Optimal. Leaf size=139 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \]
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Rubi [A]
time = 0.27, antiderivative size = 194, normalized size of antiderivative = 1.40, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6851, 93}
\begin {gather*} \frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log (-a d-(1-d) x)}{2 a \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{d}}-\sqrt [3]{x-a}\right )}{2 a \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 6851
Rubi steps
\begin {align*} \int \frac {x}{\left (x^2 (-a+x)\right )^{2/3} (-a d+(-1+d) x)} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-a d+(-1+d) x)} \, dx}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\sqrt {3} x^{4/3} (-a+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a+x}}\right )}{a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}-\frac {x^{4/3} (-a+x)^{2/3} \log (-a d-(1-d) x)}{2 a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {3 x^{4/3} (-a+x)^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{d}}-\sqrt [3]{-a+x}\right )}{2 a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 152, normalized size = 1.09 \begin {gather*} \frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{d} \sqrt [3]{-a+x}}\right )+2 \log \left (\sqrt [3]{x}-\sqrt [3]{d} \sqrt [3]{-a+x}\right )-\log \left (x^{2/3}+\sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+d^{2/3} (-a+x)^{2/3}\right )\right )}{2 a \sqrt [3]{d} \left (x^2 (-a+x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (-a d +\left (-1+d \right ) x \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 338, normalized size = 2.43 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (\frac {2 \, a d x - {\left (2 \, d + 1\right )} x^{2} - \sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} + 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x}{a d x - {\left (d - 1\right )} x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {2}{3}} x - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{2 \, a d}\right ] \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 {x}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a d + d x - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 107, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {3} {\left | d \right |}^{\frac {2}{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 d} - \frac {{\left | d \right |}^{\frac {2}{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} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{a d^{\frac {1}{3}}} \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 (a\,d-x\,\left (d-1\right )\right )\,{\left (-x^2\,\left (a-x\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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