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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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 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]

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 = 0.02, 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

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 45, normalized size = 0.83 \[ \frac {1}{9} \, \sqrt {9 \, x^{2} + 12 \, x + 8} {\left (9 \, x^{2} - 6 \, x - 4\right )} + \frac {8}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x + 8} - 2\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 45, normalized size = 0.83 \[ \frac {1}{9} \, {\left (3 \, {\left (3 \, x - 2\right )} x - 4\right )} \sqrt {9 \, x^{2} + 12 \, x + 8} + \frac {8}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x + 8} - 2\right ) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 43, normalized size = 0.80 \[ -\frac {8 \arcsinh \left (\frac {3 x}{2}+1\right )}{3}+\frac {\left (9 x^{2}+12 x +8\right )^{\frac {3}{2}}}{9}-\frac {\left (18 x +12\right ) \sqrt {9 x^{2}+12 x +8}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.87, size = 52, normalized size = 0.96 \[ \frac {1}{9} \, {\left (9 \, x^{2} + 12 \, x + 8\right )}^{\frac {3}{2}} - 2 \, \sqrt {9 \, x^{2} + 12 \, x + 8} x - \frac {4}{3} \, \sqrt {9 \, x^{2} + 12 \, x + 8} - \frac {8}{3} \, \operatorname {arsinh}\left (\frac {3}{2} \, x + 1\right ) \]

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
)

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 84, normalized size = 1.56 \[ \frac {\sqrt {9\,x^2+12\,x+8}\,\left (648\,x^2+216\,x+144\right )}{648}-\frac {4\,\ln \left (x+\frac {\sqrt {9\,x^2+12\,x+8}}{3}+\frac {2}{3}\right )}{3}-2\,\left (\frac {x}{2}+\frac {1}{3}\right )\,\sqrt {9\,x^2+12\,x+8}-\frac {4\,\ln \left (3\,x+\sqrt {9\,x^2+12\,x+8}+2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12*x + 9*x^2 + 8)^(1/2)*(216*x + 648*x^2 + 144))/648 - (4*log(x + (12*x + 9*x^2 + 8)^(1/2)/3 + 2/3))/3 - 2*(
x/2 + 1/3)*(12*x + 9*x^2 + 8)^(1/2) - (4*log(3*x + (12*x + 9*x^2 + 8)^(1/2) + 2))/3

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (3 x - 2\right ) \sqrt {9 x^{2} + 12 x + 8}\, dx \]

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)

________________________________________________________________________________________