Optimal. Leaf size=201 \[ \frac{\log \left (a d-b d x^3\right )}{3\ 2^{2/3} b^{2/3} d}+\frac{\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}-\frac{\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^{2/3} d}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d}-\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d} \]
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Rubi [C] time = 0.042767, antiderivative size = 66, normalized size of antiderivative = 0.33, 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} \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 a d \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{x \sqrt [3]{1+\frac{b x^3}{a}}}{a d-b d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 a d \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.0380564, size = 63, normalized size = 0.31 \[ \frac{x^2 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 d \left (a+b x^3\right )^{2/3}} \]
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}\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] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{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 [B] time = 1.3942, size = 869, normalized size = 4.32 \begin{align*} -\frac{2 \, \sqrt{3} 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} + \sqrt{3} x}{3 \, x}\right ) - 2 \cdot 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{2}{3}} b^{2} x^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 2 \, \sqrt{3}{\left (b^{2}\right )}^{\frac{1}{6}} b \arctan \left (\frac{{\left (\sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{\left (b^{2}\right )}^{\frac{1}{6}}}{3 \, b^{2} x}\right ) - 2 \,{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (-\frac{{\left (b^{2}\right )}^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) +{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (\frac{{\left (b^{2}\right )}^{\frac{1}{3}} b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right )}{6 \, b^{2} d} \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 \sqrt [3]{a + b x^{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{1}{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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