\(\int \frac {1+x}{(2+3 x+x^2)^{3/2}} \, dx\) [2406]

Optimal result

Integrand size = 16, antiderivative size = 17 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2 (1+x)}{\sqrt {2+3 x+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {650} \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2 (x+1)}{\sqrt {x^2+3 x+2}} \]

[In]

Int[(1 + x)/(2 + 3*x + x^2)^(3/2),x]

[Out]

(2*(1 + x))/Sqrt[2 + 3*x + x^2]

Rule 650

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*} \text {integral}& = \frac {2 (1+x)}{\sqrt {2+3 x+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {2+3 x+x^2}}{2+x} \]

[In]

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

[Out]

(2*Sqrt[2 + 3*x + x^2])/(2 + x)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {2 x +2}{\sqrt {x^{2}+3 x +2}}\)	\(16\)
trager	\(\frac {2 \sqrt {x^{2}+3 x +2}}{2+x}\)	\(18\)
gosper	\(\frac {2 \left (1+x \right )^{2} \left (2+x \right )}{\left (x^{2}+3 x +2\right )^{\frac {3}{2}}}\)	\(21\)
default	\(\frac {3+2 x}{\sqrt {x^{2}+3 x +2}}-\frac {1}{\sqrt {x^{2}+3 x +2}}\)	\(30\)

[In]

int((1+x)/(x^2+3*x+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (x + \sqrt {x^{2} + 3 \, x + 2} + 2\right )}}{x + 2} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2 \, x}{\sqrt {x^{2} + 3 \, x + 2}} + \frac {2}{\sqrt {x^{2} + 3 \, x + 2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2}{x - \sqrt {x^{2} + 3 \, x + 2} + 2} \]

[In]

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

[Out]

2/(x - sqrt(x^2 + 3*x + 2) + 2)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{\left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {2\,\sqrt {x^2+3\,x+2}}{x+2} \]

[In]

int((x + 1)/(3*x + x^2 + 2)^(3/2),x)

[Out]

(2*(3*x + x^2 + 2)^(1/2))/(x + 2)