Optimal. Leaf size=82 \[ \frac{8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac{26 \sqrt{c+d x^3}}{27 d^2}-\frac{26 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^2} \]
[Out]
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Rubi [A] time = 0.18679, antiderivative size = 82, 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 \[ \frac{8 \left (c+d x^3\right )^{3/2}}{27 d^2 \left (8 c-d x^3\right )}+\frac{26 \sqrt{c+d x^3}}{27 d^2}-\frac{26 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 21.3165, size = 71, normalized size = 0.87 \[ - \frac{26 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 d^{2}} + \frac{8 \left (c + d x^{3}\right )^{\frac{3}{2}}}{27 d^{2} \left (8 c - d x^{3}\right )} + \frac{26 \sqrt{c + d x^{3}}}{27 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)
[Out]
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Mathematica [A] time = 0.140842, size = 66, normalized size = 0.8 \[ \frac{2 \left (3 \sqrt{c+d x^3} \left (\frac{4 c}{8 c-d x^3}+1\right )-13 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{9 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^5*Sqrt[c + d*x^3])/(8*c - d*x^3)^2,x]
[Out]
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Maple [C] time = 0.016, size = 874, normalized size = 10.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(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^5/(d*x^3 - 8*c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222271, size = 1, normalized size = 0.01 \[ \left [\frac{13 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 6 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 12 \, c\right )}}{9 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}, -\frac{2 \,{\left (13 \,{\left (d x^{3} - 8 \, c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right ) - 3 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 12 \, c\right )}\right )}}{9 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^5/(d*x^3 - 8*c)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \sqrt{c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216678, size = 100, normalized size = 1.22 \[ \frac{2 \,{\left (\frac{13 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} + \frac{3 \, \sqrt{d x^{3} + c}}{d} - \frac{12 \, \sqrt{d x^{3} + c} c}{{\left (d x^{3} - 8 \, c\right )} d}\right )}}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^5/(d*x^3 - 8*c)^2,x, algorithm="giac")
[Out]