3.13.100 \(\int \frac {12+12 x+3 x^2+e^{15+6 x} (44+24 x)}{12+12 x+3 x^2} \, dx\)

Optimal. Leaf size=23 \[ x+\frac {4 e^{5-5 (-2-x)+x}}{3 (2+x)} \]

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Rubi [A]  time = 0.13, antiderivative size = 18, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 12, 6688, 2197} \begin {gather*} x+\frac {4 e^{6 x+15}}{3 (x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 + 12*x + 3*x^2 + E^(15 + 6*x)*(44 + 24*x))/(12 + 12*x + 3*x^2),x]

[Out]

x + (4*E^(15 + 6*x))/(3*(2 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, 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 {12+12 x+3 x^2+e^{15+6 x} (44+24 x)}{3 (2+x)^2} \, dx\\ &=\frac {1}{3} \int \frac {12+12 x+3 x^2+e^{15+6 x} (44+24 x)}{(2+x)^2} \, dx\\ &=\frac {1}{3} \int \left (3+\frac {4 e^{15+6 x} (11+6 x)}{(2+x)^2}\right ) \, dx\\ &=x+\frac {4}{3} \int \frac {e^{15+6 x} (11+6 x)}{(2+x)^2} \, dx\\ &=x+\frac {4 e^{15+6 x}}{3 (2+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 18, normalized size = 0.78 \begin {gather*} x+\frac {4 e^{15+6 x}}{3 (2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + 12*x + 3*x^2 + E^(15 + 6*x)*(44 + 24*x))/(12 + 12*x + 3*x^2),x]

[Out]

x + (4*E^(15 + 6*x))/(3*(2 + x))

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fricas [A]  time = 0.72, size = 24, normalized size = 1.04 \begin {gather*} \frac {3 \, x^{2} + 6 \, x + 4 \, e^{\left (6 \, x + 15\right )}}{3 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+44)*exp(6*x+15)+3*x^2+12*x+12)/(3*x^2+12*x+12),x, algorithm="fricas")

[Out]

1/3*(3*x^2 + 6*x + 4*e^(6*x + 15))/(x + 2)

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giac [A]  time = 0.27, size = 24, normalized size = 1.04 \begin {gather*} \frac {3 \, x^{2} + 6 \, x + 4 \, e^{\left (6 \, x + 15\right )}}{3 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+44)*exp(6*x+15)+3*x^2+12*x+12)/(3*x^2+12*x+12),x, algorithm="giac")

[Out]

1/3*(3*x^2 + 6*x + 4*e^(6*x + 15))/(x + 2)

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maple [A]  time = 0.25, size = 16, normalized size = 0.70

method	result	size
risch	\(x +\frac {4 \,{\mathrm e}^{6 x +15}}{3 \left (2+x \right )}\)	\(16\)
derivativedivides	\(x +\frac {5}{2}+\frac {8 \,{\mathrm e}^{6 x +15}}{6 x +12}\)	\(19\)
default	\(x +\frac {5}{2}+\frac {8 \,{\mathrm e}^{6 x +15}}{6 x +12}\)	\(19\)
norman	\(\frac {x^{2}+\frac {4 \,{\mathrm e}^{6 x +15}}{3}-4}{2+x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+44)*exp(6*x+15)+3*x^2+12*x+12)/(3*x^2+12*x+12),x,method=_RETURNVERBOSE)

[Out]

x+4/3*exp(6*x+15)/(2+x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x + \frac {4 \, x e^{\left (6 \, x + 15\right )}}{3 \, {\left (x^{2} + 4 \, x + 4\right )}} - \frac {44 \, e^{3} E_{2}\left (-6 \, x - 12\right )}{3 \, {\left (x + 2\right )}} + 8 \, \int \frac {{\left (x e^{15} - 2 \, e^{15}\right )} e^{\left (6 \, x\right )}}{6 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+44)*exp(6*x+15)+3*x^2+12*x+12)/(3*x^2+12*x+12),x, algorithm="maxima")

[Out]

x + 4/3*x*e^(6*x + 15)/(x^2 + 4*x + 4) - 44/3*e^3*exp_integral_e(2, -6*x - 12)/(x + 2) + 8*integrate(1/6*(x*e^
15 - 2*e^15)*e^(6*x)/(x^3 + 6*x^2 + 12*x + 8), x)

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mupad [B]  time = 0.97, size = 17, normalized size = 0.74 \begin {gather*} x+\frac {4\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{15}}{3\,\left (x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + exp(6*x + 15)*(24*x + 44) + 3*x^2 + 12)/(12*x + 3*x^2 + 12),x)

[Out]

x + (4*exp(6*x)*exp(15))/(3*(x + 2))

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sympy [A]  time = 0.10, size = 14, normalized size = 0.61 \begin {gather*} x + \frac {4 e^{6 x + 15}}{3 x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+44)*exp(6*x+15)+3*x**2+12*x+12)/(3*x**2+12*x+12),x)

[Out]

x + 4*exp(6*x + 15)/(3*x + 6)

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