Optimal. Leaf size=110 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3} \]
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Rubi [A] time = 0.0688638, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 55, 617, 204, 31} \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt [3]{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{9 a}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3}+\frac{b \log (x)}{6 a^{4/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3}+\frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 a^{4/3}}\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3}-\frac{b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0066085, size = 37, normalized size = 0.34 \[ \frac{b \left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{b x^3}{a}+1\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54729, size = 929, normalized size = 8.45 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b x^{3} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x^{3} - 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} a + \left (-a\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + 3 \, a}{x^{3}}\right ) + \left (-a\right )^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a\right )^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a}{18 \, a^{2} x^{3}}, -\frac{6 \, \sqrt{\frac{1}{3}} a b x^{3} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}}\right ) - \left (-a\right )^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) + 2 \, \left (-a\right )^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a}{18 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.33061, size = 39, normalized size = 0.35 \begin{align*} - \frac{\Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x^{4} \Gamma \left (\frac{7}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.95722, size = 149, normalized size = 1.35 \begin{align*} -\frac{1}{18} \, b{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{4}{3}}} - \frac{\log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{4}{3}}} + \frac{2 \, \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{a b x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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