Optimal. Leaf size=107 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2304 c^{7/2}}-\frac{7 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{192 c^3 x^3}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6} \]
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Rubi [A] time = 0.37379, antiderivative size = 107, 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 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2304 c^{7/2}}-\frac{7 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{192 c^3 x^3}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 50.6554, size = 97, normalized size = 0.91 \[ - \frac{\sqrt{c + d x^{3}}}{48 c^{2} x^{6}} + \frac{5 d \sqrt{c + d x^{3}}}{192 c^{3} x^{3}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{2304 c^{\frac{7}{2}}} - \frac{7 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{256 c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.316924, size = 332, normalized size = 3.1 \[ \frac{-\frac{40 c d^3 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{70 c d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{4 c^2}{x^6}+\frac{c d}{x^3}+5 d^2}{192 c^3 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^7*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
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Maple [C] time = 0.037, size = 540, normalized size = 5.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252336, size = 1, normalized size = 0.01 \[ \left [\frac{d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 63 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c}}{4608 \, c^{\frac{7}{2}} x^{6}}, -\frac{d^{2} x^{6} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 63 \, d^{2} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{2304 \, \sqrt{-c} c^{3} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(d*x^3 + c)*(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(1/x**7/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221481, size = 127, normalized size = 1.19 \[ \frac{1}{2304} \, d^{2}{\left (\frac{63 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )}}{c^{3} d^{2} x^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^7),x, algorithm="giac")
[Out]