Optimal. Leaf size=95 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )-\frac {x \left (a x^6-b\right )^{3/4}}{2 \left (a x^6-b-x^4\right )} \]
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Rubi [F] time = 2.94, 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 (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx &=\int \left (\frac {b+a b x^2+\left (1+3 a^2 b\right ) x^4}{a^2 \left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {1+a x^2+a^2 x^4}{a^2 \sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {b+a b x^2+\left (1+3 a^2 b\right ) x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a^2}+\frac {\int \frac {1+a x^2+a^2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a^2}\\ &=\frac {\int \left (\frac {b}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {a b x^2}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}+\frac {\left (1+3 a^2 b\right ) x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}}\right ) \, dx}{a^2}+\frac {\int \left (\frac {1}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}+\frac {a x^2}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}+\frac {a^2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )}\right ) \, dx}{a^2}\\ &=\frac {\int \frac {1}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a^2}+\frac {\int \frac {x^2}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx}{a}+\frac {b \int \frac {1}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a^2}+\frac {b \int \frac {x^2}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx}{a}+\left (\frac {1}{a^2}+3 b\right ) \int \frac {x^4}{\left (b+x^4-a x^6\right )^2 \sqrt [4]{-b+a x^6}} \, dx+\int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (2 b+a x^6\right )}{\sqrt [4]{-b+a x^6} \left (-b-x^4+a x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 15.56, size = 95, normalized size = 1.00 \begin {gather*} -\frac {x \left (-b+a x^6\right )^{3/4}}{2 \left (-b-x^4+a x^6\right )}-\frac {1}{4} \tan ^{-1}\left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {x \left (-b+a x^6\right )^{3/4}}{b-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 (a x^{6} + 2 \, b\right )} x^{4}}{{\left (a x^{6} - x^{4} - b\right )}^{2} {\left (a x^{6} - 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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{6}+2 b \right )}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} \left (a \,x^{6}-x^{4}-b \right )^{2}}\, dx\]
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 (a x^{6} + 2 \, b\right )} x^{4}}{{\left (a x^{6} - x^{4} - b\right )}^{2} {\left (a x^{6} - 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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (a\,x^6+2\,b\right )}{{\left (a\,x^6-b\right )}^{1/4}\,{\left (-a\,x^6+x^4+b\right )}^2} \,d x \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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