3.3.14 \(\int \frac {x}{(9+12 x+4 x^2)^{7/2}} \, dx\) [214]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {654, 621} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(9 + 12*x + 4*x^2)^(7/2),x]

[Out]

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

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx &=-\frac {1}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac {3}{2} \int \frac {1}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx\\ &=-\frac {1}{20 \left (9+12 x+4 x^2\right )^{5/2}}+\frac {1}{8 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.61 \begin {gather*} \frac {-1-4 x}{40 (3+2 x)^5 \sqrt {(3+2 x)^2}} \end {gather*}

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 = 0.47, size = 22, normalized size = 0.50

method	result	size
gosper	\(-\frac {\left (2 x +3\right ) \left (1+4 x \right )}{40 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}}\)	\(22\)
default	\(-\frac {\left (2 x +3\right ) \left (1+4 x \right )}{40 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}}\)	\(22\)
risch	\(\frac {64 \sqrt {\left (2 x +3\right )^{2}}\, \left (-\frac {x}{640}-\frac {1}{2560}\right )}{\left (2 x +3\right )^{7}}\)	\(24\)
meijerg	\(\frac {x^{2} \left (\frac {16}{81} x^{4}+\frac {16}{9} x^{3}+\frac {20}{3} x^{2}+\frac {40}{3} x +15\right )}{65610 \left (1+\frac {2 x}{3}\right )^{6}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x^2+12*x+9)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 24, normalized size = 0.55 \begin {gather*} -\frac {1}{20 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}}} + \frac {1}{8 \, {\left (2 \, x + 3\right )}^{6}} \end {gather*}

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 = 1.79, size = 39, normalized size = 0.89 \begin {gather*} -\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 )}} \end {gather*}

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 + 729)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {7}{2}}}\, dx \end {gather*}

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 [A]
time = 0.74, size = 22, normalized size = 0.50 \begin {gather*} -\frac {4 \, x + 1}{40 \, {\left (2 \, x + 3\right )}^{6} \mathrm {sgn}\left (2 \, x + 3\right )} \end {gather*}

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]

-1/40*(4*x + 1)/((2*x + 3)^6*sgn(2*x + 3))

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Mupad [B]
time = 0.19, size = 26, normalized size = 0.59 \begin {gather*} -\frac {\left (4\,x+1\right )\,\sqrt {4\,x^2+12\,x+9}}{40\,{\left (2\,x+3\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(12*x + 4*x^2 + 9)^(7/2),x)

[Out]

-((4*x + 1)*(12*x + 4*x^2 + 9)^(1/2))/(40*(2*x + 3)^7)

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