Optimal. Leaf size=234 \[ -\frac{\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d}+\frac{\sqrt [3]{b c-a d} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d}+\frac{\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{c} d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d} \]
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Rubi [C] time = 0.0377606, antiderivative size = 64, normalized size of antiderivative = 0.27, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 c \sqrt [3]{\frac{b x^3}{a}+1}} \]
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}}{c+d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{x \sqrt [3]{1+\frac{b x^3}{a}}}{c+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{d x^3}{c}\right )}{2 c \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.0305237, size = 65, normalized size = 0.28 \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 c \sqrt [3]{\frac{a+b x^3}{a}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{d{x}^{3}+c}\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}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89708, size = 807, normalized size = 3.45 \begin{align*} \frac{2 \, \sqrt{3} \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (b c - a d\right )} x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} c \left (\frac{b c - a d}{c}\right )^{\frac{2}{3}}}{3 \,{\left (b c - a d\right )} x}\right ) - 2 \, \sqrt{3} \left (-b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{3 \, b x}\right ) + 2 \, \left (-b\right )^{\frac{1}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 2 \, \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \log \left (-\frac{x \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - \left (-b\right )^{\frac{1}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) - \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} \left (\frac{b c - a d}{c}\right )^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} x \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, 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*} \int \frac{x \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \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}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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