3.88.2 \(\int \frac {36+18 x+3 x^2+e^{2 x} (18+12 x+2 x^2)}{9+6 x+x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{2 x}+4 x-\frac {x^2}{3+x} \]

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Rubi [A]  time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {27, 6688, 2194, 683} \begin {gather*} 3 x+e^{2 x}-\frac {9}{x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + 18*x + 3*x^2 + E^(2*x)*(18 + 12*x + 2*x^2))/(9 + 6*x + x^2),x]

[Out]

E^(2*x) + 3*x - 9/(3 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36+18 x+3 x^2+e^{2 x} \left (18+12 x+2 x^2\right )}{(3+x)^2} \, dx\\ &=\int \left (2 e^{2 x}+\frac {3 \left (12+6 x+x^2\right )}{(3+x)^2}\right ) \, dx\\ &=2 \int e^{2 x} \, dx+3 \int \frac {12+6 x+x^2}{(3+x)^2} \, dx\\ &=e^{2 x}+3 \int \left (1+\frac {3}{(3+x)^2}\right ) \, dx\\ &=e^{2 x}+3 x-\frac {9}{3+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.84 \begin {gather*} e^{2 x}+3 x-\frac {9}{3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + 18*x + 3*x^2 + E^(2*x)*(18 + 12*x + 2*x^2))/(9 + 6*x + x^2),x]

[Out]

E^(2*x) + 3*x - 9/(3 + x)

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fricas [A]  time = 0.54, size = 24, normalized size = 1.26 \begin {gather*} \frac {3 \, x^{2} + {\left (x + 3\right )} e^{\left (2 \, x\right )} + 9 \, x - 9}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+12*x+18)*exp(x)^2+3*x^2+18*x+36)/(x^2+6*x+9),x, algorithm="fricas")

[Out]

(3*x^2 + (x + 3)*e^(2*x) + 9*x - 9)/(x + 3)

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giac [A]  time = 0.13, size = 28, normalized size = 1.47 \begin {gather*} \frac {3 \, x^{2} + x e^{\left (2 \, x\right )} + 9 \, x + 3 \, e^{\left (2 \, x\right )} - 9}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+12*x+18)*exp(x)^2+3*x^2+18*x+36)/(x^2+6*x+9),x, algorithm="giac")

[Out]

(3*x^2 + x*e^(2*x) + 9*x + 3*e^(2*x) - 9)/(x + 3)

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maple [A]  time = 0.45, size = 16, normalized size = 0.84

method	result	size
default	\(-\frac {9}{3+x}+3 x +{\mathrm e}^{2 x}\)	\(16\)
risch	\(-\frac {9}{3+x}+3 x +{\mathrm e}^{2 x}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{2 x}+3 x^{2}+3 \,{\mathrm e}^{2 x}-36}{3+x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+12*x+18)*exp(x)^2+3*x^2+18*x+36)/(x^2+6*x+9),x,method=_RETURNVERBOSE)

[Out]

-9/(3+x)+3*x+exp(x)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 3 \, x + \frac {{\left (x^{2} + 6 \, x\right )} e^{\left (2 \, x\right )}}{x^{2} + 6 \, x + 9} - \frac {18 \, e^{\left (-6\right )} E_{2}\left (-2 \, x - 6\right )}{x + 3} - \frac {9}{x + 3} - 18 \, \int \frac {e^{\left (2 \, x\right )}}{x^{3} + 9 \, x^{2} + 27 \, x + 27}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+12*x+18)*exp(x)^2+3*x^2+18*x+36)/(x^2+6*x+9),x, algorithm="maxima")

[Out]

3*x + (x^2 + 6*x)*e^(2*x)/(x^2 + 6*x + 9) - 18*e^(-6)*exp_integral_e(2, -2*x - 6)/(x + 3) - 9/(x + 3) - 18*int
egrate(e^(2*x)/(x^3 + 9*x^2 + 27*x + 27), x)

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mupad [B]  time = 0.10, size = 15, normalized size = 0.79 \begin {gather*} 3\,x+{\mathrm {e}}^{2\,x}-\frac {9}{x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + exp(2*x)*(12*x + 2*x^2 + 18) + 3*x^2 + 36)/(6*x + x^2 + 9),x)

[Out]

3*x + exp(2*x) - 9/(x + 3)

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sympy [A]  time = 0.12, size = 12, normalized size = 0.63 \begin {gather*} 3 x + e^{2 x} - \frac {9}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+12*x+18)*exp(x)**2+3*x**2+18*x+36)/(x**2+6*x+9),x)

[Out]

3*x + exp(2*x) - 9/(x + 3)

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