Optimal. Leaf size=457 \[ \frac{\sqrt [3]{a} \log \left (\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{2} b^{2/3} d}-\frac{2^{2/3} \sqrt [3]{a} \log \left (\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3} d}-\frac{\sqrt [3]{a} \log \left (\frac{\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac{2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{2/3} d}+\frac{2^{2/3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} b^{2/3} d}+\frac{\sqrt [3]{a} \log \left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{2/3} d}-\frac{x^2 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 d \sqrt [3]{a+b x^3}} \]
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Rubi [C] time = 0.0465118, antiderivative size = 66, normalized size of antiderivative = 0.14, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {511, 510} \[ \frac{x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 a d \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{x \left (1+\frac{b x^3}{a}\right )^{2/3}}{a d-b d x^3} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{x^2 \left (a+b x^3\right )^{2/3} F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 a d \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0392283, size = 63, normalized size = 0.14 \[ \frac{x^2 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 d \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{-bd{x}^{3}+ad} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{x \left (a + b x^{3}\right )^{\frac{2}{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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