Optimal. Leaf size=51 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b}}{x^2}\right )+2 \tanh ^{-1}\left (\frac {x^2 \left (a x^3-b\right )^{3/4}}{b-a x^3}\right ) \]
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Rubi [F] time = 1.00, 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 \left (-8 b+5 a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x \left (-8 b+5 a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^8\right )} \, dx &=\int \left (-\frac {5 a x^4}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^8\right )}-\frac {8 b x}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^8\right )}\right ) \, dx\\ &=-\left ((5 a) \int \frac {x^4}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^8\right )} \, dx\right )-(8 b) \int \frac {x}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^8\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-8 b+5 a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.90, size = 51, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b}}{x^2}\right )+2 \tanh ^{-1}\left (\frac {x^2 \left (a x^3-b\right )^{3/4}}{b-a x^3}\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 (5 \, a x^{3} - 8 \, b\right )} x}{{\left (x^{8} - a x^{3} + b\right )} {\left (a x^{3} - b\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 = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (5 a \,x^{3}-8 b \right )}{\left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (x^{8}-a \,x^{3}+b \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 (5 \, a x^{3} - 8 \, b\right )} x}{{\left (x^{8} - a x^{3} + b\right )} {\left (a x^{3} - b\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\,\left (8\,b-5\,a\,x^3\right )}{{\left (a\,x^3-b\right )}^{1/4}\,\left (x^8-a\,x^3+b\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 \left (5 a x^{3} - 8 b\right )}{\sqrt [4]{a x^{3} - b} \left (- a x^{3} + b + x^{8}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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