Optimal. Leaf size=146 \[ \frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+d^{2/3} x^2\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [3]{d} x\right )}{a \sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} x}\right )}{a \sqrt [3]{d}} \]
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Rubi [A] time = 0.34, antiderivative size = 246, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2081, 2077, 91} \begin {gather*} \frac {\left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log (a+(d-1) x)}{2 a^3 \sqrt [3]{d} \sqrt [3]{x^2 (x-a)}}-\frac {3 \left (-a^2 x\right )^{2/3} \sqrt [3]{a^2 (x-a)} \log \left (-\frac {\sqrt [3]{\frac {2}{3}} \sqrt [3]{a^2 (x-a)}}{\sqrt [3]{d}}-\sqrt [3]{\frac {2}{3}} \sqrt [3]{-a^2 x}\right )}{2 a^3 \sqrt [3]{d} \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]{a^2 (x-a)}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a^2 x}}\right )}{a^3 \sqrt [3]{d} \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+(-1+d) x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (3 a+a (-1+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} (3 a+a (-1+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]{-a^2 (a-x)}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a^2 x}}\right )}{a^3 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}+\frac {\sqrt [3]{-a^2 (a-x)} \left (-a^2 x\right )^{2/3} \log (a-(1-d) x)}{2 a^3 \sqrt [3]{d} \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]{-a^2 (a-x)}+\sqrt [3]{d} \sqrt [3]{-a^2 x}\right )}{2 a^3 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 37, normalized size = 0.25 \begin {gather*} \frac {3 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {d x}{x-a}\right )}{a \sqrt [3]{x^2 (x-a)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 146, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{a \sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{2 a \sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 377, normalized size = 2.58 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (-\frac {{\left (d + 2\right )} x^{2} - 2 \, a x - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {2}{3}} x - \sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} d x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d x + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{{\left (d - 1\right )} x^{2} + a x}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}, -\frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d\right )^{\frac {1}{3}} x - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, x}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \left (-d\right )^{\frac {2}{3}} \log \left (\frac {\left (-d\right )^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} \left (-d\right )^{\frac {1}{3}} x + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 100, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{a d^{\frac {1}{3}}} + \frac {\log \left (d^{\frac {2}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left | -d^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\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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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{2} \left (-a +x \right )\right )^{\frac {1}{3}} \left (a +\left (-1+d \right ) x \right )}\, dx\]
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 ({\left (d - 1\right )} x + a\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+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 x - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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