Optimal. Leaf size=102 \[ -\frac{352 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}+\frac{64 c \left (c+d x^3\right )^{3/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{352 c \sqrt{c+d x^3}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]
[Out]
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Rubi [A] time = 0.256414, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{352 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^3}+\frac{64 c \left (c+d x^3\right )^{3/2}}{27 d^3 \left (8 c-d x^3\right )}+\frac{352 c \sqrt{c+d x^3}}{27 d^3}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]
Antiderivative was successfully verified.
[In] Int[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 30.5617, size = 92, normalized size = 0.9 \[ - \frac{352 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{3}} + \frac{64 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{27 d^{3} \left (8 c - d x^{3}\right )} + \frac{352 c \sqrt{c + d x^{3}}}{27 d^{3}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)
[Out]
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Mathematica [A] time = 0.154123, size = 79, normalized size = 0.77 \[ \frac{2 \left (\frac{\sqrt{c+d x^3} \left (-488 c^2+41 c d x^3+d^2 x^6\right )}{d x^3-8 c}-176 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{9 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
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Maple [C] time = 0.019, size = 892, normalized size = 8.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c)^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(sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221799, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (88 \,{\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) +{\left (d^{2} x^{6} + 41 \, c d x^{3} - 488 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{9 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, -\frac{2 \,{\left (176 \,{\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) -{\left (d^{2} x^{6} + 41 \, c d x^{3} - 488 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{9 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^8/(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+c)**(1/2)/(-d*x**3+8*c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216427, size = 126, normalized size = 1.24 \[ \frac{352 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d^{3}} - \frac{64 \, \sqrt{d x^{3} + c} c^{2}}{3 \,{\left (d x^{3} - 8 \, c\right )} d^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{6} + 48 \, \sqrt{d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^8/(d*x^3 - 8*c)^2,x, algorithm="giac")
[Out]