3.1609 \(\int \frac{d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

[Out]

-e/(20*(9 + 12*x + 4*x^2)^(5/2)) - (2*d - 3*e)/(24*(3 + 2*x)*(9 + 12*x + 4*x^2)^
(5/2))

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Rubi [A]  time = 0.0478515, 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 \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]

[Out]

-e/(20*(9 + 12*x + 4*x^2)^(5/2)) - (2*d - 3*e)/(24*(3 + 2*x)*(9 + 12*x + 4*x^2)^
(5/2))

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Rubi in Sympy [A]  time = 5.41348, size = 42, normalized size = 0.81 \[ - \frac{e}{20 \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}} - \frac{\left (\frac{d}{48} - \frac{e}{32}\right ) \left (8 x + 12\right )}{\left (4 x^{2} + 12 x + 9\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(7/2),x)

[Out]

-e/(20*(4*x**2 + 12*x + 9)**(5/2)) - (d/48 - e/32)*(8*x + 12)/(4*x**2 + 12*x + 9
)**(7/2)

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Mathematica [A]  time = 0.031262, size = 34, normalized size = 0.65 \[ \frac{-10 d-3 (4 e x+e)}{120 (2 x+3)^5 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]

[Out]

(-10*d - 3*(e + 4*e*x))/(120*(3 + 2*x)^5*Sqrt[(3 + 2*x)^2])

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Maple [A]  time = 0.007, size = 28, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 12\,ex+10\,d+3\,e \right ) }{120} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^(7/2),x)

[Out]

-1/120*(2*x+3)*(12*e*x+10*d+3*e)/((2*x+3)^2)^(7/2)

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Maxima [A]  time = 0.828535, size = 49, normalized size = 0.94 \[ -\frac{e}{20 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}}} - \frac{d}{12 \,{\left (2 \, x + 3\right )}^{6}} + \frac{e}{8 \,{\left (2 \, x + 3\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(7/2),x, algorithm="maxima")

[Out]

-1/20*e/(4*x^2 + 12*x + 9)^(5/2) - 1/12*d/(2*x + 3)^6 + 1/8*e/(2*x + 3)^6

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Fricas [A]  time = 0.200197, size = 61, normalized size = 1.17 \[ -\frac{12 \, e x + 10 \, d + 3 \, e}{120 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^(7/2),x, algorithm="fricas")

[Out]

-1/120*(12*e*x + 10*d + 3*e)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2
+ 2916*x + 729)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**(7/2),x)

[Out]

Integral((d + e*x)/((2*x + 3)**2)**(7/2), x)

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GIAC/XCAS [A]  time = 0.600044, 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)^(7/2),x, algorithm="giac")

[Out]

sage0*x