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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {650} \begin {gather*} \frac {2 (x+1)}{\sqrt {x^2+3 x+2}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.08, size = 19, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {2+3 x+x^2}}{2+x} \end {gather*}

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 = 0.66, size = 30, normalized size = 1.76

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 26, normalized size = 1.53 \begin {gather*} \frac {2 \, x}{\sqrt {x^{2} + 3 \, x + 2}} + \frac {2}{\sqrt {x^{2} + 3 \, x + 2}} \end {gather*}

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 = 2.42, size = 20, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (x + \sqrt {x^{2} + 3 \, x + 2} + 2\right )}}{x + 2} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\left (\left (x + 1\right ) \left (x + 2\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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 = 1.93, size = 19, normalized size = 1.12 \begin {gather*} \frac {2}{x - \sqrt {x^{2} + 3 \, x + 2} + 2} \end {gather*}

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)

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Mupad [B]
time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} \frac {2\,\sqrt {x^2+3\,x+2}}{x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(3*x + x^2 + 2)^(3/2),x)

[Out]

(2*(3*x + x^2 + 2)^(1/2))/(x + 2)

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