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

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=-\frac {6-x}{2 \sqrt {4+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {651} \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=-\frac {6-x}{2 \sqrt {x^2+4}} \]

[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*} \text {integral}& = -\frac {6-x}{2 \sqrt {4+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {-6+x}{2 \sqrt {4+x^2}} \]

[In]

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

[Out]

(-6 + x)/(2*Sqrt[4 + x^2])

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x^{2} + \sqrt {x^{2} + 4} {\left (x - 6\right )} + 4}{2 \, {\left (x^{2} + 4\right )}} \]

[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] (verification not implemented)

Time = 1.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x}{2 \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x}{2 \, \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x - 6}{2 \, \sqrt {x^{2} + 4}} \]

[In]

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

[Out]

1/2*(x - 6)/sqrt(x^2 + 4)

Mupad [B] (verification not implemented)

Time = 9.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x-6}{2\,\sqrt {x^2+4}} \]

[In]

int((3*x + 2)/(x^2 + 4)^(3/2),x)

[Out]

(x - 6)/(2*(x^2 + 4)^(1/2))