Integrand size = 34, antiderivative size = 26 \[ \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]
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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. 47 vs. \(2 (20) = 40\).
Time = 0.38 (sec) , antiderivative size = 139, normalized size of antiderivative = 5.35 \[ \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{18} + 6 \, a x^{13} + 6 \, a x^{9} + x^{8} + 2 \, x^{4} - 4 \, {\left (a x^{13} + x^{8} + x^{4}\right )} \sqrt {x^{5} + x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{13} - 2 \, a x^{9} + x^{8} + 2 \, x^{4} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{9} + x^{4} + 1\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{9} + a x^{5}\right )}}\right )}{a}\right ] \]
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\[ \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (5 x^{4} + 9\right )}{\sqrt {x \left (x^{4} + 1\right )} \left (a x^{9} - x^{4} - 1\right )}\, dx \]
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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 = 5.82 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {x^4 \left (9+5 x^4\right )}{\sqrt {x+x^5} \left (-1-x^4+a x^9\right )} \, dx=\frac {\ln \left (\frac {a\,x^9+x^4-2\,\sqrt {a}\,x^4\,\sqrt {x^5+x}+1}{-4\,a\,x^9+4\,x^4+4}\right )}{\sqrt {a}} \]
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