Optimal. Leaf size=708 \[ \frac{19 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{896 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{4096 c^{17/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{4096 c^{17/6}}-\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{4096 c^{17/6}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{3584 c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{11 \sqrt{c+d x^3}}{224 c x^7} \]
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Rubi [A] time = 1.07565, antiderivative size = 708, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {468, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{4096 c^{17/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{4096 c^{17/6}}-\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{4096 c^{17/6}}+\frac{19 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{896 \sqrt{2} \sqrt [4]{3} c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{3584 c^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{11 \sqrt{c+d x^3}}{224 c x^7} \]
Antiderivative was successfully verified.
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Rule 468
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{\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx &=\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}+\frac{\int \frac{66 c^2 d+\frac{105}{2} c d^2 x^3}{x^8 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{24 c d}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{\int \frac{-498 c^3 d^2-363 c^2 d^3 x^3}{x^5 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{1344 c^3 d}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}+\frac{\int \frac{3648 c^4 d^3+1245 c^3 d^4 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{43008 c^5 d}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{\int \frac{x \left (-28200 c^5 d^4+1824 c^4 d^5 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{344064 c^7 d}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{\int \left (-\frac{1824 c^4 d^4 x}{\sqrt{c+d x^3}}-\frac{13608 c^5 d^4 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{344064 c^7 d}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}+\frac{\left (19 d^3\right ) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{3584 c^3}+\frac{\left (81 d^3\right ) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{2048 c^2}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{\left (27 d^2\right ) \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}{8192 c^3}+\frac{\left (19 d^{8/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{3584 c^3}+\frac{\left (27 d^{8/3}\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}{8192 c^{8/3}}+\frac{\left (19 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{8/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{1792 c^{8/3}}-\frac{\left (81 d^{10/3}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{8192 c^{7/3}}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \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 )}{3584 c^{8/3} \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 d^{7/3} \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 )}{896 \sqrt{2} \sqrt [4]{3} c^{8/3} \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 d^{7/3}\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 )}{4096 c^{7/3}}-\frac{\left (27 d^{10/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{8192 c^{7/3}}+\frac{\left (27 d^{13/3}\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 )}{2048 c^{10/3}}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{4096 c^{17/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{4096 c^{17/6}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \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 )}{3584 c^{8/3} \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 d^{7/3} \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 )}{896 \sqrt{2} \sqrt [4]{3} c^{8/3} \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 d^{7/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{4096 c^{7/3}}\\ &=-\frac{11 \sqrt{c+d x^3}}{224 c x^7}-\frac{83 d \sqrt{c+d x^3}}{7168 c^2 x^4}-\frac{19 d^2 \sqrt{c+d x^3}}{1792 c^3 x}+\frac{19 d^{7/3} \sqrt{c+d x^3}}{1792 c^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{3 \sqrt{c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac{9 \sqrt{3} d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{4096 c^{17/6}}+\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{4096 c^{17/6}}-\frac{9 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{4096 c^{17/6}}-\frac{19 \sqrt [4]{3} \sqrt{2-\sqrt{3}} d^{7/3} \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 )}{3584 c^{8/3} \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 d^{7/3} \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 )}{896 \sqrt{2} \sqrt [4]{3} c^{8/3} \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 = 0.150801, size = 212, normalized size = 0.3 \[ -\frac{19 d^4 x^5 \sqrt{\frac{c+d x^3}{c}} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{143360 c^4 \sqrt{c+d x^3}}+\frac{1175 d^3 x^2 \sqrt{\frac{c+d x^3}{c}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{229376 c^3 \sqrt{c+d x^3}}+\sqrt{c+d x^3} \left (-\frac{3 d^3 x^2}{4096 c^3 \left (d x^3-8 c\right )}-\frac{283 d^2}{28672 c^3 x}-\frac{41 d}{7168 c^2 x^4}-\frac{1}{448 c x^7}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.014, size = 3186, normalized size = 4.5 \begin{align*} \text{output 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}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}}\,{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{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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