Optimal. Leaf size=185 \[ \frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
[Out]
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Rubi [A] time = 0.649548, 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 \[ \frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 97.0921, size = 160, normalized size = 0.86 \[ - \frac{1}{48 c^{2} x^{6} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{11 d}{3456 c^{3} x^{3} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{71 d}{13824 c^{4} x^{3} \sqrt{c + d x^{3}}} + \frac{665 d^{2}}{41472 c^{5} \sqrt{c + d x^{3}}} + \frac{13 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{497664 c^{\frac{11}{2}}} - \frac{33 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{2048 c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
[Out]
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Mathematica [C] time = 0.462981, size = 349, normalized size = 1.89 \[ \frac{\frac{\frac{8910 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+864 c^3-1836 c^2 d x^3-5107 c d^2 x^6+665 d^3 x^9}{d x^3-8 c}-\frac{5320 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{41472 c^5 x^6 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^7*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.018, size = 1106, normalized size = 6. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232975, size = 1, normalized size = 0.01 \[ \left [\frac{13 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 8019 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (665 \, d^{3} x^{9} - 5107 \, c d^{2} x^{6} - 1836 \, c^{2} d x^{3} + 864 \, c^{3}\right )} \sqrt{c}}{995328 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{13 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 8019 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (665 \, d^{3} x^{9} - 5107 \, c d^{2} x^{6} - 1836 \, c^{2} d x^{3} + 864 \, c^{3}\right )} \sqrt{-c}}{497664 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^7),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222187, size = 180, normalized size = 0.97 \[ \frac{1}{497664} \, d^{2}{\left (\frac{8019 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{5}} - \frac{13 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{5}} + \frac{12 \,{\left (341 \, d x^{3} - 2731 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c^{5}} + \frac{1296 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 4 \, \sqrt{d x^{3} + c} c\right )}}{c^{5} d^{2} x^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^7),x, algorithm="giac")
[Out]