3.1607 \(\int \frac{d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]

[Out]

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

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Rubi [A]  time = 0.0473908, antiderivative size = 52, 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 \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.29605, size = 34, normalized size = 0.65 \[ - \frac{\left (d + e x\right )^{2} \left (8 x + 12\right )}{8 \left (2 d - 3 e\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0201292, size = 34, normalized size = 0.65 \[ \frac{-2 d-e (4 x+3)}{8 (2 x+3) \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(-2*d - e*(3 + 4*x))/(8*(3 + 2*x)*Sqrt[(3 + 2*x)^2])

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Maple [A]  time = 0.006, size = 28, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 4\,ex+2\,d+3\,e \right ) }{8} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^(3/2),x)

[Out]

-1/8*(2*x+3)*(4*e*x+2*d+3*e)/((2*x+3)^2)^(3/2)

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Maxima [A]  time = 0.827836, size = 49, normalized size = 0.94 \[ -\frac{e}{4 \, \sqrt{4 \, x^{2} + 12 \, x + 9}} - \frac{d}{4 \,{\left (2 \, x + 3\right )}^{2}} + \frac{3 \, e}{8 \,{\left (2 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201526, size = 34, normalized size = 0.65 \[ -\frac{4 \, e x + 2 \, d + 3 \, e}{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="fricas")

[Out]

-1/8*(4*e*x + 2*d + 3*e)/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral((d + e*x)/((2*x + 3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.591578, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="giac")

[Out]

sage0*x