Optimal. Leaf size=128 \[ \frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{9/2}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.123816, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{9/2}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}} \]
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 ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^3 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{18 c d-\frac{5 d^2 x}{2}}{x^2 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}+\frac{\operatorname{Subst}\left (\int \frac{218 c^2 d^2-27 c d^3 x}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{384 c^4}\\ &=\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}+\frac{\operatorname{Subst}\left (\int \frac{981 c^3 d^3-\frac{245}{2} c^2 d^4 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{1728 c^6 d}\\ &=\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}+\frac{\left (109 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{1536 c^4}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{13824 c^4}\\ &=\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}+\frac{(109 d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{768 c^4}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{6912 c^4}\\ &=\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0355619, size = 91, normalized size = 0.71 \[ \frac{-d^2 x^6 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right )+981 d^2 x^6 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right )+36 c \left (9 d x^3-4 c\right )}{6912 c^4 x^6 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.029, size = 636, normalized size = 5. \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 )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41284, size = 695, normalized size = 5.43 \begin{align*} \left [\frac{{\left (d^{3} x^{9} + c 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 ) + 2943 \,{\left (d^{3} x^{9} + c 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 (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt{d x^{3} + c}}{41472 \,{\left (c^{5} d x^{9} + c^{6} x^{6}\right )}}, \frac{2943 \,{\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) -{\left (d^{3} x^{9} + c d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 12 \,{\left (245 \, c d^{2} x^{6} + 81 \, c^{2} d x^{3} - 36 \, c^{3}\right )} \sqrt{d x^{3} + c}}{20736 \,{\left (c^{5} d x^{9} + c^{6} 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 = 1.13101, size = 146, normalized size = 1.14 \begin{align*} \frac{1}{20736} \, d^{2}{\left (\frac{2943 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{4}} + \frac{1536}{\sqrt{d x^{3} + c} c^{4}} + \frac{108 \,{\left (13 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 17 \, \sqrt{d x^{3} + c} c\right )}}{c^{4} d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]