\(\int \frac {e^{-x} (-1536-288 x+e^x (-507-234 x-27 x^2))}{169+78 x+9 x^2} \, dx\) [8448]

Optimal result

Integrand size = 37, antiderivative size = 31 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=3 e^{-x} \left (\frac {8}{4-\frac {3 (1-x)}{4}}-e^x (3+x)\right ) \]

[Out]

3*(2/(3/16*x+13/16)-(3+x)*exp(x))/exp(x)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {27, 6874, 2228} \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=\frac {96 e^{-x}}{3 x+13}-3 x \]

[In]

Int[(-1536 - 288*x + E^x*(-507 - 234*x - 27*x^2))/(E^x*(169 + 78*x + 9*x^2)),x]

[Out]

-3*x + 96/(E^x*(13 + 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 2228

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{(13+3 x)^2} \, dx \\ & = \int \left (-3-\frac {96 e^{-x} (16+3 x)}{(13+3 x)^2}\right ) \, dx \\ & = -3 x-96 \int \frac {e^{-x} (16+3 x)}{(13+3 x)^2} \, dx \\ & = -3 x+\frac {96 e^{-x}}{13+3 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=-\frac {3 \left (-96 e^{-x}+(13+3 x)^2\right )}{39+9 x} \]

[In]

Integrate[(-1536 - 288*x + E^x*(-507 - 234*x - 27*x^2))/(E^x*(169 + 78*x + 9*x^2)),x]

[Out]

(-3*(-96/E^x + (13 + 3*x)^2))/(39 + 9*x)

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58

method	result	size
default	\(-3 x +\frac {96 \,{\mathrm e}^{-x}}{3 x +13}\)	\(18\)
risch	\(-3 x +\frac {96 \,{\mathrm e}^{-x}}{3 x +13}\)	\(18\)
parts	\(-3 x +\frac {96 \,{\mathrm e}^{-x}}{3 x +13}\)	\(18\)
norman	\(\frac {\left (96+169 \,{\mathrm e}^{x}-9 \,{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}}{3 x +13}\)	\(26\)
parallelrisch	\(\frac {\left (288-27 \,{\mathrm e}^{x} x^{2}+507 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{9 x +39}\)	\(27\)

[In]

int(((-27*x^2-234*x-507)*exp(x)-288*x-1536)/(9*x^2+78*x+169)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-3*x+96*exp(-x)/(3*x+13)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=-\frac {3 \, {\left ({\left (3 \, x^{2} + 13 \, x\right )} e^{x} - 32\right )} e^{\left (-x\right )}}{3 \, x + 13} \]

[In]

integrate(((-27*x^2-234*x-507)*exp(x)-288*x-1536)/(9*x^2+78*x+169)/exp(x),x, algorithm="fricas")

[Out]

-3*((3*x^2 + 13*x)*e^x - 32)*e^(-x)/(3*x + 13)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.39 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=- 3 x + \frac {96 e^{- x}}{3 x + 13} \]

[In]

integrate(((-27*x**2-234*x-507)*exp(x)-288*x-1536)/(9*x**2+78*x+169)/exp(x),x)

[Out]

-3*x + 96*exp(-x)/(3*x + 13)

Maxima [F]

\[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=\int { -\frac {3 \, {\left ({\left (9 \, x^{2} + 78 \, x + 169\right )} e^{x} + 96 \, x + 512\right )} e^{\left (-x\right )}}{9 \, x^{2} + 78 \, x + 169} \,d x } \]

[In]

integrate(((-27*x^2-234*x-507)*exp(x)-288*x-1536)/(9*x^2+78*x+169)/exp(x),x, algorithm="maxima")

[Out]

512*e^(13/3)*exp_integral_e(2, x + 13/3)/(3*x + 13) - 3*(9*x^3 + 78*x^2 - 96*x*e^(-x) + 169*x)/(9*x^2 + 78*x +
169) + 3*integrate(96*(3*x - 13)*e^(-x)/(27*x^3 + 351*x^2 + 1521*x + 2197), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=-\frac {3 \, {\left (3 \, x^{2} + 13 \, x - 32 \, e^{\left (-x\right )}\right )}}{3 \, x + 13} \]

[In]

integrate(((-27*x^2-234*x-507)*exp(x)-288*x-1536)/(9*x^2+78*x+169)/exp(x),x, algorithm="giac")

[Out]

-3*(3*x^2 + 13*x - 32*e^(-x))/(3*x + 13)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-x} \left (-1536-288 x+e^x \left (-507-234 x-27 x^2\right )\right )}{169+78 x+9 x^2} \, dx=\frac {96}{13\,{\mathrm {e}}^x+3\,x\,{\mathrm {e}}^x}-3\,x \]

[In]

int(-(exp(-x)*(288*x + exp(x)*(234*x + 27*x^2 + 507) + 1536))/(78*x + 9*x^2 + 169),x)

[Out]

96/(13*exp(x) + 3*x*exp(x)) - 3*x