, {\left (a x^{8} + 2 \, a x^{4} + a\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, {\left (3 \, a x^{4} + 3 \, a + x\right )} \sqrt {x^{5} + x}}{6 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}}, \frac {3 \, {\left (a x^{8} + 2 \, a x^{4} + a\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right ) + 2 \, {\left (3 \, a x^{4} + 3 \, a + x\right )} \sqrt {x^{5} + x}}{3 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}}\right ] \]
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Timed out. \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} x^{2}}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} x^{2}}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x} {\left (x^{4} + 1\right )}^{2}} \,d x } \]
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Time = 6.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {x^2 \left (-1+3 x^4\right )}{\left (1+x^4\right )^2 \left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=a^{3/2}\,\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\right )+\frac {2\,a\,\sqrt {x^5+x}}{x^4+1}+\frac {2\,x\,\sqrt {x^5+x}}{3\,{\left (x^4+1\right )}^2} \]
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