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

Optimal. Leaf size=42 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{2 \sqrt{5}}-\frac{\sqrt{x^4+5}}{5 x^2} \]

[Out]

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

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Rubi [A]  time = 0.107084, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{2 \sqrt{5}}-\frac{\sqrt{x^4+5}}{5 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0094, size = 39, normalized size = 0.93 \[ - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{10} - \frac{\sqrt{x^{4} + 5}}{5 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0453208, size = 42, normalized size = 1. \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{2 \sqrt{5}}-\frac{\sqrt{x^4+5}}{5 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 31, normalized size = 0.7 \[ -{\frac{1}{5\,{x}^{2}}\sqrt{{x}^{4}+5}}-{\frac{3\,\sqrt{5}}{10}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.784281, size = 63, normalized size = 1.5 \[ \frac{3}{20} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) - \frac{\sqrt{x^{4} + 5}}{5 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/20*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) - 1/5*sqr
t(x^4 + 5)/x^2

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Fricas [A]  time = 0.259365, size = 140, normalized size = 3.33 \[ \frac{3 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2}\right )} \log \left (\frac{5 \, x^{2} + \sqrt{5}{\left (x^{4} + 5\right )} - \sqrt{x^{4} + 5}{\left (\sqrt{5} x^{2} + 5\right )}}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) + 2 \, \sqrt{5}}{2 \,{\left (\sqrt{5} x^{4} - \sqrt{5} \sqrt{x^{4} + 5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.66834, size = 31, normalized size = 0.74 \[ - \frac{\sqrt{1 + \frac{5}{x^{4}}}}{5} - \frac{3 \sqrt{5} \operatorname{asinh}{\left (\frac{\sqrt{5}}{x^{2}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(1 + 5/x**4)/5 - 3*sqrt(5)*asinh(sqrt(5)/x**2)/10

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GIAC/XCAS [A]  time = 0.275485, size = 65, normalized size = 1.55 \[ -\frac{3}{20} \, \sqrt{5}{\rm ln}\left (\sqrt{5} + \sqrt{x^{4} + 5}\right ) + \frac{3}{20} \, \sqrt{5}{\rm ln}\left (-\sqrt{5} + \sqrt{x^{4} + 5}\right ) - \frac{1}{5} \, \sqrt{\frac{5}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/20*sqrt(5)*ln(sqrt(5) + sqrt(x^4 + 5)) + 3/20*sqrt(5)*ln(-sqrt(5) + sqrt(x^4
+ 5)) - 1/5*sqrt(5/x^4 + 1)