Optimal. Leaf size=638 \[ \frac{19 \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{16 d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{9 \sqrt{3} \sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{16 d^{2/3}}-\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{16 d^{2/3}}+\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{16 d^{2/3}}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.718172, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {468, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{19 \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{16 d^{2/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{9 \sqrt{3} \sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{16 d^{2/3}}-\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{16 d^{2/3}}+\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{16 d^{2/3}}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )} \]
Antiderivative was successfully verified.
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Rule 468
Rule 584
Rule 303
Rule 218
Rule 1877
Rule 486
Rule 444
Rule 63
Rule 206
Rule 2138
Rule 2145
Rule 205
Rubi steps
\begin{align*} \int \frac{x \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{\int \frac{x \left (-15 c^2 d-\frac{57}{2} c d^2 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{24 c d}\\ &=\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{\int \left (\frac{57 c d x}{2 \sqrt{c+d x^3}}-\frac{243 c^2 d x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{24 c d}\\ &=\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{19}{16} \int \frac{x}{\sqrt{c+d x^3}} \, dx-\frac{1}{8} (81 c) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx\\ &=\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{27 \int \frac{2 \sqrt [3]{c} d^{2/3}-2 d x-\frac{d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right ) \sqrt{c+d x^3}} \, dx}{32 d}+\frac{19 \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{16 \sqrt [3]{d}}-\frac{\left (27 \sqrt [3]{c}\right ) \int \frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{c+d x^3}} \, dx}{32 \sqrt [3]{d}}+\frac{\left (19 \sqrt{2-\sqrt{3}} \sqrt [3]{c}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{8 \sqrt{2} \sqrt [3]{d}}+\frac{1}{32} \left (81 c^{2/3} \sqrt [3]{d}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx\\ &=\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}-\frac{19 \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 )}{16 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{19 \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 )}{4 \sqrt{2} \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{\left (27 c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9-c x^2} \, dx,x,\frac{\left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\sqrt{c+d x^3}}\right )}{16 d^{2/3}}+\frac{1}{32} \left (27 c^{2/3} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )-\frac{\left (27 d^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{c+d x^3}}\right )}{8 \sqrt [3]{c}}\\ &=\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{9 \sqrt{3} \sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{16 d^{2/3}}-\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{16 d^{2/3}}-\frac{19 \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 )}{16 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{19 \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 )}{4 \sqrt{2} \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{\left (27 c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{16 d^{2/3}}\\ &=\frac{19 \sqrt{c+d x^3}}{8 d^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 x^2 \sqrt{c+d x^3}}{8 \left (8 c-d x^3\right )}+\frac{9 \sqrt{3} \sqrt [6]{c} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{16 d^{2/3}}-\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{16 d^{2/3}}+\frac{9 \sqrt [6]{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{16 d^{2/3}}-\frac{19 \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 )}{16 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{19 \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 )}{4 \sqrt{2} \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 = 0.182889, size = 141, normalized size = 0.22 \[ \frac{x^2 \left (-\frac{19 d x^3 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{c}-25 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+\frac{240 \left (c+d x^3\right )}{8 c-d x^3}\right )}{640 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.007, size = 873, normalized size = 1.4 \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{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{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{{\left (d x^{4} + c x\right )} \sqrt{d x^{3} + c}}{d^{2} x^{6} - 16 \, c d x^{3} + 64 \, c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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