3.1605 \(\int (d+e x) \sqrt{9+12 x+4 x^2} \, dx\)

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 = 0.0446594, 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 \[ \frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]

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

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Rubi in Sympy [A]  time = 5.43339, size = 42, normalized size = 0.84 \[ \frac{e \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}}{12} + \left (\frac{d}{16} - \frac{3 e}{32}\right ) \left (8 x + 12\right ) \sqrt{4 x^{2} + 12 x + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(4*x**2+12*x+9)**(1/2),x)

[Out]

e*(4*x**2 + 12*x + 9)**(3/2)/12 + (d/16 - 3*e/32)*(8*x + 12)*sqrt(4*x**2 + 12*x
+ 9)

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Mathematica [A]  time = 0.0244496, 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 = 0.004, size = 38, normalized size = 0.8 \[{\frac{x \left ( 4\,e{x}^{2}+6\,dx+9\,ex+18\,d \right ) }{12\,x+18}\sqrt{ \left ( 2\,x+3 \right ) ^{2}}} \]

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)*((2*x+3)^2)^(1/2)/(2*x+3)

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Maxima [A]  time = 0.823591, size = 105, normalized size = 2.1 \[ \frac{1}{12} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e + \frac{1}{2} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d x - \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e x + \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d - \frac{9}{8} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),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 = 0.199714, size = 31, normalized size = 0.62 \[ \frac{2}{3} \, e x^{3} + \frac{1}{2} \,{\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),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. \[ \int \left (d + e x\right ) \sqrt{\left (2 x + 3\right )^{2}}\, dx \]

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/XCAS [A]  time = 0.218515, size = 86, normalized size = 1.72 \[ \frac{2}{3} \, x^{3} e{\rm sign}\left (2 \, x + 3\right ) + d x^{2}{\rm sign}\left (2 \, x + 3\right ) + \frac{3}{2} \, x^{2} e{\rm sign}\left (2 \, x + 3\right ) + 3 \, d x{\rm sign}\left (2 \, x + 3\right ) + \frac{9}{8} \,{\left (2 \, d - e\right )}{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),x, algorithm="giac")

[Out]

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