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

Optimal. Leaf size=20 \[ -\frac{15-2 x^2}{10 \sqrt{x^4+5}} \]

[Out]

-(15 - 2*x^2)/(10*Sqrt[5 + x^4])

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Rubi [A]  time = 0.0505839, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{15-2 x^2}{10 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

-(15 - 2*x^2)/(10*Sqrt[5 + x^4])

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Rubi in Sympy [A]  time = 6.2442, size = 17, normalized size = 0.85 \[ - \frac{- 2 x^{2} + 15}{10 \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x**2 + 15)/(10*sqrt(x**4 + 5))

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Mathematica [A]  time = 0.0157912, size = 20, normalized size = 1. \[ \frac{2 x^2-15}{10 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

(-15 + 2*x^2)/(10*Sqrt[5 + x^4])

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Maple [A]  time = 0.006, size = 17, normalized size = 0.9 \[{\frac{2\,{x}^{2}-15}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.776759, size = 30, normalized size = 1.5 \[ \frac{x^{2}}{5 \, \sqrt{x^{4} + 5}} - \frac{3}{2 \, \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.308302, size = 50, normalized size = 2.5 \[ \frac{3 \, x^{2} - 3 \, \sqrt{x^{4} + 5} + 2}{2 \,{\left (x^{4} - \sqrt{x^{4} + 5} x^{2} + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.1521, size = 24, normalized size = 1.2 \[ \frac{x^{2}}{5 \sqrt{x^{4} + 5}} - \frac{3}{2 \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270664, size = 22, normalized size = 1.1 \[ \frac{2 \, x^{2} - 15}{10 \, \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(2*x^2 - 15)/sqrt(x^4 + 5)