3.68.60 \(\int \frac {e^{-x} (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 (324-9 x-14 x^2-x^3))}{81+18 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{-x} (3-x) \left (-e^2-2 x+\frac {25}{9+x}\right ) \]

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Rubi [B]  time = 0.17, antiderivative size = 81, normalized size of antiderivative = 3.00, number of steps used = 12, number of rules used = 6, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {27, 2199, 2194, 2176, 2177, 2178} \begin {gather*} 2 e^{-x} x^2+4 e^{-x} x-\left (10-e^2\right ) e^{-x} x+4 e^{-x}+\frac {300 e^{-x}}{x+9}-\left (19+4 e^2\right ) e^{-x}-\left (10-e^2\right ) e^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1461 + 852*x + 37*x^2 - 26*x^3 - 2*x^4 + E^2*(324 - 9*x - 14*x^2 - x^3))/(E^x*(81 + 18*x + x^2)),x]

[Out]

4/E^x - (10 - E^2)/E^x - (19 + 4*E^2)/E^x + (4*x)/E^x - ((10 - E^2)*x)/E^x + (2*x^2)/E^x + 300/(E^x*(9 + 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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-1461+852 x+37 x^2-26 x^3-2 x^4+e^2 \left (324-9 x-14 x^2-x^3\right )\right )}{(9+x)^2} \, dx\\ &=\int \left (19 e^{-x} \left (1+\frac {4 e^2}{19}\right )-e^{-x} \left (-10+e^2\right ) x-2 e^{-x} x^2-\frac {300 e^{-x}}{(9+x)^2}-\frac {300 e^{-x}}{9+x}\right ) \, dx\\ &=-\left (2 \int e^{-x} x^2 \, dx\right )-300 \int \frac {e^{-x}}{(9+x)^2} \, dx-300 \int \frac {e^{-x}}{9+x} \, dx+\left (10-e^2\right ) \int e^{-x} x \, dx+\left (19+4 e^2\right ) \int e^{-x} \, dx\\ &=-e^{-x} \left (19+4 e^2\right )-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x}-300 e^9 \text {Ei}(-9-x)-4 \int e^{-x} x \, dx+300 \int \frac {e^{-x}}{9+x} \, dx+\left (10-e^2\right ) \int e^{-x} \, dx\\ &=-e^{-x} \left (10-e^2\right )-e^{-x} \left (19+4 e^2\right )+4 e^{-x} x-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x}-4 \int e^{-x} \, dx\\ &=4 e^{-x}-e^{-x} \left (10-e^2\right )-e^{-x} \left (19+4 e^2\right )+4 e^{-x} x-e^{-x} \left (10-e^2\right ) x+2 e^{-x} x^2+\frac {300 e^{-x}}{9+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 31, normalized size = 1.15 \begin {gather*} \frac {e^{-x} (-3+x) \left (-25+18 x+2 x^2+e^2 (9+x)\right )}{9+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1461 + 852*x + 37*x^2 - 26*x^3 - 2*x^4 + E^2*(324 - 9*x - 14*x^2 - x^3))/(E^x*(81 + 18*x + x^2)),x
]

[Out]

((-3 + x)*(-25 + 18*x + 2*x^2 + E^2*(9 + x)))/(E^x*(9 + x))

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fricas [A]  time = 0.57, size = 36, normalized size = 1.33 \begin {gather*} \frac {{\left (2 \, x^{3} + 12 \, x^{2} + {\left (x^{2} + 6 \, x - 27\right )} e^{2} - 79 \, x + 75\right )} e^{\left (-x\right )}}{x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-14*x^2-9*x+324)*exp(2)-2*x^4-26*x^3+37*x^2+852*x-1461)/(x^2+18*x+81)/exp(x),x, algorithm="fri
cas")

[Out]

(2*x^3 + 12*x^2 + (x^2 + 6*x - 27)*e^2 - 79*x + 75)*e^(-x)/(x + 9)

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giac [B]  time = 0.13, size = 65, normalized size = 2.41 \begin {gather*} \frac {2 \, x^{3} e^{\left (-x\right )} + 12 \, x^{2} e^{\left (-x\right )} + x^{2} e^{\left (-x + 2\right )} - 79 \, x e^{\left (-x\right )} + 6 \, x e^{\left (-x + 2\right )} + 75 \, e^{\left (-x\right )} - 27 \, e^{\left (-x + 2\right )}}{x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-14*x^2-9*x+324)*exp(2)-2*x^4-26*x^3+37*x^2+852*x-1461)/(x^2+18*x+81)/exp(x),x, algorithm="gia
c")

[Out]

(2*x^3*e^(-x) + 12*x^2*e^(-x) + x^2*e^(-x + 2) - 79*x*e^(-x) + 6*x*e^(-x + 2) + 75*e^(-x) - 27*e^(-x + 2))/(x
+ 9)

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maple [A]  time = 0.10, size = 38, normalized size = 1.41

method	result	size
norman	\(\frac {\left (\left ({\mathrm e}^{2}+12\right ) x^{2}+\left (6 \,{\mathrm e}^{2}-79\right ) x +2 x^{3}+75-27 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-x}}{x +9}\)	\(38\)
gosper	\(\frac {\left (x^{2} {\mathrm e}^{2}+2 x^{3}+6 \,{\mathrm e}^{2} x +12 x^{2}-27 \,{\mathrm e}^{2}-79 x +75\right ) {\mathrm e}^{-x}}{x +9}\)	\(41\)
risch	\(\frac {\left (x^{2} {\mathrm e}^{2}+2 x^{3}+6 \,{\mathrm e}^{2} x +12 x^{2}-27 \,{\mathrm e}^{2}-79 x +75\right ) {\mathrm e}^{-x}}{x +9}\)	\(41\)
default	\(\frac {300 \,{\mathrm e}^{-x}}{x +9}-37 \,{\mathrm e}^{-x}+26 \left (x -17\right ) {\mathrm e}^{-x}+2 \left (x^{2}-16 x +227\right ) {\mathrm e}^{-x}+324 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{-x}}{x +9}+{\mathrm e}^{9} \expIntegralEi \left (1, x +9\right )\right )-14 \,{\mathrm e}^{2} \left (-{\mathrm e}^{-x}-\frac {81 \,{\mathrm e}^{-x}}{x +9}+99 \,{\mathrm e}^{9} \expIntegralEi \left (1, x +9\right )\right )-{\mathrm e}^{2} \left (-\left (x -17\right ) {\mathrm e}^{-x}+\frac {729 \,{\mathrm e}^{-x}}{x +9}-972 \,{\mathrm e}^{9} \expIntegralEi \left (1, x +9\right )\right )-9 \,{\mathrm e}^{2} \left (\frac {9 \,{\mathrm e}^{-x}}{x +9}-10 \,{\mathrm e}^{9} \expIntegralEi \left (1, x +9\right )\right )\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-14*x^2-9*x+324)*exp(2)-2*x^4-26*x^3+37*x^2+852*x-1461)/(x^2+18*x+81)/exp(x),x,method=_RETURNVERBOSE
)

[Out]

((exp(2)+12)*x^2+(6*exp(2)-79)*x+2*x^3+75-27*exp(2))/(x+9)/exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {{\left (2 \, x^{4} + x^{3} {\left (e^{2} + 30\right )} + x^{2} {\left (15 \, e^{2} + 29\right )} + 3 \, x {\left (9 \, e^{2} - 212\right )}\right )} e^{\left (-x\right )}}{x^{2} + 18 \, x + 81} - \frac {324 \, e^{11} E_{2}\left (x + 9\right )}{x + 9} + \frac {1461 \, e^{9} E_{2}\left (x + 9\right )}{x + 9} - \int \frac {3 \, {\left (x {\left (27 \, e^{2} - 262\right )} + 81 \, e^{2} - 1908\right )} e^{\left (-x\right )}}{x^{3} + 27 \, x^{2} + 243 \, x + 729}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-14*x^2-9*x+324)*exp(2)-2*x^4-26*x^3+37*x^2+852*x-1461)/(x^2+18*x+81)/exp(x),x, algorithm="max
ima")

[Out]

(2*x^4 + x^3*(e^2 + 30) + x^2*(15*e^2 + 29) + 3*x*(9*e^2 - 212))*e^(-x)/(x^2 + 18*x + 81) - 324*e^11*exp_integ
ral_e(2, x + 9)/(x + 9) + 1461*e^9*exp_integral_e(2, x + 9)/(x + 9) - integrate(3*(x*(27*e^2 - 262) + 81*e^2 -
1908)*e^(-x)/(x^3 + 27*x^2 + 243*x + 729), x)

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mupad [B]  time = 0.12, size = 31, normalized size = 1.15 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,\left (x-3\right )\,\left (18\,x+9\,{\mathrm {e}}^2+x\,{\mathrm {e}}^2+2\,x^2-25\right )}{x+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(exp(2)*(9*x + 14*x^2 + x^3 - 324) - 852*x - 37*x^2 + 26*x^3 + 2*x^4 + 1461))/(18*x + x^2 + 81),
x)

[Out]

(exp(-x)*(x - 3)*(18*x + 9*exp(2) + x*exp(2) + 2*x^2 - 25))/(x + 9)

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sympy [B]  time = 0.17, size = 39, normalized size = 1.44 \begin {gather*} \frac {\left (2 x^{3} + x^{2} e^{2} + 12 x^{2} - 79 x + 6 x e^{2} - 27 e^{2} + 75\right ) e^{- x}}{x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-14*x**2-9*x+324)*exp(2)-2*x**4-26*x**3+37*x**2+852*x-1461)/(x**2+18*x+81)/exp(x),x)

[Out]

(2*x**3 + x**2*exp(2) + 12*x**2 - 79*x + 6*x*exp(2) - 27*exp(2) + 75)*exp(-x)/(x + 9)

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