Optimal. Leaf size=140 \[ \frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^3-a x^2}\right )}{a d^{2/3}}-\frac {\log \left (d^{2/3} \left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^3-a x^2}+x^2\right )}{2 a d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^3-a x^2}+x}\right )}{a d^{2/3}} \]
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Rubi [A] time = 0.38, antiderivative size = 248, normalized size of antiderivative = 1.77, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2081, 2077, 91} \begin {gather*} -\frac {\left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log ((d-1) x-a d)}{2 a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}}+\frac {3 \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log \left (-\sqrt [3]{\frac {2}{3}} \sqrt [3]{d} \sqrt [3]{a^2 (x-a)}-\sqrt [3]{\frac {2}{3}} \sqrt [3]{-a^2 x}\right )}{2 a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}}+\frac {\sqrt {3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a^2 (x-a)}}{\sqrt {3} \sqrt [3]{-a^2 x}}\right )}{a^3 d^{2/3} \sqrt [3]{x^2 (x-a)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 2077
Rule 2081
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x^2 (-a+x)} (-a d+(-1+d) x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (a (-1+d)-3 a d)+(-1+d) x\right ) \sqrt [3]{-\frac {2 a^3}{27}-\frac {a^2 x}{3}+x^3}} \, dx,x,-\frac {a}{3}+x\right )\\ &=\frac {\left (2^{2/3} \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (-a+x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {2 a^3}{9}-\frac {2 a^2 x}{3}\right )^{2/3} \sqrt [3]{-\frac {2 a^3}{9}+\frac {a^2 x}{3}} \left (\frac {1}{3} (a (-1+d)-3 a d)+(-1+d) x\right )} \, dx,x,-\frac {a}{3}+x\right )}{3 \sqrt [3]{-a x^2+x^3}}\\ &=\frac {\sqrt {3} \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{-a^2 (a-x)}}{\sqrt {3} \sqrt [3]{-a^2 x}}\right )}{a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}-\frac {\sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log (a d+(1-d) x)}{2 a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}+\frac {3 \sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log \left (\sqrt [3]{d} \sqrt [3]{-a^2 (a-x)}+\sqrt [3]{-a^2 x}\right )}{2 a^3 d^{2/3} \sqrt [3]{-a x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 42, normalized size = 0.30 \begin {gather*} -\frac {3 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x}{d (x-a)}\right )}{a d \sqrt [3]{x^2 (x-a)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 140, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{a d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 a d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 153, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left (2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d + {\left (d^{2}\right )}^{\frac {1}{3}} x\right )}}{3 \, d x}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d - {\left (d^{2}\right )}^{\frac {1}{3}} x}{x}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d x + {\left (d^{2}\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{2 \, a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 103, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{a {\left | d \right |}^{\frac {2}{3}}} - \frac {{\left | d \right |}^{\frac {4}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{2 \, a d^{2}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{a d^{\frac {2}{3}}} \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 \[\int \frac {1}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (-a d +\left (-1+d \right ) x \right )}\, dx\]
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 {1}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}} {\left (a d - {\left (d - 1\right )} x\right )}}\,{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 {1}{\left (a\,d-x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (- a + x\right )} \left (- a d + d x - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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