3.1312 \(\int \frac{x^{13/2}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

[Out]

(x^(5/2)*Sqrt[1 + x^5])/5 - ArcSinh[x^(5/2)]/5

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Rubi [A]  time = 0.0345988, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/Sqrt[1 + x^5],x]

[Out]

(x^(5/2)*Sqrt[1 + x^5])/5 - ArcSinh[x^(5/2)]/5

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Rubi in Sympy [A]  time = 4.94909, size = 22, normalized size = 0.76 \[ \frac{x^{\frac{5}{2}} \sqrt{x^{5} + 1}}{5} - \frac{\operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/(x**5+1)**(1/2),x)

[Out]

x**(5/2)*sqrt(x**5 + 1)/5 - asinh(x**(5/2))/5

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Mathematica [A]  time = 0.0532852, size = 29, normalized size = 1. \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/Sqrt[1 + x^5],x]

[Out]

(x^(5/2)*Sqrt[1 + x^5])/5 - ArcSinh[x^(5/2)]/5

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Maple [A]  time = 0.049, size = 39, normalized size = 1.3 \[{\frac{1}{5}{x}^{{\frac{5}{2}}}\sqrt{{x}^{5}+1}}-{\frac{1}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{x \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(x^5+1)^(1/2),x)

[Out]

1/5*x^(5/2)*(x^5+1)^(1/2)-1/5*arcsinh(x^(5/2))*(x*(x^5+1))^(1/2)/x^(1/2)/(x^5+1)
^(1/2)

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Maxima [A]  time = 1.4995, size = 78, normalized size = 2.69 \[ \frac{\sqrt{x^{5} + 1}}{5 \, x^{\frac{5}{2}}{\left (\frac{x^{5} + 1}{x^{5}} - 1\right )}} - \frac{1}{10} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) + \frac{1}{10} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

1/5*sqrt(x^5 + 1)/(x^(5/2)*((x^5 + 1)/x^5 - 1)) - 1/10*log(sqrt(x^5 + 1)/x^(5/2)
+ 1) + 1/10*log(sqrt(x^5 + 1)/x^(5/2) - 1)

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Fricas [A]  time = 0.2659, size = 47, normalized size = 1.62 \[ \frac{1}{5} \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + \frac{1}{10} \, \log \left (-2 \, x^{5} + 2 \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

1/5*sqrt(x^5 + 1)*x^(5/2) + 1/10*log(-2*x^5 + 2*sqrt(x^5 + 1)*x^(5/2) - 1)

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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**(13/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230682, size = 39, normalized size = 1.34 \[ \frac{1}{5} \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + \frac{1}{5} \,{\rm ln}\left (-x^{\frac{5}{2}} + \sqrt{x^{5} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

1/5*sqrt(x^5 + 1)*x^(5/2) + 1/5*ln(-x^(5/2) + sqrt(x^5 + 1))