3.46 \(\int \frac{x^3 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]

[Out]

-(2 + 3*x^2)/(2*Sqrt[5 + x^4]) + (3*ArcSinh[x^2/Sqrt[5]])/2

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Rubi [A]  time = 0.0831988, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(2 + 3*x^2))/(5 + x^4)^(3/2),x]

[Out]

-(2 + 3*x^2)/(2*Sqrt[5 + x^4]) + (3*ArcSinh[x^2/Sqrt[5]])/2

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Rubi in Sympy [A]  time = 8.20522, size = 31, normalized size = 0.89 \[ - \frac{15 x^{2} + 10}{10 \sqrt{x^{4} + 5}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(3*x**2+2)/(x**4+5)**(3/2),x)

[Out]

-(15*x**2 + 10)/(10*sqrt(x**4 + 5)) + 3*asinh(sqrt(5)*x**2/5)/2

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Mathematica [A]  time = 0.028633, size = 35, normalized size = 1. \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{-3 x^2-2}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(2 + 3*x^2))/(5 + x^4)^(3/2),x]

[Out]

(-2 - 3*x^2)/(2*Sqrt[5 + x^4]) + (3*ArcSinh[x^2/Sqrt[5]])/2

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Maple [A]  time = 0.01, size = 34, normalized size = 1. \[ -{\frac{1}{\sqrt{{x}^{4}+5}}}-{\frac{3\,{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3}{2}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(3*x^2+2)/(x^4+5)^(3/2),x)

[Out]

-1/(x^4+5)^(1/2)-3/2*x^2/(x^4+5)^(1/2)+3/2*arcsinh(1/5*5^(1/2)*x^2)

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Maxima [A]  time = 0.787725, size = 73, normalized size = 2.09 \[ -\frac{3 \, x^{2}}{2 \, \sqrt{x^{4} + 5}} - \frac{1}{\sqrt{x^{4} + 5}} + \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="maxima")

[Out]

-3/2*x^2/sqrt(x^4 + 5) - 1/sqrt(x^4 + 5) + 3/4*log(sqrt(x^4 + 5)/x^2 + 1) - 3/4*
log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.306065, size = 95, normalized size = 2.71 \[ \frac{2 \, x^{2} - 3 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) - 2 \, \sqrt{x^{4} + 5} - 15}{2 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2} + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="fricas")

[Out]

1/2*(2*x^2 - 3*(x^4 - sqrt(x^4 + 5)*x^2 + 5)*log(-x^2 + sqrt(x^4 + 5)) - 2*sqrt(
x^4 + 5) - 15)/(x^4 - sqrt(x^4 + 5)*x^2 + 5)

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Sympy [A]  time = 16.0662, size = 39, normalized size = 1.11 \[ - \frac{3 x^{2}}{2 \sqrt{x^{4} + 5}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} - \frac{1}{\sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(3*x**2+2)/(x**4+5)**(3/2),x)

[Out]

-3*x**2/(2*sqrt(x**4 + 5)) + 3*asinh(sqrt(5)*x**2/5)/2 - 1/sqrt(x**4 + 5)

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GIAC/XCAS [A]  time = 0.274721, size = 45, normalized size = 1.29 \[ -\frac{3 \, x^{2} + 2}{2 \, \sqrt{x^{4} + 5}} - \frac{3}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^3/(x^4 + 5)^(3/2),x, algorithm="giac")

[Out]

-1/2*(3*x^2 + 2)/sqrt(x^4 + 5) - 3/2*ln(-x^2 + sqrt(x^4 + 5))