Optimal. Leaf size=193 \[ \frac {\sqrt {2} \left (-4 a+\sqrt {1-4 a}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {2 a-\sqrt {1-4 a}-1}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {2 a-\sqrt {1-4 a}-1}}+\frac {\sqrt {2} \left (4 a+\sqrt {1-4 a}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {2 a+\sqrt {1-4 a}-1}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {2 a+\sqrt {1-4 a}-1}} \]
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Rubi [A] time = 0.38, antiderivative size = 81, normalized size of antiderivative = 0.42, number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6715, 826, 1164, 628} \begin {gather*} \frac {\log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}-\sqrt {x^5-1}\right )}{5 \sqrt {a}}-\frac {\log \left (-\sqrt {a} \left (1-x^5\right )+\sqrt {a}+\sqrt {x^5-1}\right )}{5 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 826
Rule 1164
Rule 6715
Rubi steps
\begin {align*} \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {-2+x}{\sqrt {-1+x} \left (1-x+a x^2\right )} \, dx,x,x^5\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {-1+x^2}{a+(-1+2 a) x^2+a x^4} \, dx,x,\sqrt {-1+x^5}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{\sqrt {a}}+2 x}{-1-\frac {x}{\sqrt {a}}-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{\sqrt {a}}-2 x}{-1+\frac {x}{\sqrt {a}}-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ &=\frac {\log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )-\sqrt {-1+x^5}\right )}{5 \sqrt {a}}-\frac {\log \left (\sqrt {a}-\sqrt {a} \left (1-x^5\right )+\sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 158, normalized size = 0.82 \begin {gather*} \frac {\left (\sqrt {1-4 a}-1\right ) \sqrt {-2 a+\sqrt {1-4 a}+1} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {-2 a+\sqrt {1-4 a}+1}}\right )-\left (\sqrt {1-4 a}+1\right ) \sqrt {-2 a-\sqrt {1-4 a}+1} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {-2 a-\sqrt {1-4 a}+1}}\right )}{5 \sqrt {2} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 193, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \left (-4 a+\sqrt {1-4 a}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {2 a-\sqrt {1-4 a}-1}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {2 a-\sqrt {1-4 a}-1}}+\frac {\sqrt {2} \left (4 a+\sqrt {1-4 a}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x^5-1}}{\sqrt {2 a+\sqrt {1-4 a}-1}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {2 a+\sqrt {1-4 a}-1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 74, normalized size = 0.38 \begin {gather*} \left [\frac {\log \left (\frac {a x^{10} - 2 \, \sqrt {x^{5} - 1} \sqrt {a} x^{5} + x^{5} - 1}{a x^{10} - x^{5} + 1}\right )}{5 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{5}}{\sqrt {x^{5} - 1}}\right )}{5 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (x^{5}-2\right )}{\sqrt {x^{5}-1}\, \left (a \,x^{10}-x^{5}+1\right )}\, dx \end {gather*}
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 (x^{5} - 2\right )} x^{4}}{{\left (a x^{10} - x^{5} + 1\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 47, normalized size = 0.24 \begin {gather*} \frac {\ln \left (\frac {a\,x^{10}+x^5-2\,\sqrt {a}\,x^5\,\sqrt {x^5-1}-1}{4\,a\,x^{10}-4\,x^5+4}\right )}{5\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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