3.128 \(\int \frac{1}{\left (2+3 x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + x^2]

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Rubi [A]  time = 0.0107709, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x + x^2)^(-3/2),x]

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + x^2]

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Rubi in Sympy [A]  time = 1.19142, size = 17, normalized size = 0.89 \[ - \frac{4 x + 6}{\sqrt{x^{2} + 3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*x + 6)/sqrt(x**2 + 3*x + 2)

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Mathematica [A]  time = 0.0122003, size = 19, normalized size = 1. \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + x^2]

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Maple [A]  time = 0.005, size = 24, normalized size = 1.3 \[ -2\,{\frac{ \left ( 2+x \right ) \left ( 1+x \right ) \left ( 2\,x+3 \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+3*x+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.745951, size = 35, normalized size = 1.84 \[ -\frac{4 \, x}{\sqrt{x^{2} + 3 \, x + 2}} - \frac{6}{\sqrt{x^{2} + 3 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x/sqrt(x^2 + 3*x + 2) - 6/sqrt(x^2 + 3*x + 2)

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Fricas [A]  time = 0.221172, size = 42, normalized size = 2.21 \[ \frac{2}{2 \, x^{2} - \sqrt{x^{2} + 3 \, x + 2}{\left (2 \, x + 3\right )} + 6 \, x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(2*x^2 - sqrt(x^2 + 3*x + 2)*(2*x + 3) + 6*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x^{2} + 3 x + 2\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**2 + 3*x + 2)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.213333, size = 23, normalized size = 1.21 \[ -\frac{2 \,{\left (2 \, x + 3\right )}}{\sqrt{x^{2} + 3 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)/sqrt(x^2 + 3*x + 2)