Optimal. Leaf size=711 \[ \frac{d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{2688 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}-\frac{13 d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12288 \sqrt{3} c^{23/6}}+\frac{13 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{36864 c^{23/6}}-\frac{13 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{36864 c^{23/6}}-\frac{\sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{3584\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 1.07148, antiderivative size = 711, 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 = {469, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}-\frac{13 d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{12288 \sqrt{3} c^{23/6}}+\frac{13 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{36864 c^{23/6}}-\frac{13 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{36864 c^{23/6}}+\frac{d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{2688 \sqrt{2} \sqrt [4]{3} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{3584\ 3^{3/4} c^{11/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )} \]
Antiderivative was successfully verified.
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Rule 469
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{\sqrt{c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx &=\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{\int \frac{-10 c-\frac{17 d x^3}{2}}{x^8 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{24 c}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac{\int \frac{106 c^2 d+55 c d^2 x^3}{x^5 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{1344 c^3}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{\int \frac{-64 c^3 d^2-265 c^2 d^3 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{43008 c^5}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac{\int \frac{x \left (2440 c^4 d^3-32 c^3 d^4 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{344064 c^7}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac{\int \left (\frac{32 c^3 d^3 x}{\sqrt{c+d x^3}}+\frac{2184 c^4 d^3 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{344064 c^7}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}+\frac{d^3 \int \frac{x}{\sqrt{c+d x^3}} \, dx}{10752 c^4}+\frac{\left (13 d^3\right ) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{2048 c^3}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{\left (13 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}{24576 c^4}+\frac{d^{8/3} \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{10752 c^4}+\frac{\left (13 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}{24576 c^{11/3}}+\frac{\left (\sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{8/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{5376 c^{11/3}}-\frac{\left (13 d^{10/3}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{8192 c^{10/3}}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{\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\ 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{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 )}{2688 \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{\left (13 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 )}{12288 c^{10/3}}-\frac{\left (13 d^{10/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{24576 c^{10/3}}+\frac{\left (13 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 )}{6144 c^{13/3}}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{13 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 )}{12288 \sqrt{3} c^{23/6}}+\frac{13 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 )}{36864 c^{23/6}}-\frac{\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\ 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{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 )}{2688 \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{\left (13 d^{7/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{12288 c^{10/3}}\\ &=-\frac{5 \sqrt{c+d x^3}}{672 c^2 x^7}-\frac{53 d \sqrt{c+d x^3}}{21504 c^3 x^4}-\frac{d^2 \sqrt{c+d x^3}}{5376 c^4 x}+\frac{d^{7/3} \sqrt{c+d x^3}}{5376 c^4 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt{c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac{13 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 )}{12288 \sqrt{3} c^{23/6}}+\frac{13 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 )}{36864 c^{23/6}}-\frac{13 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{36864 c^{23/6}}-\frac{\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\ 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{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 )}{2688 \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 = 0.14255, size = 209, normalized size = 0.29 \[ \frac{1525 c d^3 x^9 \left (8 c-d x^3\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-8 \left (d^4 x^{12} \left (8 c-d x^3\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+20 c \left (243 c^2 d^2 x^6+648 c^3 d x^3+384 c^4-25 c d^3 x^9-4 d^4 x^{12}\right )\right )}{3440640 c^5 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.013, size = 3169, 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{\sqrt{d x^{3} + c}}{{\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{\sqrt{d x^{3} + c}}{{\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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