3.2.28 \(\int \frac {1}{(2+3 x+x^2)^{3/2}} \, dx\) [128]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627} \begin {gather*} -\frac {2 (2 x+3)}{\sqrt {x^2+3 x+2}} \end {gather*}

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]

Rule 627

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

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2 (3+2 x)}{\sqrt {2+3 x+x^2}} \end {gather*}

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 = 0.46, size = 18, normalized size = 0.95

method	result	size
default	\(-\frac {2 \left (2 x +3\right )}{\sqrt {x^{2}+3 x +2}}\)	\(18\)
trager	\(-\frac {2 \left (2 x +3\right )}{\sqrt {x^{2}+3 x +2}}\)	\(18\)
risch	\(-\frac {2 \left (2 x +3\right )}{\sqrt {x^{2}+3 x +2}}\)	\(18\)
gosper	\(-\frac {2 \left (2+x \right ) \left (x +1\right ) \left (2 x +3\right )}{\left (x^{2}+3 x +2\right )^{\frac {3}{2}}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 26, normalized size = 1.37 \begin {gather*} -\frac {4 \, x}{\sqrt {x^{2} + 3 \, x + 2}} - \frac {6}{\sqrt {x^{2} + 3 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (17) = 34\).
time = 1.33, size = 38, normalized size = 2.00 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} + \sqrt {x^{2} + 3 \, x + 2} {\left (2 \, x + 3\right )} + 6 \, x + 4\right )}}{x^{2} + 3 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)/(x^2 + 3*x + 2)

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

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 [A]
time = 1.38, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (2 \, x + 3\right )}}{\sqrt {x^{2} + 3 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*x + 3)/sqrt(x^2 + 3*x + 2)

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Mupad [B]
time = 0.05, size = 15, normalized size = 0.79 \begin {gather*} -\frac {4\,\left (x+\frac {3}{2}\right )}{\sqrt {x^2+3\,x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x + x^2 + 2)^(3/2),x)

[Out]

-(4*(x + 3/2))/(3*x + x^2 + 2)^(1/2)

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