Optimal. Leaf size=732 \[ \frac{5179 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt{3}\right )}{72576 \sqrt{2} \sqrt [4]{3} c^{17/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}-\frac{7 d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{331776 \sqrt{3} c^{35/6}}+\frac{7 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{995328 c^{35/6}}-\frac{7 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{995328 c^{35/6}}-\frac{5179 \sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{96768\ 3^{3/4} c^{17/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.17801, antiderivative size = 732, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.518, Rules used = {472, 579, 583, 584, 303, 218, 1877, 486, 444, 63, 206, 2138, 2145, 205} \[ \frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}-\frac{7 d^{7/3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{331776 \sqrt{3} c^{35/6}}+\frac{7 d^{7/3} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{995328 c^{35/6}}-\frac{7 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{995328 c^{35/6}}+\frac{5179 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{72576 \sqrt{2} \sqrt [4]{3} c^{17/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{5179 \sqrt{2-\sqrt{3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{96768\ 3^{3/4} c^{17/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 472
Rule 579
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^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{\int \frac{34 c d+\frac{23 d^2 x^3}{2}}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx}{216 c^2 d}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{\int \frac{-1719 c^2 d^2+\frac{765}{4} c d^3 x^3}{x^8 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{2916 c^4 d^2}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{\int \frac{-74313 c^3 d^3+\frac{18909}{2} c^2 d^4 x^3}{x^5 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{163296 c^6 d^2}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{\int \frac{-1491552 c^4 d^4+\frac{371565}{2} c^3 d^5 x^3}{x^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{5225472 c^8 d^2}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{\int \frac{x \left (5971500 c^5 d^5-745776 c^4 d^6 x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{41803776 c^{10} d^2}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{\int \left (\frac{745776 c^4 d^5 x}{\sqrt{c+d x^3}}+\frac{5292 c^5 d^5 x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}\right ) \, dx}{41803776 c^{10} d^2}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{\left (5179 d^3\right ) \int \frac{x}{\sqrt{c+d x^3}} \, dx}{290304 c^6}+\frac{\left (7 d^3\right ) \int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{55296 c^5}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}-\frac{\left (7 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}{663552 c^6}+\frac{\left (5179 d^{8/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{290304 c^6}+\frac{\left (7 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}{663552 c^{17/3}}+\frac{\left (5179 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} d^{8/3}\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{145152 c^{17/3}}-\frac{\left (7 d^{10/3}\right ) \int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx}{221184 c^{16/3}}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{5179 \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 )}{96768\ 3^{3/4} c^{17/3} \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{5179 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 )}{72576 \sqrt{2} \sqrt [4]{3} c^{17/3} \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 (7 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 )}{331776 c^{16/3}}-\frac{\left (7 d^{10/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{663552 c^{16/3}}+\frac{\left (7 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 )}{165888 c^{19/3}}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{7 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 )}{331776 \sqrt{3} c^{35/6}}+\frac{7 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 )}{995328 c^{35/6}}-\frac{5179 \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 )}{96768\ 3^{3/4} c^{17/3} \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{5179 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 )}{72576 \sqrt{2} \sqrt [4]{3} c^{17/3} \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 (7 d^{7/3}\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{331776 c^{16/3}}\\ &=\frac{5}{648 c^3 x^7 \sqrt{c+d x^3}}+\frac{1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{191 \sqrt{c+d x^3}}{18144 c^4 x^7}+\frac{8257 d \sqrt{c+d x^3}}{580608 c^5 x^4}-\frac{5179 d^2 \sqrt{c+d x^3}}{145152 c^6 x}+\frac{5179 d^{7/3} \sqrt{c+d x^3}}{145152 c^6 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{7 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 )}{331776 \sqrt{3} c^{35/6}}+\frac{7 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 )}{995328 c^{35/6}}-\frac{7 d^{7/3} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{995328 c^{35/6}}-\frac{5179 \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 )}{96768\ 3^{3/4} c^{17/3} \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{5179 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 )}{72576 \sqrt{2} \sqrt [4]{3} c^{17/3} \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 = 0.165154, size = 210, normalized size = 0.29 \[ \frac{829375 c d^3 x^9 \left (8 c-d x^3\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-8 \left (5179 d^4 x^{12} \left (8 c-d x^3\right ) \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )+20 c \left (101817 c^2 d^2 x^6-18792 c^3 d x^3+10368 c^4+153269 c d^3 x^9-20716 d^4 x^{12}\right )\right )}{92897280 c^7 x^7 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.014, size = 3299, normalized size = 4.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{2} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )}^{2} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]