Optimal. Leaf size=136 \[ -\frac {\log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{3 a^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2 \sqrt [3]{a x^3-b}+\sqrt [3]{a} x}\right )}{\sqrt {3} a^{2/3}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{6 a^{2/3}} \]
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Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {331, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}\right )}{3 a^{2/3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}+\frac {\log \left (\frac {a^{2/3} x^2}{\left (a x^3-b\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1\right )}{6 a^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 331
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x}{\left (-b+a x^3\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{1-a x^3} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{a} x} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{3 \sqrt [3]{a}}-\frac {\operatorname {Subst}\left (\int \frac {1-\sqrt [3]{a} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{3 \sqrt [3]{a}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{a}+2 a^{2/3} x}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{a} x+a^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-b+a x^3}}\right )}{2 \sqrt [3]{a}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\log \left (1+\frac {a^{2/3} x^2}{\left (-b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{a^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{3 a^{2/3}}+\frac {\log \left (1+\frac {a^{2/3} x^2}{\left (-b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{6 a^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 44, normalized size = 0.32 \begin {gather*} \frac {x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {a x^3}{a x^3-b}\right )}{2 \left (a x^3-b\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 136, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{-b+a x^3}}\right )}{\sqrt {3} a^{2/3}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 a^{2/3}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 a^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 181, normalized size = 1.33 \begin {gather*} \frac {2 \, \sqrt {3} a \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a x - 2 \, \sqrt {3} {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2} x}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {1}{3}} a}{x}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-a^{2}\right )^{\frac {1}{3}} a x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} x - {\left (a x^{3} - b\right )}^{\frac {2}{3}} a}{x^{2}}\right )}{6 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (a x^{3} - b\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a \,x^{3}-b \right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 108, normalized size = 0.79 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} + \frac {\log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{\frac {2}{3}}} - \frac {\log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{3 \, a^{\frac {2}{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\,x^3-b\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.94, size = 41, normalized size = 0.30 \begin {gather*} \frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 b^{\frac {2}{3}} \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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