Optimal. Leaf size=28 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^5-x}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]
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Rubi [F] time = 1.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.54, size = 28, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 146, normalized size = 5.21 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (5 x^{4}-9\right )}{\sqrt {x^{5}-x}\, \left (a \,x^{9}-x^{4}+1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 48, normalized size = 1.71 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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