Optimal. Leaf size=124 \[ \frac {a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{5/3}}-\frac {1}{2 b x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 124, 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 = {263, 325, 200, 31, 634, 617, 204, 628} \[ \frac {a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{5/3}}-\frac {1}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 263
Rule 325
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^6} \, dx &=\int \frac {1}{x^3 \left (b+a x^3\right )} \, dx\\ &=-\frac {1}{2 b x^2}-\frac {a \int \frac {1}{b+a x^3} \, dx}{b}\\ &=-\frac {1}{2 b x^2}-\frac {a \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 b^{5/3}}-\frac {a \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 b^{5/3}}\\ &=-\frac {1}{2 b x^2}-\frac {a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac {a^{2/3} \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 b^{5/3}}-\frac {a \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 b^{4/3}}\\ &=-\frac {1}{2 b x^2}-\frac {a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac {a^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{5/3}}-\frac {a^{2/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{b^{5/3}}\\ &=-\frac {1}{2 b x^2}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{5/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac {a^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 119, normalized size = 0.96 \[ \frac {a^{2/3} x^2 \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 a^{2/3} x^2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt {3} a^{2/3} x^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-3 b^{2/3}}{6 b^{5/3} x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 143, normalized size = 1.15 \[ \frac {2 \, \sqrt {3} x^{2} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - x^{2} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a^{2} x^{2} + a b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + b^{2} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x^{2} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x - b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 3}{6 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 115, normalized size = 0.93 \[ \frac {a \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2}} - \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, b^{2}} - \frac {1}{2 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 99, normalized size = 0.80 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} b}-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {b}{a}\right )^{\frac {2}{3}} b}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {b}{a}\right )^{\frac {2}{3}} b}-\frac {1}{2 b \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 106, normalized size = 0.85 \[ -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {1}{2 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 128, normalized size = 1.03 \[ \frac {a^{2/3}\,\ln \left ({\left (-b\right )}^{7/3}-a^{1/3}\,b^2\,x\right )}{3\,{\left (-b\right )}^{5/3}}-\frac {1}{2\,b\,x^2}-\frac {a^{2/3}\,\ln \left (3\,a^3\,b^2\,x+3\,a^{8/3}\,{\left (-b\right )}^{7/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-b\right )}^{5/3}}+\frac {a^{2/3}\,\ln \left (3\,a^3\,b^2\,x-9\,a^{8/3}\,{\left (-b\right )}^{7/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{{\left (-b\right )}^{5/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 32, normalized size = 0.26 \[ \operatorname {RootSum} {\left (27 t^{3} b^{5} + a^{2}, \left (t \mapsto t \log {\left (- \frac {3 t b^{2}}{a} + x \right )} \right )\right )} - \frac {1}{2 b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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