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

Optimal. Leaf size=58 \[ -\sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{15}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{4} \sqrt{x^4+5} \left (3 x^2+4\right ) \]

[Out]

((4 + 3*x^2)*Sqrt[5 + x^4])/4 + (15*ArcSinh[x^2/Sqrt[5]])/4 - Sqrt[5]*ArcTanh[Sq
rt[5 + x^4]/Sqrt[5]]

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Rubi [A]  time = 0.143838, antiderivative size = 58, 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 \[ -\sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{15}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{4} \sqrt{x^4+5} \left (3 x^2+4\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4 + 3*x^2)*Sqrt[5 + x^4])/4 + (15*ArcSinh[x^2/Sqrt[5]])/4 - Sqrt[5]*ArcTanh[Sq
rt[5 + x^4]/Sqrt[5]]

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Rubi in Sympy [A]  time = 13.8267, size = 53, normalized size = 0.91 \[ \frac{\left (3 x^{2} + 4\right ) \sqrt{x^{4} + 5}}{4} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} - \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0938827, size = 57, normalized size = 0.98 \[ -\sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{15}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\sqrt{x^4+5} \left (\frac{3 x^2}{4}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1 + (3*x^2)/4)*Sqrt[5 + x^4] + (15*ArcSinh[x^2/Sqrt[5]])/4 - Sqrt[5]*ArcTanh[Sq
rt[5 + x^4]/Sqrt[5]]

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Maple [A]  time = 0.017, size = 49, normalized size = 0.8 \[ \sqrt{{x}^{4}+5}-\sqrt{5}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) +{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5}}+{\frac{15}{4}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^4+5)^(1/2)-5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))+3/4*x^2*(x^4+5)^(1/2)+15/4*
arcsinh(1/5*5^(1/2)*x^2)

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Maxima [A]  time = 0.782895, size = 134, normalized size = 2.31 \[ \frac{1}{2} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \sqrt{x^{4} + 5} + \frac{15 \, \sqrt{x^{4} + 5}}{4 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} + \frac{15}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{15}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272107, size = 247, normalized size = 4.26 \[ -\frac{6 \, x^{8} + 8 \, x^{6} + 30 \, x^{4} + 40 \, x^{2} + 15 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 5} x^{2} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 4 \,{\left (2 \, \sqrt{5} \sqrt{x^{4} + 5} x^{2} - \sqrt{5}{\left (2 \, x^{4} + 5\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (6 \, x^{6} + 8 \, x^{4} + 15 \, x^{2} + 20\right )} \sqrt{x^{4} + 5}}{4 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 5} x^{2} + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(6*x^8 + 8*x^6 + 30*x^4 + 40*x^2 + 15*(2*x^4 - 2*sqrt(x^4 + 5)*x^2 + 5)*log
(-x^2 + sqrt(x^4 + 5)) + 4*(2*sqrt(5)*sqrt(x^4 + 5)*x^2 - sqrt(5)*(2*x^4 + 5))*l
og((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 + 8*x^4 + 15*x^2 + 20)*sqrt(x^4 + 5))/(2*x^4 - 2*sqrt(x^4 + 5)*x^
2 + 5)

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Sympy [A]  time = 12.2978, size = 83, normalized size = 1.43 \[ \frac{3 x^{6}}{4 \sqrt{x^{4} + 5}} + \frac{15 x^{2}}{4 \sqrt{x^{4} + 5}} + \sqrt{x^{4} + 5} + \frac{\sqrt{5} \log{\left (x^{4} \right )}}{2} - \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**6/(4*sqrt(x**4 + 5)) + 15*x**2/(4*sqrt(x**4 + 5)) + sqrt(x**4 + 5) + sqrt(5
)*log(x**4)/2 - sqrt(5)*log(sqrt(x**4/5 + 1) + 1) + 15*asinh(sqrt(5)*x**2/5)/4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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