Integrand size = 36, antiderivative size = 28 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]
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\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {x^{7/2} \left (-9+5 x^4\right )}{\sqrt {-1+x^4} \left (1-x^4+a x^9\right )} \, dx}{\sqrt {-x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^8 \left (-9+5 x^8\right )}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\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 {9 x^8}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )}+\frac {5 x^{16}}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}} \\ & = \frac {\left (10 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^{16}}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}-\frac {\left (18 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}} \\ \end{align*}
\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \]
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\[\int \frac {x^{4} \left (5 x^{4}-9\right )}{\sqrt {x^{5}-x}\, \left (a \,x^{9}-x^{4}+1\right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.36 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{18} + 6 \, a x^{13} - 6 \, a x^{9} + x^{8} - 2 \, x^{4} - 4 \, {\left (a x^{13} + x^{8} - x^{4}\right )} \sqrt {x^{5} - x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{13} + 2 \, a x^{9} + x^{8} - 2 \, x^{4} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{9} + x^{4} - 1\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{9} - a x^{5}\right )}}\right )}{a}\right ] \]
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\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (5 x^{4} - 9\right )}{\sqrt {x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (a x^{9} - x^{4} + 1\right )}\, dx \]
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\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]
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\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]
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Time = 6.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\frac {\ln \left (\frac {a\,x^9+x^4-2\,\sqrt {a}\,x^4\,\sqrt {x\,\left (x^4-1\right )}-1}{4\,a\,x^9-4\,x^4+4}\right )}{\sqrt {a}} \]
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