3.85.48 $\int \frac{e^{-x}(-1536-288x+e^x(-507-234x-27x^2))}{169+78x+9x^2}dx$

Optimal. Leaf size=31 $$3e^{-x}(\frac{8}{4-\frac{3(1-x)}{4}}-e^x(3+x))$$

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Rubi [A]  time = 0.22, antiderivative size = 18, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 3, integrand size = 37, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.081, Rules used = {27, 6742, 2197} $$\begin{array}{c}\frac{96e^{-x}}{3x+13}-3x\end{array}$$

Antiderivative was successfully verified.

[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 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 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{-x}(-1536-288x+e^x(-507-234x-27x^2))}{(13+3x{)}^2}dx\\ & =\int (-3-\frac{96e^{-x}(16+3x)}{(13+3x{)}^2})dx\\ & =-3x-96\int \frac{e^{-x}(16+3x)}{(13+3x{)}^2}dx\\ & =-3x+\frac{96e^{-x}}{13+3x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.10, size = 18, normalized size = 0.58 $$\begin{array}{c}-3(x-\frac{32e^{-x}}{13+3x})\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

-3*(x - 32/(E^x*(13 + 3*x)))

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fricas [A]  time = 0.94, size = 27, normalized size = 0.87 $$\begin{array}{c}-\frac{3((3x^2+13x)e^x-32)e^{(-x)}}{3x+13}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.14, size = 24, normalized size = 0.77 $$\begin{array}{c}-\frac{3(3x^2+13x-32e^{(-x)})}{3x+13}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.42, size = 18, normalized size = 0.58

method	result	size
default	$-3x+\frac{96{\mathrm{e}}^{-x}}{3x+13}$	$18$
risch	$-3x+\frac{96{\mathrm{e}}^{-x}}{3x+13}$	$18$
norman	$\frac{(96+169{\mathrm{e}}^x-9{\mathrm{e}}^x{x}^2){\mathrm{e}}^{-x}}{3x+13}$	$26$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\frac{512e^{\frac{13}{3}}{E}_2(x+\frac{13}{3})}{3x+13}-\frac{3(9x^3+78x^2-96x{e}^{(-x)}+169x)}{9x^2+78x+169}+3\int \frac{96(3x-13)e^{(-x)}}{27x^3+351x^2+1521x+2197}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.11, size = 18, normalized size = 0.58 $$\begin{array}{c}\frac{96}{13{\mathrm{e}}^x+3x{\mathrm{e}}^x}-3x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.12, size = 12, normalized size = 0.39 $$\begin{array}{c}-3x+\frac{96e^{-x}}{3x+13}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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