Optimal. Leaf size=185 \[ -\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}+\frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.162809, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {446, 103, 151, 152, 156, 63, 208, 206} \[ -\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}+\frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^3 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{17 c d-\frac{7 d^2 x}{2}}{x^2 (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{\operatorname{Subst}\left (\int \frac{198 c^2 d^2-\frac{85}{2} c d^3 x}{x (8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{384 c^4}\\ &=-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{-1782 c^3 d^3+213 c^2 d^4 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{27648 c^6 d}\\ &=\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{-8019 c^4 d^4+\frac{1995}{2} c^3 d^5 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{124416 c^8 d^2}\\ &=\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{\left (33 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{4096 c^5}+\frac{\left (13 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{331776 c^5}\\ &=\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{(33 d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{2048 c^5}+\frac{\left (13 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{165888 c^5}\\ &=\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0543834, size = 135, normalized size = 0.73 \[ \frac{13 d^2 x^6 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right )-3 \left (4 c \left (288 c^2-612 c d x^3+71 d^2 x^6\right )+891 d^2 x^6 \left (d x^3-8 c\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right )\right )}{165888 c^5 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.015, size = 1106, normalized size = 6. \begin{align*} \text{result 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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.66096, size = 919, normalized size = 4.97 \begin{align*} \left [\frac{13 \,{\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt{c} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 8019 \,{\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 24 \,{\left (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt{d x^{3} + c}}{995328 \,{\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}, \frac{8019 \,{\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) - 13 \,{\left (d^{4} x^{12} - 7 \, c d^{3} x^{9} - 8 \, c^{2} d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 12 \,{\left (665 \, c d^{3} x^{9} - 5107 \, c^{2} d^{2} x^{6} - 1836 \, c^{3} d x^{3} + 864 \, c^{4}\right )} \sqrt{d x^{3} + c}}{497664 \,{\left (c^{6} d^{2} x^{12} - 7 \, c^{7} d x^{9} - 8 \, c^{8} x^{6}\right )}}\right ] \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 [A] time = 1.1429, size = 180, normalized size = 0.97 \begin{align*} \frac{1}{497664} \, d^{2}{\left (\frac{8019 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{5}} - \frac{13 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{5}} + \frac{12 \,{\left (341 \, d x^{3} - 2731 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c^{5}} + \frac{1296 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 4 \, \sqrt{d x^{3} + c} c\right )}}{c^{5} d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]