3.1313 \(\int \frac{x^{3/2}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=10 \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

[Out]

(2*ArcSinh[x^(5/2)])/5

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Rubi [A]  time = 0.0232676, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/Sqrt[1 + x^5],x]

[Out]

(2*ArcSinh[x^(5/2)])/5

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Rubi in Sympy [A]  time = 3.91108, size = 8, normalized size = 0.8 \[ \frac{2 \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(x**5+1)**(1/2),x)

[Out]

2*asinh(x**(5/2))/5

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Mathematica [A]  time = 0.0438722, size = 10, normalized size = 1. \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/Sqrt[1 + x^5],x]

[Out]

(2*ArcSinh[x^(5/2)])/5

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Maple [A]  time = 0.043, size = 7, normalized size = 0.7 \[{\frac{2}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(x^5+1)^(1/2),x)

[Out]

2/5*arcsinh(x^(5/2))

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Maxima [A]  time = 1.43817, size = 45, normalized size = 4.5 \[ \frac{1}{5} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) - \frac{1}{5} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.265184, size = 30, normalized size = 3. \[ \frac{1}{5} \, \log \left (2 \, x^{5} + 2 \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.55956, size = 8, normalized size = 0.8 \[ \frac{2 \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(x**5+1)**(1/2),x)

[Out]

2*asinh(x**(5/2))/5

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GIAC/XCAS [A]  time = 0.228861, size = 34, normalized size = 3.4 \[ \frac{1}{5} \,{\rm ln}\left (\sqrt{\frac{1}{x^{5}} + 1} + 1\right ) - \frac{1}{5} \,{\rm ln}\left (\sqrt{\frac{1}{x^{5}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

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