3.2395 \(\int \frac{1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 (x+1)}{\sqrt{x^2+3 x+2}} \]

[Out]

(2*(1 + x))/Sqrt[2 + 3*x + x^2]

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Rubi [A]  time = 0.0197919, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2 (x+1)}{\sqrt{x^2+3 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(1 + x))/Sqrt[2 + 3*x + x^2]

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Rubi in Sympy [A]  time = 3.97071, size = 15, normalized size = 0.88 \[ \frac{2 \left (x + 1\right )}{\sqrt{x^{2} + 3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)/sqrt(x**2 + 3*x + 2)

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Mathematica [A]  time = 0.0208744, size = 19, normalized size = 1.12 \[ \frac{2 \sqrt{x^2+3 x+2}}{x+2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x + x^2])/(2 + x)

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Maple [A]  time = 0.005, size = 21, normalized size = 1.2 \[ 2\,{\frac{ \left ( 1+x \right ) ^{2} \left ( 2+x \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2+3*x+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.67451, size = 35, normalized size = 2.06 \[ \frac{2 \, x}{\sqrt{x^{2} + 3 \, x + 2}} + \frac{2}{\sqrt{x^{2} + 3 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 + 3*x + 2)^(3/2),x, algorithm="maxima")

[Out]

2*x/sqrt(x^2 + 3*x + 2) + 2/sqrt(x^2 + 3*x + 2)

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Fricas [A]  time = 0.214325, size = 26, normalized size = 1.53 \[ \frac{2}{x - \sqrt{x^{2} + 3 \, x + 2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 + 3*x + 2)^(3/2),x, algorithm="fricas")

[Out]

2/(x - sqrt(x^2 + 3*x + 2) + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)/((x + 1)*(x + 2))**(3/2), x)

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GIAC/XCAS [A]  time = 0.208836, size = 26, normalized size = 1.53 \[ \frac{2}{x - \sqrt{x^{2} + 3 \, x + 2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 + 3*x + 2)^(3/2),x, algorithm="giac")

[Out]

2/(x - sqrt(x^2 + 3*x + 2) + 2)