Optimal. Leaf size=50 \[ \frac{\sqrt{x^5+1} \sqrt{a x^{13}}}{5 x^4}-\frac{\sqrt{a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]
[Out]
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Rubi [A] time = 0.0400788, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\sqrt{x^5+1} \sqrt{a x^{13}}}{5 x^4}-\frac{\sqrt{a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^13]/Sqrt[1 + x^5],x]
[Out]
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Rubi in Sympy [A] time = 10.2477, size = 42, normalized size = 0.84 \[ \frac{\sqrt{a x^{13}} \sqrt{x^{5} + 1}}{5 x^{4}} - \frac{\sqrt{a x^{13}} \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5 x^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**13)**(1/2)/(x**5+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0880008, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{1+x^5}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a*x^13]/Sqrt[1 + x^5],x]
[Out]
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Maple [A] time = 0.066, size = 57, normalized size = 1.1 \[{\frac{1}{5\,{x}^{4}}\sqrt{a{x}^{13}}\sqrt{{x}^{5}+1}}-{\frac{1}{5\,{x}^{7}}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{13}}\sqrt{ax \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^13)^(1/2)/(x^5+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{x^{5} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.392661, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} x^{4} \log \left (-\frac{8 \, a x^{14} + 8 \, a x^{9} + a x^{4} - 4 \, \sqrt{a x^{13}}{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{a}}{x^{4}}\right ) + 4 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}{20 \, x^{4}}, \frac{\sqrt{-a} x^{4} \arctan \left (\frac{{\left (2 \, x^{9} + x^{4}\right )} \sqrt{-a}}{2 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}\right ) + 2 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}{10 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**13)**(1/2)/(x**5+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282395, size = 92, normalized size = 1.84 \[ \frac{a^{\frac{11}{2}}{\rm ln}\left (-\sqrt{a x} a^{\frac{5}{2}} x^{2} + \sqrt{a^{6} x^{5} + a^{6}}\right )}{5 \,{\left | a \right |}^{5}} + \frac{\sqrt{a^{6} x^{5} + a^{6}} \sqrt{a x} x^{2}}{5 \, a^{2}{\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="giac")
[Out]