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

Optimal. Leaf size=13 \[ -\frac{1}{3 \left (x^2+4\right )^{3/2}} \]

[Out]

-1/(3*(4 + x^2)^(3/2))

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Rubi [A]  time = 0.00546531, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{3 \left (x^2+4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(4 + x^2)^(5/2),x]

[Out]

-1/(3*(4 + x^2)^(3/2))

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Rubi in Sympy [A]  time = 0.763352, size = 12, normalized size = 0.92 \[ - \frac{1}{3 \left (x^{2} + 4\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*(x**2 + 4)**(3/2))

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Mathematica [A]  time = 0.00290577, size = 13, normalized size = 1. \[ -\frac{1}{3 \left (x^2+4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(4 + x^2)^(5/2),x]

[Out]

-1/(3*(4 + x^2)^(3/2))

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Maple [A]  time = 0.004, size = 10, normalized size = 0.8 \[ -{\frac{1}{3} \left ({x}^{2}+4 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+4)^(5/2),x)

[Out]

-1/3/(x^2+4)^(3/2)

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Maxima [A]  time = 1.32695, size = 12, normalized size = 0.92 \[ -\frac{1}{3 \,{\left (x^{2} + 4\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(x^2 + 4)^(3/2)

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Fricas [A]  time = 0.200361, size = 82, normalized size = 6.31 \[ \frac{x^{3} - \sqrt{x^{2} + 4}{\left (x^{2} + 1\right )} + 3 \, x}{3 \,{\left (x^{6} + 9 \, x^{4} + 24 \, x^{2} -{\left (x^{5} + 7 \, x^{3} + 12 \, x\right )} \sqrt{x^{2} + 4} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^3 - sqrt(x^2 + 4)*(x^2 + 1) + 3*x)/(x^6 + 9*x^4 + 24*x^2 - (x^5 + 7*x^3 +
12*x)*sqrt(x^2 + 4) + 16)

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Sympy [A]  time = 4.08395, size = 26, normalized size = 2. \[ - \frac{1}{3 x^{2} \sqrt{x^{2} + 4} + 12 \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+4)**(5/2),x)

[Out]

-1/(3*x**2*sqrt(x**2 + 4) + 12*sqrt(x**2 + 4))

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GIAC/XCAS [A]  time = 0.200548, size = 12, normalized size = 0.92 \[ -\frac{1}{3 \,{\left (x^{2} + 4\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(x^2 + 4)^(3/2)