### 3.2367 $$\int (-2+3 x) \sqrt{8+12 x+9 x^2} \, dx$$

Optimal. Leaf size=54 $\frac{1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{3 x}{2}+1\right )$

[Out]

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

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Rubi [A]  time = 0.0180984, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {640, 612, 619, 215} $\frac{1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{3 x}{2}+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(-2 + 3*x)*Sqrt[8 + 12*x + 9*x^2],x]

[Out]

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

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 612

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] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int (-2+3 x) \sqrt{8+12 x+9 x^2} \, dx &=\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-4 \int \sqrt{8+12 x+9 x^2} \, dx\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-8 \int \frac{1}{\sqrt{8+12 x+9 x^2}} \, dx\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{144}}} \, dx,x,12+18 x\right )\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac{8}{3} \sinh ^{-1}\left (1+\frac{3 x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0203861, size = 40, normalized size = 0.74 $\frac{1}{9} \left (\left (9 x^2-6 x-4\right ) \sqrt{9 x^2+12 x+8}-24 \sinh ^{-1}\left (\frac{3 x}{2}+1\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-2 + 3*x)*Sqrt[8 + 12*x + 9*x^2],x]

[Out]

((-4 - 6*x + 9*x^2)*Sqrt[8 + 12*x + 9*x^2] - 24*ArcSinh[1 + (3*x)/2])/9

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Maple [A]  time = 0.052, size = 43, normalized size = 0.8 \begin{align*} -{\frac{18\,x+12}{9}\sqrt{9\,{x}^{2}+12\,x+8}}-{\frac{8}{3}{\it Arcsinh} \left ( 1+{\frac{3\,x}{2}} \right ) }+{\frac{1}{9} \left ( 9\,{x}^{2}+12\,x+8 \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-2+3*x)*(9*x^2+12*x+8)^(1/2),x)

[Out]

-1/9*(18*x+12)*(9*x^2+12*x+8)^(1/2)-8/3*arcsinh(1+3/2*x)+1/9*(9*x^2+12*x+8)^(3/2)

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Maxima [A]  time = 1.49423, size = 70, normalized size = 1.3 \begin{align*} \frac{1}{9} \,{\left (9 \, x^{2} + 12 \, x + 8\right )}^{\frac{3}{2}} - 2 \, \sqrt{9 \, x^{2} + 12 \, x + 8} x - \frac{4}{3} \, \sqrt{9 \, x^{2} + 12 \, x + 8} - \frac{8}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+3*x)*(9*x^2+12*x+8)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.26475, size = 123, normalized size = 2.28 \begin{align*} \frac{1}{9} \, \sqrt{9 \, x^{2} + 12 \, x + 8}{\left (9 \, x^{2} - 6 \, x - 4\right )} + \frac{8}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+3*x)*(9*x^2+12*x+8)^(1/2),x, algorithm="fricas")

[Out]

1/9*sqrt(9*x^2 + 12*x + 8)*(9*x^2 - 6*x - 4) + 8/3*log(-3*x + sqrt(9*x^2 + 12*x + 8) - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (3 x - 2\right ) \sqrt{9 x^{2} + 12 x + 8}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+3*x)*(9*x**2+12*x+8)**(1/2),x)

[Out]

Integral((3*x - 2)*sqrt(9*x**2 + 12*x + 8), x)

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Giac [A]  time = 1.0945, size = 61, normalized size = 1.13 \begin{align*} \frac{1}{9} \,{\left (3 \,{\left (3 \, x - 2\right )} x - 4\right )} \sqrt{9 \, x^{2} + 12 \, x + 8} + \frac{8}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+3*x)*(9*x^2+12*x+8)^(1/2),x, algorithm="giac")

[Out]

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