Optimal. Leaf size=675 \[ -\frac{113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3456 \sqrt{3} c^{23/6}}+\frac{d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{10368 c^{23/6}}-\frac{d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{23/6}}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}} \]
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Rubi [A] time = 0.920685, antiderivative size = 675, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {472, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3456 \sqrt{3} c^{23/6}}+\frac{d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{10368 c^{23/6}}-\frac{d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{23/6}}-\frac{113 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{216 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{288\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 472
Rule 583
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{1}{x^5 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{2 \int \frac{-\frac{91 c d}{2}+\frac{11 d^2 x^3}{2}}{x^5 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{27 c^2 d}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{\int \frac{-904 c^2 d^2+\frac{455}{4} c d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{432 c^4 d}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{\int \frac{x \left (3610 c^3 d^3-452 c^2 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{3456 c^6 d}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{\int \left (\frac{452 c^2 d^3 x}{\sqrt{c+d x^3}}-\frac{6 c^3 d^3 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{3456 c^6 d}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{\left (113 d^2\right ) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{864 c^4}+\frac{d^2 \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{576 c^3}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{d \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}{6912 c^4}-\frac{\left (113 d^{5/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{864 c^4}+\frac{d^{5/3} \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}{6912 c^{11/3}}-\frac{\left (113 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{5/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{432 c^{11/3}}-\frac{d^{7/3} \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{2304 c^{10/3}}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \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 )}{288\ 3^{3/4} c^{11/3} \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{113 d^{4/3} \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 )}{216 \sqrt{2} \sqrt [4]{3} c^{11/3} \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{d^{4/3} \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 )}{3456 c^{10/3}}-\frac{d^{7/3} \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{6912 c^{10/3}}+\frac{d^{10/3} \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 )}{1728 c^{13/3}}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3456 \sqrt{3} c^{23/6}}+\frac{d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{10368 c^{23/6}}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \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 )}{288\ 3^{3/4} c^{11/3} \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{113 d^{4/3} \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 )}{216 \sqrt{2} \sqrt [4]{3} c^{11/3} \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{d^{4/3} \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{3456 c^{10/3}}\\ &=\frac{2}{27 c^2 x^4 \sqrt{c+d x^3}}-\frac{91 \sqrt{c+d x^3}}{864 c^3 x^4}+\frac{113 d \sqrt{c+d x^3}}{432 c^4 x}-\frac{113 d^{4/3} \sqrt{c+d x^3}}{432 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^{4/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3456 \sqrt{3} c^{23/6}}+\frac{d^{4/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{10368 c^{23/6}}-\frac{d^{4/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{23/6}}+\frac{113 \sqrt{2-\sqrt{3}} d^{4/3} \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 )}{288\ 3^{3/4} c^{11/3} \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{113 d^{4/3} \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 )}{216 \sqrt{2} \sqrt [4]{3} c^{11/3} \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 = 0.0869576, size = 153, normalized size = 0.23 \[ \frac{452 d^3 x^9 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-9025 c d^2 x^6 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+160 c \left (-27 c^2+135 c d x^3+226 d^2 x^6\right )}{138240 c^5 x^4 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.026, size = 1864, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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