Optimal. Leaf size=43 \[ 2 \tan ^{-1}\left (x \sqrt [4]{a x^6-b x^3}\right )-2 \tanh ^{-1}\left (x \sqrt [4]{a x^6-b x^3}\right ) \]
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Rubi [F] time = 2.79, 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^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-b+a x^3}\right ) \int \frac {x^{17/4} \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-1-b x^7+a x^{10}\right )} \, dx}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (-7 b+10 a x^{12}\right )}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{-b+a x^{12}} \left (1+b x^{28}-a x^{40}\right )}+\frac {10 a x^{32}}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (40 a x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{32}}{\sqrt [4]{-b+a x^{12}} \left (-1-b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (28 b x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-b+a x^{12}} \left (1+b x^{28}-a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ \end {align*}
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Mathematica [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (-7 b+10 a x^3\right )}{\sqrt [4]{-b x^3+a x^6} \left (-1-b x^7+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 18.47, size = 43, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (x \sqrt [4]{-b x^3+a x^6}\right )-2 \tanh ^{-1}\left (x \sqrt [4]{-b x^3+a x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (10 \, a x^{3} - 7 \, b\right )} x^{5}}{{\left (a x^{10} - b x^{7} - 1\right )} {\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (10 a \,x^{3}-7 b \right )}{\left (a \,x^{6}-b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{7}-1\right )}\, dx\]
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 (10 \, a x^{3} - 7 \, b\right )} x^{5}}{{\left (a x^{10} - b x^{7} - 1\right )} {\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5\,\left (7\,b-10\,a\,x^3\right )}{{\left (a\,x^6-b\,x^3\right )}^{1/4}\,\left (-a\,x^{10}+b\,x^7+1\right )} \,d x \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^{5} \left (10 a x^{3} - 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{3} - b\right )} \left (a x^{10} - b x^{7} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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