Optimal. Leaf size=659 \[ \frac{2 \sqrt{2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}-\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}+\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3\ 2^{2/3} d^{2/3}} \]
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Rubi [A] time = 0.167457, antiderivative size = 659, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {489, 303, 218, 1877, 484} \[ \frac{2 \sqrt{2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}-\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}+\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3\ 2^{2/3} d^{2/3}} \]
Antiderivative was successfully verified.
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Rule 489
Rule 303
Rule 218
Rule 1877
Rule 484
Rubi steps
\begin{align*} \int \frac{x \sqrt{c+d x^3}}{4 c+d x^3} \, dx &=-\left ((3 c) \int \frac{x}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\right )+\int \frac{x}{\sqrt{c+d x^3}} \, dx\\ &=\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}-\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}+\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3\ 2^{2/3} d^{2/3}}+\frac{\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{\sqrt [3]{d}}+\frac{\left (\sqrt{2 \left (2-\sqrt{3}\right )} \sqrt [3]{c}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{\sqrt [3]{d}}\\ &=\frac{2 \sqrt{c+d x^3}}{d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}-\frac{\sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} d^{2/3}}+\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{2^{2/3} d^{2/3}}-\frac{\sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}\\ \end{align*}
Mathematica [C] time = 0.0295845, size = 63, normalized size = 0.1 \[ \frac{x^2 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{8 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.007, size = 848, normalized size = 1.3 \begin{align*} \text{result too large to display} \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{\sqrt{d x^{3} + c} x}{d x^{3} + 4 \, c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{3} + c} x}{d x^{3} + 4 \, c}, x\right ) \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{c + d x^{3}}}{4 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{\sqrt{d x^{3} + c} x}{d x^{3} + 4 \, c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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