Optimal. Leaf size=146 \[ -\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {290, 325, 292, 31, 634, 617, 204, 628} \[ -\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 290
Rule 292
Rule 325
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3 a x \left (a+b x^3\right )}+\frac {4 \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{3 a}\\ &=-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )}-\frac {(4 b) \int \frac {x}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )}+\frac {\left (4 b^{2/3}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac {\left (4 b^{2/3}\right ) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{7/3}}\\ &=-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}-\frac {\left (2 \sqrt [3]{b}\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{7/3}}-\frac {\left (2 b^{2/3}\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^2}\\ &=-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}-\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{7/3}}\\ &=-\frac {4}{3 a^2 x}+\frac {1}{3 a x \left (a+b x^3\right )}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}-\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 131, normalized size = 0.90 \[ \frac {-2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {3 \sqrt [3]{a} b x^2}{a+b x^3}+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {9 \sqrt [3]{a}}{x}}{9 a^{7/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 146, normalized size = 1.00 \[ -\frac {12 \, b x^{3} + 4 \, \sqrt {3} {\left (b x^{4} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (b x^{4} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 4 \, {\left (b x^{4} + a x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 9 \, a}{9 \, {\left (a^{2} b x^{4} + a^{3} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 139, normalized size = 0.95 \[ \frac {4 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} + \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} - \frac {4 \, b x^{3} + 3 \, a}{3 \, {\left (b x^{4} + a x\right )} a^{2}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 117, normalized size = 0.80 \[ -\frac {b \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {1}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 126, normalized size = 0.86 \[ -\frac {4 \, b x^{3} + 3 \, a}{3 \, {\left (a^{2} b x^{4} + a^{3} x\right )}} - \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {4 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 120, normalized size = 0.82 \[ \frac {4\,b^{1/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{9\,a^{7/3}}-\frac {\frac {1}{a}+\frac {4\,b\,x^3}{3\,a^2}}{b\,x^4+a\,x}-\frac {4\,b^{1/3}\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}+\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{7/3}}+\frac {b^{1/3}\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}-\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 56, normalized size = 0.38 \[ \frac {- 3 a - 4 b x^{3}}{3 a^{3} x + 3 a^{2} b x^{4}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} - 64 b, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{5}}{16 b} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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