3.1608 \(\int \frac{d+e x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{e}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]

[Out]

-e/(12*(9 + 12*x + 4*x^2)^(3/2)) - (2*d - 3*e)/(16*(3 + 2*x)*(9 + 12*x + 4*x^2)^
(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0459352, 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 \[ -\frac{2 d-3 e}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{e}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

-e/(12*(9 + 12*x + 4*x^2)^(3/2)) - (2*d - 3*e)/(16*(3 + 2*x)*(9 + 12*x + 4*x^2)^
(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.44487, size = 44, normalized size = 0.85 \[ - \frac{e}{12 \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}} - \frac{\left (\frac{d}{32} - \frac{3 e}{64}\right ) \left (8 x + 12\right )}{\left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(5/2),x)

[Out]

-e/(12*(4*x**2 + 12*x + 9)**(3/2)) - (d/32 - 3*e/64)*(8*x + 12)/(4*x**2 + 12*x +
9)**(5/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0294397, size = 34, normalized size = 0.65 \[ \frac{-6 d-e (8 x+3)}{48 (2 x+3)^3 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

(-6*d - e*(3 + 8*x))/(48*(3 + 2*x)^3*Sqrt[(3 + 2*x)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 28, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 8\,ex+6\,d+3\,e \right ) }{48} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^(5/2),x)

[Out]

-1/48*(2*x+3)*(8*e*x+6*d+3*e)/((2*x+3)^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.830398, size = 49, normalized size = 0.94 \[ -\frac{e}{12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}}} - \frac{d}{8 \,{\left (2 \, x + 3\right )}^{4}} + \frac{3 \, e}{16 \,{\left (2 \, x + 3\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(5/2),x, algorithm="maxima")

[Out]

-1/12*e/(4*x^2 + 12*x + 9)^(3/2) - 1/8*d/(2*x + 3)^4 + 3/16*e/(2*x + 3)^4

_______________________________________________________________________________________

Fricas [A]  time = 0.202298, size = 47, normalized size = 0.9 \[ -\frac{8 \, e x + 6 \, d + 3 \, e}{48 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(5/2),x, algorithm="fricas")

[Out]

-1/48*(8*e*x + 6*d + 3*e)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**(5/2),x)

[Out]

Integral((d + e*x)/((2*x + 3)**2)**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.611408, 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)^(5/2),x, algorithm="giac")

[Out]

sage0*x