3.1311 \(\int \frac{x^{23/2}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3}{20} \sinh ^{-1}\left (x^{5/2}\right )+\frac{1}{10} \sqrt{x^5+1} x^{15/2}-\frac{3}{20} \sqrt{x^5+1} x^{5/2} \]

[Out]

(-3*x^(5/2)*Sqrt[1 + x^5])/20 + (x^(15/2)*Sqrt[1 + x^5])/10 + (3*ArcSinh[x^(5/2)
])/20

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Rubi [A]  time = 0.0470733, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3}{20} \sinh ^{-1}\left (x^{5/2}\right )+\frac{1}{10} \sqrt{x^5+1} x^{15/2}-\frac{3}{20} \sqrt{x^5+1} x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(-3*x^(5/2)*Sqrt[1 + x^5])/20 + (x^(15/2)*Sqrt[1 + x^5])/10 + (3*ArcSinh[x^(5/2)
])/20

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Rubi in Sympy [A]  time = 6.02558, size = 41, normalized size = 0.87 \[ \frac{x^{\frac{15}{2}} \sqrt{x^{5} + 1}}{10} - \frac{3 x^{\frac{5}{2}} \sqrt{x^{5} + 1}}{20} + \frac{3 \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(23/2)/(x**5+1)**(1/2),x)

[Out]

x**(15/2)*sqrt(x**5 + 1)/10 - 3*x**(5/2)*sqrt(x**5 + 1)/20 + 3*asinh(x**(5/2))/2
0

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Mathematica [A]  time = 0.0727955, size = 35, normalized size = 0.74 \[ \frac{1}{20} \left (3 \sinh ^{-1}\left (x^{5/2}\right )+\sqrt{x^5+1} \left (2 x^5-3\right ) x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5) + 3*ArcSinh[x^(5/2)])/20

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Maple [A]  time = 0.073, size = 46, normalized size = 1. \[{\frac{2\,{x}^{5}-3}{20}{x}^{{\frac{5}{2}}}\sqrt{{x}^{5}+1}}+{\frac{3}{20}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{x \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(23/2)/(x^5+1)^(1/2),x)

[Out]

1/20*x^(5/2)*(2*x^5-3)*(x^5+1)^(1/2)+3/20*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 = 1.43045, size = 116, normalized size = 2.47 \[ -\frac{\frac{5 \, \sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - \frac{3 \,{\left (x^{5} + 1\right )}^{\frac{3}{2}}}{x^{\frac{15}{2}}}}{20 \,{\left (\frac{2 \,{\left (x^{5} + 1\right )}}{x^{5}} - \frac{{\left (x^{5} + 1\right )}^{2}}{x^{10}} - 1\right )}} + \frac{3}{40} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) - \frac{3}{40} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

-1/20*(5*sqrt(x^5 + 1)/x^(5/2) - 3*(x^5 + 1)^(3/2)/x^(15/2))/(2*(x^5 + 1)/x^5 -
(x^5 + 1)^2/x^10 - 1) + 3/40*log(sqrt(x^5 + 1)/x^(5/2) + 1) - 3/40*log(sqrt(x^5
+ 1)/x^(5/2) - 1)

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Fricas [A]  time = 0.269195, size = 62, normalized size = 1.32 \[ \frac{1}{20} \,{\left (2 \, x^{7} - 3 \, x^{2}\right )} \sqrt{x^{5} + 1} \sqrt{x} + \frac{3}{40} \, \log \left (2 \, x^{5} + 2 \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

1/20*(2*x^7 - 3*x^2)*sqrt(x^5 + 1)*sqrt(x) + 3/40*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**(23/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233545, size = 49, normalized size = 1.04 \[ \frac{1}{20} \,{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1} x^{\frac{5}{2}} - \frac{3}{20} \,{\rm ln}\left (-x^{\frac{5}{2}} + \sqrt{x^{5} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

1/20*(2*x^5 - 3)*sqrt(x^5 + 1)*x^(5/2) - 3/20*ln(-x^(5/2) + sqrt(x^5 + 1))