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

Optimal. Leaf size=35 \[ \frac{1}{4} \left (3 x^2+4\right ) \sqrt{x^4+5}-\frac{15}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0872677, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1}{4} \left (3 x^2+4\right ) \sqrt{x^4+5}-\frac{15}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.99697, size = 31, normalized size = 0.89 \[ \frac{\left (3 x^{2} + 4\right ) \sqrt{x^{4} + 5}}{4} - \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0287086, size = 34, normalized size = 0.97 \[ \frac{1}{4} \left (\left (3 x^2+4\right ) \sqrt{x^4+5}-15 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 32, normalized size = 0.9 \[ \sqrt{{x}^{4}+5}+{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5}}-{\frac{15}{4}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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 = 0.782286, size = 88, normalized size = 2.51 \[ \sqrt{x^{4} + 5} + \frac{15 \, \sqrt{x^{4} + 5}}{4 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} - \frac{15}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{15}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 = 0.266802, size = 142, normalized size = 4.06 \[ -\frac{6 \, x^{8} + 8 \, x^{6} + 30 \, x^{4} + 40 \, x^{2} - 15 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 5} x^{2} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (6 \, x^{6} + 8 \, x^{4} + 15 \, x^{2} + 20\right )} \sqrt{x^{4} + 5}}{4 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 5} x^{2} + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^3/sqrt(x^4 + 5),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)) - (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 = 8.27188, size = 53, normalized size = 1.51 \[ \frac{3 x^{6}}{4 \sqrt{x^{4} + 5}} + \frac{15 x^{2}}{4 \sqrt{x^{4} + 5}} + \sqrt{x^{4} + 5} - \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**6/(4*sqrt(x**4 + 5)) + 15*x**2/(4*sqrt(x**4 + 5)) + sqrt(x**4 + 5) - 15*asi
nh(sqrt(5)*x**2/5)/4

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GIAC/XCAS [A]  time = 0.265266, size = 45, normalized size = 1.29 \[ \frac{1}{4} \, \sqrt{x^{4} + 5}{\left (3 \, x^{2} + 4\right )} + \frac{15}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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