3.14.20 \(\int \frac {d+e x}{(9+12 x+4 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac {2 d-3 e}{8 (2 x+3) \sqrt {4 x^2+12 x+9}}-\frac {e}{4 \sqrt {4 x^2+12 x+9}} \]

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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} \begin {gather*} -\frac {2 d-3 e}{8 (2 x+3) \sqrt {4 x^2+12 x+9}}-\frac {e}{4 \sqrt {4 x^2+12 x+9}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

-e/(4*Sqrt[9 + 12*x + 4*x^2]) - (2*d - 3*e)/(8*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^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 )^{3/2}} \, dx &=-\frac {e}{4 \sqrt {9+12 x+4 x^2}}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx\\ &=-\frac {e}{4 \sqrt {9+12 x+4 x^2}}-\frac {2 d-3 e}{8 (3+2 x) \sqrt {9+12 x+4 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 0.65 \begin {gather*} \frac {-2 d-e (4 x+3)}{8 (2 x+3) \sqrt {(2 x+3)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(-2*d - e*(3 + 4*x))/(8*(3 + 2*x)*Sqrt[(3 + 2*x)^2])

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IntegrateAlgebraic [A]  time = 0.34, size = 56, normalized size = 1.08 \begin {gather*} \frac {d-\frac {3 e}{2}}{\left (\sqrt {4 x^2+12 x+9}-2 x-3\right )^2}-\frac {e}{2 \left (\sqrt {4 x^2+12 x+9}-2 x-3\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

IntegrateAlgebraic[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(d - (3*e)/2)/(-3 - 2*x + Sqrt[9 + 12*x + 4*x^2])^2 - e/(2*(-3 - 2*x + Sqrt[9 + 12*x + 4*x^2]))

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fricas [A]  time = 0.41, size = 25, normalized size = 0.48 \begin {gather*} -\frac {4 \, e x + 2 \, d + 3 \, e}{8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.04, size = 28, normalized size = 0.54 \begin {gather*} -\frac {\left (2 x +3\right ) \left (4 e x +2 d +3 e \right )}{8 \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(4*x^2+12*x+9)^(3/2),x)

[Out]

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

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maxima [A]  time = 2.34, size = 36, normalized size = 0.69 \begin {gather*} -\frac {e}{4 \, \sqrt {4 \, x^{2} + 12 \, x + 9}} - \frac {d}{4 \, {\left (2 \, x + 3\right )}^{2}} + \frac {3 \, e}{8 \, {\left (2 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 32, normalized size = 0.62 \begin {gather*} -\frac {\left (2\,d+3\,e+4\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{8\,{\left (2\,x+3\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(12*x + 4*x^2 + 9)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)

[Out]

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

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