\(\int \frac {d+e x}{(9+12 x+4 x^2)^{3/2}} \, dx\) [1618]

Optimal result

Integrand size = 20, antiderivative size = 52 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {e}{4 \sqrt {9+12 x+4 x^2}}-\frac {2 d-3 e}{8 (3+2 x) \sqrt {9+12 x+4 x^2}} \]

[Out]

-1/4*e/((3+2*x)^2)^(1/2)+1/8*(-2*d+3*e)/(3+2*x)/((3+2*x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {654, 621} \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {2 d-3 e}{8 (2 x+3) \sqrt {4 x^2+12 x+9}}-\frac {e}{4 \sqrt {4 x^2+12 x+9}} \]

[In]

Int[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

-1/4*e/Sqrt[9 + 12*x + 4*x^2] - (2*d - 3*e)/(8*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {e}{4 \sqrt {9+12 x+4 x^2}}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx \\ & = -\frac {e}{4 \sqrt {9+12 x+4 x^2}}-\frac {2 d-3 e}{8 (3+2 x) \sqrt {9+12 x+4 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.73 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=\frac {(d+e x)^2}{2 (-2 d+3 e) (3+2 x) \sqrt {(3+2 x)^2}} \]

[In]

Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(d + e*x)^2/(2*(-2*d + 3*e)*(3 + 2*x)*Sqrt[(3 + 2*x)^2])

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.54

method	result	size
gosper	\(-\frac {\left (2 x +3\right ) \left (4 e x +2 d +3 e \right )}{8 \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}\)	\(28\)
default	\(-\frac {\left (2 x +3\right ) \left (4 e x +2 d +3 e \right )}{8 \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}}\)	\(28\)
risch	\(\frac {4 \sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {1}{8} e x -\frac {1}{16} d -\frac {3}{32} e \right )}{\left (2 x +3\right )^{3}}\)	\(30\)
meijerg	\(\frac {e \,x^{2}}{54 \left (1+\frac {2 x}{3}\right )^{2}}+\frac {d x \left (\frac {2 x}{3}+2\right )}{54 \left (1+\frac {2 x}{3}\right )^{2}}\)	\(31\)

[In]

int((e*x+d)/(4*x^2+12*x+9)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/8*(2*x+3)*(4*e*x+2*d+3*e)/((2*x+3)^2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.48 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, e x + 2 \, d + 3 \, e}{8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="fricas")

[Out]

-1/8*(4*e*x + 2*d + 3*e)/(4*x^2 + 12*x + 9)

Sympy [F]

\[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=\int \frac {d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral((d + e*x)/((2*x + 3)**2)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.69 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {e}{4 \, \sqrt {4 \, x^{2} + 12 \, x + 9}} - \frac {d}{4 \, {\left (2 \, x + 3\right )}^{2}} + \frac {3 \, e}{8 \, {\left (2 \, x + 3\right )}^{2}} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="maxima")

[Out]

-1/4*e/sqrt(4*x^2 + 12*x + 9) - 1/4*d/(2*x + 3)^2 + 3/8*e/(2*x + 3)^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, e x + 2 \, d + 3 \, e}{8 \, {\left (2 \, x + 3\right )}^{2} \mathrm {sgn}\left (2 \, x + 3\right )} \]

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="giac")

[Out]

-1/8*(4*e*x + 2*d + 3*e)/((2*x + 3)^2*sgn(2*x + 3))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx=-\frac {\left (2\,d+3\,e+4\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{8\,{\left (2\,x+3\right )}^3} \]

[In]

int((d + e*x)/(12*x + 4*x^2 + 9)^(3/2),x)

[Out]

-((2*d + 3*e + 4*e*x)*(12*x + 4*x^2 + 9)^(1/2))/(8*(2*x + 3)^3)