\(\int (d+e x) \sqrt {9+12 x+4 x^2} \, dx\) [1616]

Optimal result

Integrand size = 20, antiderivative size = 50 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {1}{8} (2 d-3 e) (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, 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, 623} \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {1}{8} (2 x+3) \sqrt {4 x^2+12 x+9} (2 d-3 e)+\frac {1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]

[In]

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

[Out]

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

Rule 623

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1)
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-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 {1}{12} e \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{2} (2 d-3 e) \int \sqrt {9+12 x+4 x^2} \, dx \\ & = \frac {1}{8} (2 d-3 e) (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {x \sqrt {(3+2 x)^2} (6 d (3+x)+e x (9+4 x))}{6 (3+2 x)} \]

[In]

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

[Out]

(x*Sqrt[(3 + 2*x)^2]*(6*d*(3 + x) + e*x*(9 + 4*x)))/(6*(3 + 2*x))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 2.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {\operatorname {csgn}\left (2 x +3\right ) \left (2 x +3\right )^{2} \left (4 e x +6 d -3 e \right )}{24}\)	\(27\)
gosper	\(\frac {x \left (4 e \,x^{2}+6 d x +9 e x +18 d \right ) \sqrt {\left (2 x +3\right )^{2}}}{12 x +18}\)	\(38\)
risch	\(\frac {2 \sqrt {\left (2 x +3\right )^{2}}\, e \,x^{3}}{3 \left (2 x +3\right )}+\frac {\sqrt {\left (2 x +3\right )^{2}}\, \left (2 d +3 e \right ) x^{2}}{4 x +6}+\frac {3 \sqrt {\left (2 x +3\right )^{2}}\, d x}{2 x +3}\)	\(72\)

[In]

int((e*x+d)*(4*x^2+12*x+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/24*csgn(2*x+3)*(2*x+3)^2*(4*e*x+6*d-3*e)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.46 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {2}{3} \, e x^{3} + \frac {1}{2} \, {\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \]

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\sqrt {4 x^{2} + 12 x + 9} \cdot \left (\frac {3 d}{4} + \frac {e x^{2}}{3} - \frac {3 e}{8} + x \left (\frac {d}{2} + \frac {e}{4}\right )\right ) \]

[In]

integrate((e*x+d)*(4*x**2+12*x+9)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.56 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {1}{12} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} e + \frac {1}{2} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d x - \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e x + \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d - \frac {9}{8} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \]

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {2}{3} \, e x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + d x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {3}{2} \, e x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + 3 \, d x \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {9}{8} \, {\left (2 \, d - e\right )} \mathrm {sgn}\left (2 \, x + 3\right ) \]

[In]

integrate((e*x+d)*(4*x^2+12*x+9)^(1/2),x, algorithm="giac")

[Out]

2/3*e*x^3*sgn(2*x + 3) + d*x^2*sgn(2*x + 3) + 3/2*e*x^2*sgn(2*x + 3) + 3*d*x*sgn(2*x + 3) + 9/8*(2*d - e)*sgn(
2*x + 3)

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.60 \[ \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx=\frac {\left (2\,x+3\right )\,\left (6\,d-3\,e+4\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{24} \]

[In]

int((d + e*x)*(12*x + 4*x^2 + 9)^(1/2),x)

[Out]

((2*x + 3)*(6*d - 3*e + 4*e*x)*(12*x + 4*x^2 + 9)^(1/2))/24