3.6.90 \(\int \frac {2+3 x}{(4+x^2)^{3/2}} \, dx\) [590]

Optimal. Leaf size=18 \[ -\frac {6-x}{2 \sqrt {4+x^2}} \]

[Out]

1/2*(-6+x)/(x^2+4)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {651} \begin {gather*} -\frac {6-x}{2 \sqrt {x^2+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)/(4 + x^2)^(3/2),x]

[Out]

-1/2*(6 - x)/Sqrt[4 + x^2]

Rule 651

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

Rubi steps

\begin {align*} \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx &=-\frac {6-x}{2 \sqrt {4+x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 16, normalized size = 0.89 \begin {gather*} \frac {-6+x}{2 \sqrt {4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)/(4 + x^2)^(3/2),x]

[Out]

(-6 + x)/(2*Sqrt[4 + x^2])

________________________________________________________________________________________

Maple [A]
time = 0.38, size = 21, normalized size = 1.17

method	result	size
gosper	\(\frac {x -6}{2 \sqrt {x^{2}+4}}\)	\(13\)
trager	\(\frac {x -6}{2 \sqrt {x^{2}+4}}\)	\(13\)
risch	\(\frac {x -6}{2 \sqrt {x^{2}+4}}\)	\(13\)
default	\(\frac {x}{2 \sqrt {x^{2}+4}}-\frac {3}{\sqrt {x^{2}+4}}\)	\(21\)
meijerg	\(\frac {x}{4 \sqrt {1+\frac {x^{2}}{4}}}+\frac {\frac {3 \sqrt {\pi }}{2}-\frac {3 \sqrt {\pi }}{2 \sqrt {1+\frac {x^{2}}{4}}}}{\sqrt {\pi }}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)/(x^2+4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x/(x^2+4)^(1/2)-3/(x^2+4)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 20, normalized size = 1.11 \begin {gather*} \frac {x}{2 \, \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="maxima")

[Out]

1/2*x/sqrt(x^2 + 4) - 3/sqrt(x^2 + 4)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
time = 2.79, size = 25, normalized size = 1.39 \begin {gather*} \frac {x^{2} + \sqrt {x^{2} + 4} {\left (x - 6\right )} + 4}{2 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x^2 + sqrt(x^2 + 4)*(x - 6) + 4)/(x^2 + 4)

________________________________________________________________________________________

Sympy [A]
time = 1.70, size = 20, normalized size = 1.11 \begin {gather*} \frac {x}{2 \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x**2+4)**(3/2),x)

[Out]

x/(2*sqrt(x**2 + 4)) - 3/sqrt(x**2 + 4)

________________________________________________________________________________________

Giac [A]
time = 2.30, size = 12, normalized size = 0.67 \begin {gather*} \frac {x - 6}{2 \, \sqrt {x^{2} + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="giac")

[Out]

1/2*(x - 6)/sqrt(x^2 + 4)

________________________________________________________________________________________

Mupad [B]
time = 0.34, size = 12, normalized size = 0.67 \begin {gather*} \frac {x-6}{2\,\sqrt {x^2+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)/(x^2 + 4)^(3/2),x)

[Out]

(x - 6)/(2*(x^2 + 4)^(1/2))

________________________________________________________________________________________