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

Optimal. Leaf size=44 \[ \frac{1}{8 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{1}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0319145, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{8 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{1}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.72289, size = 36, normalized size = 0.82 \[ \frac{8 x + 12}{32 \left (4 x^{2} + 12 x + 9\right )^{\frac{7}{2}}} - \frac{1}{20 \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(8*x + 12)/(32*(4*x**2 + 12*x + 9)**(7/2)) - 1/(20*(4*x**2 + 12*x + 9)**(5/2))

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Mathematica [A]  time = 0.0156324, size = 27, normalized size = 0.61 \[ \frac{-4 x-1}{40 (2 x+3)^5 \sqrt{(2 x+3)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-1 - 4*x)/(40*(3 + 2*x)^5*Sqrt[(3 + 2*x)^2])

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Maple [A]  time = 0.007, size = 22, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 1+4\,x \right ) }{40} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(4*x^2+12*x+9)^(7/2),x)

[Out]

-1/40*(2*x+3)*(1+4*x)/((2*x+3)^2)^(7/2)

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Maxima [A]  time = 0.769815, size = 32, normalized size = 0.73 \[ -\frac{1}{20 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}}} + \frac{1}{8 \,{\left (2 \, x + 3\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217465, size = 53, normalized size = 1.2 \[ -\frac{4 \, x + 1}{40 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(4*x + 1)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 72
9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((2*x + 3)**2)**(7/2), x)

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GIAC/XCAS [A]  time = 0.562673, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x