3.1620 \(\int \frac {d+e x}{(9+12 x+4 x^2)^{7/2}} \, dx\)

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]

-1/20*e/(4*x^2+12*x+9)^(5/2)+1/24*(-2*d+3*e)/(3+2*x)/(4*x^2+12*x+9)^(5/2)

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Rubi [A]  time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {640, 607} \[ -\frac {2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac {e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]

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))

Rule 607

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 640

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 {d+e x}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx &=-\frac {e}{20 \left (9+12 x+4 x^2\right )^{5/2}}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx\\ &=-\frac {e}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac {2 d-3 e}{24 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.73, size = 45, normalized size = 0.87 \[ -\frac {12 \, e x + 10 \, d + 3 \, e}{120 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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

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maple [A]  time = 0.04, size = 28, normalized size = 0.54 \[ -\frac {\left (2 x +3\right ) \left (12 e x +10 d +3 e \right )}{120 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}} \]

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 = 2.35, size = 36, normalized size = 0.69 \[ -\frac {e}{20 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}}} - \frac {d}{12 \, {\left (2 \, x + 3\right )}^{6}} + \frac {e}{8 \, {\left (2 \, x + 3\right )}^{6}} \]

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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mupad [B]  time = 0.58, size = 32, normalized size = 0.62 \[ -\frac {\left (10\,d+3\,e+12\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{120\,{\left (2\,x+3\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((10*d + 3*e + 12*e*x)*(12*x + 4*x^2 + 9)^(1/2))/(120*(2*x + 3)^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {7}{2}}}\, dx \]

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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