Optimal. Leaf size=152 \[ -\frac {\log \left (a^2 d^{2/3} \left (x^3-a x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2 \sqrt [3]{x^3-a x^2}+a^2 x^4\right )}{2 d^{2/3}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x^2}\right )}{d^{2/3}} \]
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Rubi [F] time = 1.90, 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 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (a d-d x+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x (-4 a+3 x)}{\sqrt [3]{x^2 (-a+x)} \left (a d-d x+x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {\sqrt [3]{x} (-4 a+3 x)}{\sqrt [3]{-a+x} \left (a d-d x+x^4\right )} \, dx}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-4 a+3 x^3\right )}{\sqrt [3]{-a+x^3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-a+x}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 a x^3}{\sqrt [3]{-a+x^3} \left (a d-d x^3+x^{12}\right )}+\frac {3 x^6}{\sqrt [3]{-a+x^3} \left (a d-d x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ &=\frac {\left (9 x^{2/3} \sqrt [3]{-a+x}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-a+x^3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}-\frac {\left (12 a x^{2/3} \sqrt [3]{-a+x}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ \end {align*}
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Mathematica [C] time = 1.43, size = 646, normalized size = 4.25 \begin {gather*} \frac {a x \left (4 \text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-\sqrt [3]{\text {$\#$1}} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt [3]{\text {$\#$1}} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )-2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )+6 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]+5 \text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-2 \text {$\#$1}^{4/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{4/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1} \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]-\text {RootSum}\left [\text {$\#$1}^4 \left (-a^3\right )+\text {$\#$1}^3 d-3 \text {$\#$1}^2 d+3 \text {$\#$1} d-d\&,\frac {-2 \text {$\#$1}^{7/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x}{x-a}}\right )+\text {$\#$1}^{7/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x}{x-a}}+\left (\frac {x}{x-a}\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {x}{x-a}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-6 \text {$\#$1}^2 \sqrt [3]{\frac {x}{x-a}}}{4 \text {$\#$1}^3 a^3-3 \text {$\#$1}^2 d+6 \text {$\#$1} d-3 d}\&\right ]\right )}{2 \sqrt [3]{\frac {x}{x-a}} \sqrt [3]{x^2 (x-a)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.61, size = 78, normalized size = 0.51 \begin {gather*} -\frac {a \text {RootSum}\left [a^3-d \text {$\#$1}^3+3 d \text {$\#$1}^6-3 d \text {$\#$1}^9+d \text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-a x^2+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ]}{d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 156, normalized size = 1.03 \begin {gather*} \frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{2}}{x^{4}}\right )}{2 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x \left (-4 a +3 x \right )}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (x^{4}+a d -d x \right )}\, dx\]
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 (4 \, a - 3 \, x\right )} x}{{\left (x^{4} + a d - d x\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}}\,{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\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (x^4-d\,x+a\,d\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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