### 3.2406 $$\int \frac{1+x}{(2+3 x+x^2)^{3/2}} \, dx$$

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 = 0.0045294, 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, Rules used = {636} $\frac{2 (x+1)}{\sqrt{x^2+3 x+2}}$

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]

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx &=\frac{2 (1+x)}{\sqrt{2+3 x+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0395123, 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 = 0.041, size = 21, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( 1+x \right ) ^{2} \left ( 2+x \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \end{align*}

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 = 0.96237, size = 35, normalized size = 2.06 \begin{align*} \frac{2 \, x}{\sqrt{x^{2} + 3 \, x + 2}} + \frac{2}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(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 = 2.01473, size = 55, normalized size = 3.24 \begin{align*} \frac{2 \,{\left (x + \sqrt{x^{2} + 3 \, x + 2} + 2\right )}}{x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 [A]  time = 1.27919, size = 26, normalized size = 1.53 \begin{align*} \frac{2}{x - \sqrt{x^{2} + 3 \, x + 2} + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(x - sqrt(x^2 + 3*x + 2) + 2)