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

Optimal. Leaf size=45 \[ 3 \sqrt{x^4+5}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (3 x^2+2\right ) x^2}{2 \sqrt{x^4+5}} \]

[Out]

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

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Rubi [A]  time = 0.120434, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 3 \sqrt{x^4+5}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (3 x^2+2\right ) x^2}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.5176, size = 41, normalized size = 0.91 \[ - \frac{x^{2} \left (30 x^{2} + 20\right )}{20 \sqrt{x^{4} + 5}} + 3 \sqrt{x^{4} + 5} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*(30*x**2 + 20)/(20*sqrt(x**4 + 5)) + 3*sqrt(x**4 + 5) + asinh(sqrt(5)*x**2
/5)

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Mathematica [A]  time = 0.0475588, size = 36, normalized size = 0.8 \[ \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{3 x^4-2 x^2+30}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

(30 - 2*x^2 + 3*x^4)/(2*Sqrt[5 + x^4]) + ArcSinh[x^2/Sqrt[5]]

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Maple [A]  time = 0.018, size = 37, normalized size = 0.8 \[ -{{x}^{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) +{\frac{3\,{x}^{4}+30}{2}{\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x^2/(x^4+5)^(1/2)+arcsinh(1/5*5^(1/2)*x^2)+3/2/(x^4+5)^(1/2)*(x^4+10)

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Maxima [A]  time = 0.783187, size = 85, normalized size = 1.89 \[ -\frac{x^{2}}{\sqrt{x^{4} + 5}} + \frac{3}{2} \, \sqrt{x^{4} + 5} + \frac{15}{2 \, \sqrt{x^{4} + 5}} + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.326741, size = 151, normalized size = 3.36 \[ -\frac{6 \, x^{8} + 75 \, x^{4} + 10 \, x^{2} + 2 \,{\left (2 \, x^{6} + 10 \, x^{2} -{\left (2 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) - 2 \,{\left (3 \, x^{6} + 30 \, x^{2} + 5\right )} \sqrt{x^{4} + 5} + 150}{2 \,{\left (2 \, x^{6} + 10 \, x^{2} -{\left (2 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*x^8 + 75*x^4 + 10*x^2 + 2*(2*x^6 + 10*x^2 - (2*x^4 + 5)*sqrt(x^4 + 5))*l
og(-x^2 + sqrt(x^4 + 5)) - 2*(3*x^6 + 30*x^2 + 5)*sqrt(x^4 + 5) + 150)/(2*x^6 +
10*x^2 - (2*x^4 + 5)*sqrt(x^4 + 5))

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Sympy [A]  time = 20.5688, size = 48, normalized size = 1.07 \[ \frac{3 x^{4}}{2 \sqrt{x^{4} + 5}} - \frac{x^{2}}{\sqrt{x^{4} + 5}} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} + \frac{15}{\sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269716, size = 53, normalized size = 1.18 \[ \frac{{\left (3 \, x^{2} - 2\right )} x^{2} + 30}{2 \, \sqrt{x^{4} + 5}} -{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((3*x^2 - 2)*x^2 + 30)/sqrt(x^4 + 5) - ln(-x^2 + sqrt(x^4 + 5))