Optimal. Leaf size=202 \[ -\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1355, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1355
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac {\left (a b+b^2 x^3\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{3 \sqrt [3]{a} b \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{3 \sqrt [3]{a} b \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{6 \sqrt [3]{a} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{2 b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {\left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 109, normalized size = 0.54 \begin {gather*} \frac {\left (a+b x^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt {\left (a+b x^3\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 8.36, size = 135, normalized size = 0.67 \begin {gather*} \frac {\left (a+b x^3\right ) \left (\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}\right )}{\sqrt {\left (a+b x^3\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.76, size = 304, normalized size = 1.50 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, a b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{6 \, a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 124, normalized size = 0.61 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}}} - \frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 97, normalized size = 0.48 \begin {gather*} -\frac {\left (b \,x^{3}+a \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )-\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {a}{b}\right )^{\frac {1}{3}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 98, normalized size = 0.49 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\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 = -0.00 \begin {gather*} \int \frac {x}{\sqrt {{\left (b\,x^3+a\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 0.12 \begin {gather*} \operatorname {RootSum} {\left (27 t^{3} a b^{2} + 1, \left (t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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