Optimal. Leaf size=146 \[ \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b}}{x}\right )+\tanh ^{-1}\left (\frac {x \left (a x^3-b\right )^{3/4}}{b-a x^3}\right )-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}+x^2}\right )}{\sqrt {2}} \]
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Rubi [F] time = 2.38, 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 (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx &=\int \left (\frac {4 b-a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {-4 b+a x^3}{2 \sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx+\frac {1}{2} \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )}-\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3-x^4\right )} \, dx\right )+\frac {1}{2} a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-(2 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (-4 b+a x^3\right )}{\sqrt [4]{-b+a x^3} \left (-b^2+2 a b x^3-a^2 x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.77, size = 146, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {\sqrt [4]{-b+a x^3}}{x}\right )-\frac {\tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )}{\sqrt {2}}+\tanh ^{-1}\left (\frac {x \left (-b+a x^3\right )^{3/4}}{b-a x^3}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right )}{\sqrt {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 {{\left (a x^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{3}-4 b \right )}{\left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (-a^{2} x^{6}+x^{8}+2 a b \,x^{3}-b^{2}\right )}\, dx\]
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 (a x^{3} - 4 \, b\right )} x^{4}}{{\left (a^{2} x^{6} - x^{8} - 2 \, a b x^{3} + b^{2}\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 = -0.01 \begin {gather*} -\int -\frac {x^4\,\left (4\,b-a\,x^3\right )}{{\left (a\,x^3-b\right )}^{1/4}\,\left (a^2\,x^6-2\,a\,b\,x^3+b^2-x^8\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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