Optimal. Leaf size=229 \[ \frac{a \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{4/3} d}-\frac{a \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{4/3} d}+\frac{5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{4/3} d}-\frac{5 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{4/3} d}+\frac{2^{2/3} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d}-\frac{x \left (a+b x^3\right )^{2/3}}{3 b d} \]
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Rubi [C] time = 0.0634338, antiderivative size = 66, normalized size of antiderivative = 0.29, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {511, 510} \[ \frac{x^4 \left (a+b x^3\right )^{2/3} F_1\left (\frac{4}{3};-\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{4 a d \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^3 \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^3 \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^4 \left (a+b x^3\right )^{2/3} F_1\left (\frac{4}{3};-\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{4 a d \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.227412, size = 216, normalized size = 0.94 \[ \frac{\frac{15 x^4 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac{2^{2/3} a \left (\log \left (\frac{2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{b^{4/3}}-\frac{12 x \left (a+b x^3\right )^{2/3}}{b}}{36 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{-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^{3}}{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 [A] time = 1.69662, size = 1786, normalized size = 7.8 \begin{align*} \text{result too large to display} \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^{3} \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^{3}}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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