Optimal. Leaf size=75 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (a x^6-b\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \]
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Rubi [F] time = 1.09, 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 {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-b+a x^6}}+\frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-b+a x^6}} \, dx+\int \frac {3 b+2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx\\ &=\frac {\sqrt [4]{1-\frac {a x^6}{b}} \int \frac {1}{\sqrt [4]{1-\frac {a x^6}{b}}} \, dx}{\sqrt [4]{-b+a x^6}}+\int \left (-\frac {3 b}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}}+\frac {2 x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {a x^6}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};\frac {a x^6}{b}\right )}{\sqrt [4]{-b+a x^6}}+2 \int \frac {x^4}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx-(3 b) \int \frac {1}{\left (b+2 x^4-a x^6\right ) \sqrt [4]{-b+a x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 b+a x^6}{\sqrt [4]{-b+a x^6} \left (-b-2 x^4+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.20, size = 75, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x \left (-b+a x^6\right )^{3/4}}{b-a x^6}\right )}{\sqrt [4]{2}} \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 {a x^{6} + 2 \, b}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+2 b}{\left (a \,x^{6}-b \right )^{\frac {1}{4}} \left (a \,x^{6}-2 x^{4}-b \right )}\, dx\]
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 {a x^{6} + 2 \, b}{{\left (a x^{6} - 2 \, x^{4} - b\right )} {\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 {a\,x^6+2\,b}{{\left (a\,x^6-b\right )}^{1/4}\,\left (-a\,x^6+2\,x^4+b\right )} \,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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