3.2356 \(\int (-2+3 x) \sqrt{8+12 x+9 x^2} \, dx\)

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*ArcSin
h[1 + (3*x)/2])/3

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Rubi [A]  time = 0.0516609, 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 \[ \frac{1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{3 x}{2}+1\right ) \]

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*ArcSin
h[1 + (3*x)/2])/3

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Rubi in Sympy [A]  time = 6.04017, size = 60, normalized size = 1.11 \[ - \frac{\left (18 x + 12\right ) \sqrt{9 x^{2} + 12 x + 8}}{9} + \frac{\left (9 x^{2} + 12 x + 8\right )^{\frac{3}{2}}}{9} - \frac{8 \operatorname{atanh}{\left (\frac{18 x + 12}{6 \sqrt{9 x^{2} + 12 x + 8}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2+3*x)*(9*x**2+12*x+8)**(1/2),x)

[Out]

-(18*x + 12)*sqrt(9*x**2 + 12*x + 8)/9 + (9*x**2 + 12*x + 8)**(3/2)/9 - 8*atanh(
(18*x + 12)/(6*sqrt(9*x**2 + 12*x + 8)))/3

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Mathematica [A]  time = 0.0319698, size = 42, normalized size = 0.78 \[ \left (x^2-\frac{2 x}{3}-\frac{4}{9}\right ) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 43, normalized size = 0.8 \[ -{\frac{18\,x+12}{9}\sqrt{9\,{x}^{2}+12\,x+8}}-{\frac{8}{3}{\it Arcsinh} \left ( 1+{\frac{3\,x}{2}} \right ) }+{\frac{1}{9} \left ( 9\,{x}^{2}+12\,x+8 \right ) ^{{\frac{3}{2}}}} \]

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 = 0.749989, size = 70, normalized size = 1.3 \[ \frac{1}{9} \,{\left (9 \, x^{2} + 12 \, x + 8\right )}^{\frac{3}{2}} - 2 \, \sqrt{9 \, x^{2} + 12 \, x + 8} x - \frac{4}{3} \, \sqrt{9 \, x^{2} + 12 \, x + 8} - \frac{8}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 8)*(3*x - 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 = 0.225328, size = 231, normalized size = 4.28 \[ -\frac{729 \, x^{6} + 1458 \, x^{5} + 729 \, x^{4} - 972 \, x^{3} - 1512 \, x^{2} - 24 \,{\left (27 \, x^{3} + 54 \, x^{2} - \sqrt{9 \, x^{2} + 12 \, x + 8}{\left (9 \, x^{2} + 12 \, x + 5\right )} + 45 \, x + 14\right )} \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\right ) -{\left (243 \, x^{5} + 324 \, x^{4} - 27 \, x^{3} - 342 \, x^{2} - 240 \, x - 46\right )} \sqrt{9 \, x^{2} + 12 \, x + 8} - 774 \, x - 132}{9 \,{\left (27 \, x^{3} + 54 \, x^{2} - \sqrt{9 \, x^{2} + 12 \, x + 8}{\left (9 \, x^{2} + 12 \, x + 5\right )} + 45 \, x + 14\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 8)*(3*x - 2),x, algorithm="fricas")

[Out]

-1/9*(729*x^6 + 1458*x^5 + 729*x^4 - 972*x^3 - 1512*x^2 - 24*(27*x^3 + 54*x^2 -
sqrt(9*x^2 + 12*x + 8)*(9*x^2 + 12*x + 5) + 45*x + 14)*log(-3*x + sqrt(9*x^2 + 1
2*x + 8) - 2) - (243*x^5 + 324*x^4 - 27*x^3 - 342*x^2 - 240*x - 46)*sqrt(9*x^2 +
12*x + 8) - 774*x - 132)/(27*x^3 + 54*x^2 - sqrt(9*x^2 + 12*x + 8)*(9*x^2 + 12*
x + 5) + 45*x + 14)

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

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)

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GIAC/XCAS [A]  time = 0.206737, size = 61, normalized size = 1.13 \[ \frac{1}{9} \,{\left (3 \,{\left (3 \, x - 2\right )} x - 4\right )} \sqrt{9 \, x^{2} + 12 \, x + 8} + \frac{8}{3} \,{\rm ln}\left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 12*x + 8)*(3*x - 2),x, algorithm="giac")

[Out]

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