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

Optimal. Leaf size=51 \[ \frac{1}{2} \sqrt{x^4+5} x^4-\frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{1}{2} \left (10-x^2\right ) \sqrt{x^4+5} \]

[Out]

(x^4*Sqrt[5 + x^4])/2 - ((10 - x^2)*Sqrt[5 + x^4])/2 - (5*ArcSinh[x^2/Sqrt[5]])/
2

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Rubi [A]  time = 0.134439, antiderivative size = 51, 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 \[ \frac{1}{2} \sqrt{x^4+5} x^4-\frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{1}{2} \left (10-x^2\right ) \sqrt{x^4+5} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[5 + x^4])/2 - ((10 - x^2)*Sqrt[5 + x^4])/2 - (5*ArcSinh[x^2/Sqrt[5]])/
2

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Rubi in Sympy [A]  time = 10.6654, size = 44, normalized size = 0.86 \[ \frac{x^{4} \sqrt{x^{4} + 5}}{2} - \frac{\left (- 6 x^{2} + 60\right ) \sqrt{x^{4} + 5}}{12} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(x**4 + 5)/2 - (-6*x**2 + 60)*sqrt(x**4 + 5)/12 - 5*asinh(sqrt(5)*x**2/
5)/2

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Mathematica [A]  time = 0.029074, size = 36, normalized size = 0.71 \[ \frac{1}{2} \sqrt{x^4+5} \left (x^4+x^2-10\right )-\frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(-10 + x^2 + x^4))/2 - (5*ArcSinh[x^2/Sqrt[5]])/2

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Maple [A]  time = 0.016, size = 39, normalized size = 0.8 \[{\frac{{x}^{2}}{2}\sqrt{{x}^{4}+5}}-{\frac{5}{2}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{{x}^{4}-10}{2}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.778367, size = 103, normalized size = 2.02 \[ \frac{1}{2} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - \frac{15}{2} \, \sqrt{x^{4} + 5} + \frac{5 \, \sqrt{x^{4} + 5}}{2 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} - \frac{5}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{5}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.296281, size = 190, normalized size = 3.73 \[ -\frac{4 \, x^{12} + 4 \, x^{10} - 15 \, x^{8} + 25 \, x^{6} - 225 \, x^{4} + 25 \, x^{2} - 5 \,{\left (4 \, x^{6} + 15 \, x^{2} -{\left (4 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (4 \, x^{10} + 4 \, x^{8} - 25 \, x^{6} + 15 \, x^{4} - 150 \, x^{2}\right )} \sqrt{x^{4} + 5} - 250}{2 \,{\left (4 \, x^{6} + 15 \, x^{2} -{\left (4 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*x^12 + 4*x^10 - 15*x^8 + 25*x^6 - 225*x^4 + 25*x^2 - 5*(4*x^6 + 15*x^2 -
(4*x^4 + 5)*sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4 + 5)) - (4*x^10 + 4*x^8 - 25*x^6
+ 15*x^4 - 150*x^2)*sqrt(x^4 + 5) - 250)/(4*x^6 + 15*x^2 - (4*x^4 + 5)*sqrt(x^4
+ 5))

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Sympy [A]  time = 10.5183, size = 66, normalized size = 1.29 \[ \frac{x^{6}}{2 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{5 x^{2}}{2 \sqrt{x^{4} + 5}} - 5 \sqrt{x^{4} + 5} - \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(2*sqrt(x**4 + 5)) + x**4*sqrt(x**4 + 5)/2 + 5*x**2/(2*sqrt(x**4 + 5)) - 5*
sqrt(x**4 + 5) - 5*asinh(sqrt(5)*x**2/5)/2

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GIAC/XCAS [A]  time = 0.264367, size = 50, normalized size = 0.98 \[ \frac{1}{2} \, \sqrt{x^{4} + 5}{\left ({\left (x^{2} + 1\right )} x^{2} - 10\right )} + \frac{5}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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