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

Optimal. Leaf size=58 \[ -\frac{\sqrt{x^4+5}}{10 x^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{10 \sqrt{5}}-\frac{3 \sqrt{x^4+5}}{10 x^2} \]

[Out]

-Sqrt[5 + x^4]/(10*x^4) - (3*Sqrt[5 + x^4])/(10*x^2) + ArcTanh[Sqrt[5 + x^4]/Sqr
t[5]]/(10*Sqrt[5])

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Rubi [A]  time = 0.14823, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\sqrt{x^4+5}}{10 x^4}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{10 \sqrt{5}}-\frac{3 \sqrt{x^4+5}}{10 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[5 + x^4]/(10*x^4) - (3*Sqrt[5 + x^4])/(10*x^2) + ArcTanh[Sqrt[5 + x^4]/Sqr
t[5]]/(10*Sqrt[5])

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Rubi in Sympy [A]  time = 12.361, size = 51, normalized size = 0.88 \[ \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{50} - \frac{3 \sqrt{x^{4} + 5}}{10 x^{2}} - \frac{\sqrt{x^{4} + 5}}{10 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)/50 - 3*sqrt(x**4 + 5)/(10*x**2) - sqrt(x
**4 + 5)/(10*x**4)

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Mathematica [A]  time = 0.0496393, size = 51, normalized size = 0.88 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{10 \sqrt{5}}+\sqrt{x^4+5} \left (-\frac{1}{10 x^4}-\frac{3}{10 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1/(10*x^4) - 3/(10*x^2))*Sqrt[5 + x^4] + ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]/(10*Sq
rt[5])

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Maple [A]  time = 0.018, size = 43, normalized size = 0.7 \[ -{\frac{1}{10\,{x}^{4}}\sqrt{{x}^{4}+5}}+{\frac{\sqrt{5}}{50}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) }-{\frac{3}{10\,{x}^{2}}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(x^4+5)^(1/2)/x^4+1/50*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))-3/10*(x^4+5)
^(1/2)/x^2

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Maxima [A]  time = 0.781845, size = 80, normalized size = 1.38 \[ -\frac{1}{100} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) - \frac{3 \, \sqrt{x^{4} + 5}}{10 \, x^{2}} - \frac{\sqrt{x^{4} + 5}}{10 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/100*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) - 3/10*
sqrt(x^4 + 5)/x^2 - 1/10*sqrt(x^4 + 5)/x^4

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Fricas [A]  time = 0.314246, size = 216, normalized size = 3.72 \[ \frac{\sqrt{5}{\left (2 \, x^{4} + 15 \, x^{2} + 5\right )} \sqrt{x^{4} + 5} -{\left (2 \, x^{8} - 2 \, \sqrt{x^{4} + 5} x^{6} + 5 \, x^{4}\right )} \log \left (-\frac{5 \, x^{2} - \sqrt{5}{\left (x^{4} + 5\right )} + \sqrt{x^{4} + 5}{\left (\sqrt{5} x^{2} - 5\right )}}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) - \sqrt{5}{\left (2 \, x^{6} + 15 \, x^{4} + 10 \, x^{2}\right )}}{10 \,{\left (2 \, \sqrt{5} \sqrt{x^{4} + 5} x^{6} - \sqrt{5}{\left (2 \, x^{8} + 5 \, x^{4}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(sqrt(5)*(2*x^4 + 15*x^2 + 5)*sqrt(x^4 + 5) - (2*x^8 - 2*sqrt(x^4 + 5)*x^6
+ 5*x^4)*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)) - sqrt(5)*(2*x^6 + 15*x^4 + 10*x^2))/(2*sqrt(5)*sqrt(x^4
+ 5)*x^6 - sqrt(5)*(2*x^8 + 5*x^4))

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Sympy [A]  time = 7.18127, size = 88, normalized size = 1.52 \[ \frac{\sqrt{5} \left (- \frac{\log{\left (\sqrt{\frac{x^{4}}{5} + 1} - 1 \right )}}{4} + \frac{\log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{4} - \frac{1}{4 \left (\sqrt{\frac{x^{4}}{5} + 1} + 1\right )} - \frac{1}{4 \left (\sqrt{\frac{x^{4}}{5} + 1} - 1\right )}\right )}{25} - \frac{3 \sqrt{5} \sqrt{5 x^{4} + 25}}{50 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5)*(-log(sqrt(x**4/5 + 1) - 1)/4 + log(sqrt(x**4/5 + 1) + 1)/4 - 1/(4*(sqrt
(x**4/5 + 1) + 1)) - 1/(4*(sqrt(x**4/5 + 1) - 1)))/25 - 3*sqrt(5)*sqrt(5*x**4 +
25)/(50*x**2)

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GIAC/XCAS [A]  time = 0.27716, size = 72, normalized size = 1.24 \[ -\frac{1}{10} \,{\left (\frac{1}{x^{2}} + 3\right )} \sqrt{\frac{5}{x^{4}} + 1} + \frac{1}{100} \, \sqrt{5}{\rm ln}\left (\sqrt{5} + \sqrt{x^{4} + 5}\right ) - \frac{1}{100} \, \sqrt{5}{\rm ln}\left (-\sqrt{5} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(1/x^2 + 3)*sqrt(5/x^4 + 1) + 1/100*sqrt(5)*ln(sqrt(5) + sqrt(x^4 + 5)) -
1/100*sqrt(5)*ln(-sqrt(5) + sqrt(x^4 + 5))