### 3.1617 $$\int \frac{d+e x}{\sqrt{9+12 x+4 x^2}} \, dx$$

Optimal. Leaf size=56 $\frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0156218, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {640, 608, 31} $\frac{(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt{4 x^2+12 x+9}}+\frac{1}{4} e \sqrt{4 x^2+12 x+9}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 608

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{d+e x}{\sqrt{9+12 x+4 x^2}} \, dx &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{1}{2} (2 d-3 e) \int \frac{1}{\sqrt{9+12 x+4 x^2}} \, dx\\ &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{((2 d-3 e) (6+4 x)) \int \frac{1}{6+4 x} \, dx}{2 \sqrt{9+12 x+4 x^2}}\\ &=\frac{1}{4} e \sqrt{9+12 x+4 x^2}+\frac{(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt{9+12 x+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0153117, size = 42, normalized size = 0.75 $\frac{(2 x+3) ((2 d-3 e) \log (2 x+3)+e (2 x+3))}{4 \sqrt{(2 x+3)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.12, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( 3+2\,x \right ) \left ( 2\,\ln \left ( 3+2\,x \right ) d-3\,e\ln \left ( 3+2\,x \right ) +2\,ex \right ) }{4}{\frac{1}{\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]

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

________________________________________________________________________________________

Maxima [A]  time = 1.55833, size = 41, normalized size = 0.73 \begin{align*} \frac{1}{2} \, d \log \left (x + \frac{3}{2}\right ) - \frac{3}{4} \, e \log \left (x + \frac{3}{2}\right ) + \frac{1}{4} \, \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]

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

________________________________________________________________________________________

Fricas [A]  time = 1.60149, size = 54, normalized size = 0.96 \begin{align*} \frac{1}{2} \, e x + \frac{1}{4} \,{\left (2 \, d - 3 \, e\right )} \log \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="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\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)

________________________________________________________________________________________

Giac [A]  time = 1.1582, size = 62, normalized size = 1.11 \begin{align*} -\frac{1}{4} \,{\left (2 \, d - 3 \, e\right )} \log \left ({\left | -2 \, x + \sqrt{4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) + \frac{1}{4} \, \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="giac")

[Out]

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