Optimal. Leaf size=107 \[ \frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}-\frac{\sqrt [3]{a+b x^3}}{3 x^3} \]
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Rubi [A] time = 0.0644006, antiderivative size = 107, 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, 47, 57, 617, 204, 31} \[ \frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}-\frac{\sqrt [3]{a+b x^3}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt [3]{a+b x^3}}{3 x^3}+\frac{1}{9} b \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt [3]{a+b x^3}}{3 x^3}-\frac{b \log (x)}{6 a^{2/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\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 \sqrt [3]{a}}\\ &=-\frac{\sqrt [3]{a+b x^3}}{3 x^3}-\frac{b \log (x)}{6 a^{2/3}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}+\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^{2/3}}\\ &=-\frac{\sqrt [3]{a+b x^3}}{3 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^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0080772, size = 37, normalized size = 0.35 \[ \frac{b \left (a+b x^3\right )^{4/3} \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{b x^3}{a}+1\right )}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\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 = 1.73168, size = 436, normalized size = 4.07 \begin{align*} -\frac{2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a b x^{3} \arctan \left (\frac{{\left (a^{2}\right )}^{\frac{1}{6}}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) +{\left (a^{2}\right )}^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \,{\left (a^{2}\right )}^{\frac{2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2}}{18 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.7844, size = 41, normalized size = 0.38 \begin{align*} - \frac{\sqrt [3]{b} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.22434, size = 143, normalized size = 1.34 \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{2}{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{2}{3}}} - \frac{2 \, \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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