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

Optimal. Leaf size=59 \[ -\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\sqrt{x^4+5} \left (2-3 x^2\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.147822, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\sqrt{x^4+5} \left (2-3 x^2\right )}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.02, size = 56, normalized size = 0.95 \[ \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{2} - \frac{\left (- 3 x^{2} + 2\right ) \sqrt{x^{4} + 5}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)*(x**4+5)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0914243, size = 55, normalized size = 0.93 \[ -\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\sqrt{x^4+5} \left (\frac{3}{2}-\frac{1}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 61, normalized size = 1. \[ -{\frac{1}{5\,{x}^{2}} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{5}\sqrt{{x}^{4}+5}}+{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) +{\frac{3}{2}\sqrt{{x}^{4}+5}}-{\frac{3\,\sqrt{5}}{2}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)*(x^4+5)^(1/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.78202, size = 119, normalized size = 2.02 \[ \frac{3}{4} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \frac{3}{2} \, \sqrt{x^{4} + 5} - \frac{\sqrt{x^{4} + 5}}{x^{2}} + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) + 3/2*sqrt
(x^4 + 5) - sqrt(x^4 + 5)/x^2 + 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 = 0.302511, size = 250, normalized size = 4.24 \[ -\frac{6 \, x^{8} + 30 \, x^{4} - 10 \, x^{2} + 2 \,{\left (2 \, x^{6} - 2 \, \sqrt{x^{4} + 5} x^{4} + 5 \, x^{2}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 3 \,{\left (2 \, \sqrt{5} \sqrt{x^{4} + 5} x^{4} - \sqrt{5}{\left (2 \, x^{6} + 5 \, x^{2}\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (6 \, x^{6} + 15 \, x^{2} - 10\right )} \sqrt{x^{4} + 5}}{2 \,{\left (2 \, x^{6} - 2 \, \sqrt{x^{4} + 5} x^{4} + 5 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*x^8 + 30*x^4 - 10*x^2 + 2*(2*x^6 - 2*sqrt(x^4 + 5)*x^4 + 5*x^2)*log(-x^2
+ sqrt(x^4 + 5)) + 3*(2*sqrt(5)*sqrt(x^4 + 5)*x^4 - sqrt(5)*(2*x^6 + 5*x^2))*lo
g((x^4 + sqrt(5)*x^2 - sqrt(x^4 + 5)*(x^2 + sqrt(5)) + 5)/(x^4 - sqrt(x^4 + 5)*x
^2)) - (6*x^6 + 15*x^2 - 10)*sqrt(x^4 + 5))/(2*x^6 - 2*sqrt(x^4 + 5)*x^4 + 5*x^2
)

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Sympy [A]  time = 11.9394, size = 83, normalized size = 1.41 \[ - \frac{x^{2}}{\sqrt{x^{4} + 5}} + \frac{3 \sqrt{x^{4} + 5}}{2} + \frac{3 \sqrt{5} \log{\left (x^{4} \right )}}{4} - \frac{3 \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{2} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} - \frac{5}{x^{2} \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)*(x**4+5)**(1/2)/x**3,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{4} + 5}{\left (3 \, x^{2} + 2\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^4 + 5)*(3*x^2 + 2)/x^3, x)