Optimal. Leaf size=294 \[ -\frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [12]{d} x\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [3]{x^3-a x^2}+\sqrt [12]{d} x\right )}{4 a d^{5/6}}+\frac {\log \left (\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{4 a d^{5/6}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \left (x^3-a x^2\right )^{2/3}}\right )}{4 a d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{2 \left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2}\right )}{4 a d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} \left (x^3-a x^2\right )^{2/3}}{(x-a) \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x^2}\right )}{4 a d^{5/6}} \]
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Rubi [F] time = 1.79, 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^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-a+x} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) 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^9}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9+(-1+d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 219, normalized size = 0.74 \begin {gather*} \frac {x \left (-\log \left (\sqrt [3]{d} \left (\frac {x}{x-a}\right )^{4/3}-\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1\right )+\log \left (\sqrt [3]{d} \left (\frac {x}{x-a}\right )^{4/3}+\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}+1}{\sqrt {3}}\right )+4 \tanh ^{-1}\left (\sqrt [6]{d} \left (\frac {x}{x-a}\right )^{2/3}\right )\right )}{8 a d^{5/6} \sqrt [3]{\frac {x}{x-a}} \sqrt [3]{x^2 (x-a)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.31, size = 426, normalized size = 1.45 \begin {gather*} -\frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [12]{d} x\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [3]{x^3-a x^2}+\sqrt [12]{d} x\right )}{4 a d^{5/6}}+\frac {\log \left (\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{4 a d^{5/6}}+\frac {\log \left (-\sqrt [12]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{8 a d^{5/6}}+\frac {\log \left (\sqrt [12]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{8 a d^{5/6}}-\frac {\log \left (-\sqrt {3} \sqrt [12]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{8 a d^{5/6}}-\frac {\log \left (\sqrt {3} \sqrt [12]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [6]{d} x^2\right )}{8 a d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} x^2 \left (x^3-a x^2\right )^{2/3}}{\left (x^3-a x^2\right )^{4/3}-\sqrt [3]{d} x^4}\right )}{4 a d^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 557, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a^{5} d^{4} x^{2} \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - \sqrt {3} x^{2}}{3 \, x^{2}}\right ) - \frac {1}{2} \, \sqrt {3} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} a^{5} d^{4} x^{2} \sqrt {\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a^{5} d^{4} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {5}{6}} + \sqrt {3} x^{2}}{3 \, x^{2}}\right ) + \frac {1}{8} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}\right ) - \frac {1}{8} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a^{2} d^{2} x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{3}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} a d \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )}}{x^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 209, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x -6 a^{2} x^{2}+4 a \,x^{3}+\left (-1+d \right ) x^{4}\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 {x^{3}}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 6 \, a^{2} x^{2} + 4 \, a x^{3}\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 = -0.00 \begin {gather*} \int \frac {x^3}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+\left (d-1\right )\,x^4\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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