### 3.1616 $$\int (d+e x) \sqrt{9+12 x+4 x^2} \, dx$$

Optimal. Leaf size=50 $\frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2}$

[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

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Rubi [A]  time = 0.0120345, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {640, 609} $\frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[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 640

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]

Rule 609

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)]

Rubi steps

\begin{align*} \int (d+e x) \sqrt{9+12 x+4 x^2} \, dx &=\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]  time = 0.0115699, size = 38, normalized size = 0.76 $\frac{x \sqrt{(2 x+3)^2} (6 d (x+3)+e x (4 x+9))}{6 (2 x+3)}$

Antiderivative was successfully veriﬁed.

[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))

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Maple [A]  time = 0.079, size = 38, normalized size = 0.8 \begin{align*}{\frac{x \left ( 4\,e{x}^{2}+6\,dx+9\,ex+18\,d \right ) }{18+12\,x}\sqrt{ \left ( 3+2\,x \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(4*e*x^2+6*d*x+9*e*x+18*d)*((3+2*x)^2)^(1/2)/(3+2*x)

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Maxima [A]  time = 1.59159, size = 105, normalized size = 2.1 \begin{align*} \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 \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.55504, size = 55, normalized size = 1.1 \begin{align*} \frac{2}{3} \, e x^{3} + \frac{1}{2} \,{\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right ) \sqrt{\left (2 x + 3\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18821, size = 86, normalized size = 1.72 \begin{align*} \frac{2}{3} \, x^{3} e \mathrm{sgn}\left (2 \, x + 3\right ) + d x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{3}{2} \, x^{2} e \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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