Optimal. Leaf size=115 \[ \frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {263, 292, 31, 634, 617, 204, 628} \[ \frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 263
Rule 292
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx &=\int \frac {x}{b+a x^3} \, dx\\ &=-\frac {\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\\ &=-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{a}}+\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{2/3} \sqrt [3]{b}}\\ &=-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{2/3} \sqrt [3]{b}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 89, normalized size = 0.77 \[ \frac {\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{6 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 304, normalized size = 2.64 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (-a^{2} b\right )^{\frac {2}{3}} x}{a x^{3} + b}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, a x + \left (-a^{2} b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a}\right ) + \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac {1}{3}} a x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac {1}{3}}\right )}{6 \, a^{2} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 112, normalized size = 0.97 \[ -\frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 0.79 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {b}{a}\right )^{\frac {1}{3}} a}-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {b}{a}\right )^{\frac {1}{3}} a}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {b}{a}\right )^{\frac {1}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 98, normalized size = 0.85 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 111, normalized size = 0.97 \[ \frac {\ln \left (a^{1/3}\,x-{\left (-b\right )}^{1/3}\right )}{3\,a^{2/3}\,{\left (-b\right )}^{1/3}}+\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}}-\frac {\ln \left (a\,x-\frac {a^{2/3}\,{\left (-b\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,{\left (-b\right )}^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 24, normalized size = 0.21 \[ \operatorname {RootSum} {\left (27 t^{3} a^{2} b + 1, \left (t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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