Optimal. Leaf size=83 \[ \frac{64 c \sqrt{c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]
[Out]
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Rubi [A] time = 0.228534, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{64 c \sqrt{c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac{2 \sqrt{c+d x^3}}{3 d^3}-\frac{224 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^3} \]
Antiderivative was successfully verified.
[In] Int[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
[Out]
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Rubi in Sympy [A] time = 26.1207, size = 73, normalized size = 0.88 \[ - \frac{224 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 d^{3}} + \frac{64 c \sqrt{c + d x^{3}}}{27 d^{3} \left (8 c - d x^{3}\right )} + \frac{2 \sqrt{c + d x^{3}}}{3 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.152742, size = 66, normalized size = 0.8 \[ \frac{2 \left (3 \sqrt{c+d x^3} \left (\frac{32 c}{8 c-d x^3}+9\right )-112 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{81 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^8/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
[Out]
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Maple [C] time = 0.017, size = 874, normalized size = 10.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227659, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (56 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \,{\left (9 \, d x^{3} - 104 \, c\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, -\frac{2 \,{\left (112 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \,{\left (9 \, d x^{3} - 104 \, c\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),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(x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215078, size = 93, normalized size = 1.12 \[ \frac{224 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{3}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, d^{3}} - \frac{64 \, \sqrt{d x^{3} + c} c}{27 \,{\left (d x^{3} - 8 \, c\right )} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="giac")
[Out]