Optimal. Leaf size=104 \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]
[Out]
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Rubi [A] time = 0.419337, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 54.5681, size = 94, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}}}{48 x^{6}} - \frac{11 d \sqrt{c + d x^{3}}}{192 c x^{3}} + \frac{9 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{256 c^{\frac{3}{2}}} - \frac{37 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{768 c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(3/2)/x**7/(-d*x**3+8*c),x)
[Out]
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Mathematica [C] time = 0.327822, size = 332, normalized size = 3.19 \[ \frac{\frac{132 d^3 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{185 d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{33 d^2}{2 c}-\frac{6 c}{x^6}-\frac{45 d}{2 x^3}}{288 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)),x]
[Out]
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Maple [C] time = 0.038, size = 617, normalized size = 5.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(3/2)/x^7/(-d*x^3+8*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254815, size = 1, normalized size = 0.01 \[ \left [\frac{27 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 37 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 8 \,{\left (11 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c}}{1536 \, c^{\frac{3}{2}} x^{6}}, -\frac{27 \, d^{2} x^{6} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 37 \, d^{2} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 4 \,{\left (11 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{768 \, \sqrt{-c} c x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*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((d*x**3+c)**(3/2)/x**7/(-d*x**3+8*c),x)
[Out]
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GIAC/XCAS [A] time = 0.221336, size = 127, normalized size = 1.22 \[ \frac{1}{768} \, d^{2}{\left (\frac{37 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{4 \,{\left (11 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 7 \, \sqrt{d x^{3} + c} c\right )}}{c d^{2} x^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7),x, algorithm="giac")
[Out]