3.89.60 $\int \frac{e^{-x}(-3+3x-19x^2-10x^3)}{9-54x+21x^2+180x^3+100x^4}dx$

Optimal. Leaf size=22 $$\frac{e^{-x}x}{-3-9(-1-x)x+x^2}$$

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Rubi [C]  time = 1.18, antiderivative size = 503, normalized size of antiderivative = 22.86, number of steps used = 25, number of rules used = 5, integrand size = 43, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.116, Rules used = {6688, 6742, 2177, 2178, 2270} $$\begin{array}{c}\frac{3(9+\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x-\sqrt{201}-9)\right)}{1340}-\frac{(201-11\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x-\sqrt{201}-9)\right)}{4020}-\frac{40e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x-\sqrt{201}-9)\right)}{67\sqrt{201}}-\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x-\sqrt{201}-9)\right)+\frac{2}{67}{e}^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x-\sqrt{201}-9)\right)-\frac{(201+11\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x+\sqrt{201}-9)\right)}{4020}+\frac{3(9-\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x+\sqrt{201}-9)\right)}{1340}+\frac{40e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x+\sqrt{201}-9)\right)}{67\sqrt{201}}+\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x+\sqrt{201}-9)\right)+\frac{2}{67}{e}^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-20x+\sqrt{201}-9)\right)+\frac{3(9-\sqrt{201})e^{-x}}{67(20x-\sqrt{201}+9)}+\frac{40e^{-x}}{67(20x-\sqrt{201}+9)}+\frac{3(9+\sqrt{201})e^{-x}}{67(20x+\sqrt{201}+9)}+\frac{40e^{-x}}{67(20x+\sqrt{201}+9)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-3 + 3*x - 19*x^2 - 10*x^3)/(E^x*(9 - 54*x + 21*x^2 + 180*x^3 + 100*x^4)),x]

[Out]

40/(67*E^x*(9 - Sqrt[201] + 20*x)) + (3*(9 - Sqrt[201]))/(67*E^x*(9 - Sqrt[201] + 20*x)) + 40/(67*E^x*(9 + Sqr
t[201] + 20*x)) + (3*(9 + Sqrt[201]))/(67*E^x*(9 + Sqrt[201] + 20*x)) + (2*E^((9 + Sqrt[201])/20)*ExpIntegralE
i[(-9 - Sqrt[201] - 20*x)/20])/67 - (9*Sqrt[3/67]*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)
/20])/67 - (40*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/(67*Sqrt[201]) - ((201 - 11*S
qrt[201])*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/4020 + (3*(9 + Sqrt[201])*E^((9 +
Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/1340 + (2*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 +
Sqrt[201] - 20*x)/20])/67 + (9*Sqrt[3/67]*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/6
7 + (40*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/(67*Sqrt[201]) + (3*(9 - Sqrt[201])*
E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/1340 - ((201 + 11*Sqrt[201])*E^((9 - Sqrt[20
1])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/4020

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 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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}(-3+3x-19x^2-10x^3)}{{(3-9x-10x^2)}^2}dx\\ & =\int (\frac{3e^{-x}(-2+3x)}{{(-3+9x+10x^2)}^2}+\frac{e^{-x}(-1-x)}{-3+9x+10x^2})dx\\ & =3\int \frac{e^{-x}(-2+3x)}{{(-3+9x+10x^2)}^2}dx+\int \frac{e^{-x}(-1-x)}{-3+9x+10x^2}dx\\ & =3\int (-\frac{2e^{-x}}{{(-3+9x+10x^2)}^2}+\frac{3e^{-x}x}{{(-3+9x+10x^2)}^2})dx+\int (\frac{(-1-\frac{11}{\sqrt{201}})e^{-x}}{9-\sqrt{201}+20x}+\frac{(-1+\frac{11}{\sqrt{201}})e^{-x}}{9+\sqrt{201}+20x})dx\\ & =-(6\int \frac{e^{-x}}{{(-3+9x+10x^2)}^2}dx)+9\int \frac{e^{-x}x}{{(-3+9x+10x^2)}^2}dx+\frac{1}{201}(-201+11\sqrt{201})\int \frac{e^{-x}}{9+\sqrt{201}+20x}dx-\frac{1}{201}(201+11\sqrt{201})\int \frac{e^{-x}}{9-\sqrt{201}+20x}dx\\ & =-\frac{(201-11\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{4020}-\frac{(201+11\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{4020}-6\int (\frac{400e^{-x}}{201{(-9+\sqrt{201}-20x)}^2}+\frac{400e^{-x}}{201\sqrt{201}(-9+\sqrt{201}-20x)}+\frac{400e^{-x}}{201{(9+\sqrt{201}+20x)}^2}+\frac{400e^{-x}}{201\sqrt{201}(9+\sqrt{201}+20x)})dx+9\int (\frac{20(-9+\sqrt{201})e^{-x}}{201{(-9+\sqrt{201}-20x)}^2}-\frac{20\sqrt{\frac{3}{67}}{e}^{-x}}{67(-9+\sqrt{201}-20x)}+\frac{20(-9-\sqrt{201})e^{-x}}{201{(9+\sqrt{201}+20x)}^2}-\frac{20\sqrt{\frac{3}{67}}{e}^{-x}}{67(9+\sqrt{201}+20x)})dx\\ & =-\frac{(201-11\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{4020}-\frac{(201+11\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{4020}-\frac{800}{67}\int \frac{e^{-x}}{{(-9+\sqrt{201}-20x)}^2}dx-\frac{800}{67}\int \frac{e^{-x}}{{(9+\sqrt{201}+20x)}^2}dx-\frac{1}{67}\left(180\sqrt{\frac{3}{67}}\right)\int \frac{e^{-x}}{-9+\sqrt{201}-20x}dx-\frac{1}{67}\left(180\sqrt{\frac{3}{67}}\right)\int \frac{e^{-x}}{9+\sqrt{201}+20x}dx-\frac{800\int \frac{e^{-x}}{-9+\sqrt{201}-20x}dx}{67\sqrt{201}}-\frac{800\int \frac{e^{-x}}{9+\sqrt{201}+20x}dx}{67\sqrt{201}}-\frac{1}{67}\left(60(9-\sqrt{201})\right)\int \frac{e^{-x}}{{(-9+\sqrt{201}-20x)}^2}dx-\frac{1}{67}\left(60(9+\sqrt{201})\right)\int \frac{e^{-x}}{{(9+\sqrt{201}+20x)}^2}dx\\ & =\frac{40e^{-x}}{67(9-\sqrt{201}+20x)}+\frac{3(9-\sqrt{201})e^{-x}}{67(9-\sqrt{201}+20x)}+\frac{40e^{-x}}{67(9+\sqrt{201}+20x)}+\frac{3(9+\sqrt{201})e^{-x}}{67(9+\sqrt{201}+20x)}-\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)-\frac{40e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{67\sqrt{201}}-\frac{(201-11\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{4020}+\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)+\frac{40e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{67\sqrt{201}}-\frac{(201+11\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{4020}-\frac{40}{67}\int \frac{e^{-x}}{-9+\sqrt{201}-20x}dx+\frac{40}{67}\int \frac{e^{-x}}{9+\sqrt{201}+20x}dx-\frac{1}{67}\left(3(9-\sqrt{201})\right)\int \frac{e^{-x}}{-9+\sqrt{201}-20x}dx+\frac{1}{67}\left(3(9+\sqrt{201})\right)\int \frac{e^{-x}}{9+\sqrt{201}+20x}dx\\ & =\frac{40e^{-x}}{67(9-\sqrt{201}+20x)}+\frac{3(9-\sqrt{201})e^{-x}}{67(9-\sqrt{201}+20x)}+\frac{40e^{-x}}{67(9+\sqrt{201}+20x)}+\frac{3(9+\sqrt{201})e^{-x}}{67(9+\sqrt{201}+20x)}+\frac{2}{67}{e}^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)-\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)-\frac{40e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{67\sqrt{201}}-\frac{(201-11\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{4020}+\frac{3(9+\sqrt{201})e^{\frac{1}{20}(9+\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9-\sqrt{201}-20x)\right)}{1340}+\frac{2}{67}{e}^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)+\frac{9}{67}\sqrt{\frac{3}{67}}{e}^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)+\frac{40e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{67\sqrt{201}}+\frac{3(9-\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{1340}-\frac{(201+11\sqrt{201})e^{\frac{1}{20}(9-\sqrt{201})}\text{Ei}\left(\frac{1}{20}(-9+\sqrt{201}-20x)\right)}{4020}\end{array}\end{array}$$

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Mathematica [A]  time = 0.27, size = 19, normalized size = 0.86 $$\begin{array}{c}\frac{e^{-x}x}{-3+9x+10x^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 3*x - 19*x^2 - 10*x^3)/(E^x*(9 - 54*x + 21*x^2 + 180*x^3 + 100*x^4)),x]

[Out]

x/(E^x*(-3 + 9*x + 10*x^2))

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fricas [A]  time = 0.48, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{x{e}^{(-x)}}{10x^2+9x-3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="fricas")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

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giac [A]  time = 0.13, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{x{e}^{(-x)}}{10x^2+9x-3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="giac")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

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maple [A]  time = 0.07, size = 19, normalized size = 0.86

method	result	size
gosper	$\frac{x{\mathrm{e}}^{-x}}{10x^2+9x-3}$	$19$
norman	$\frac{x{\mathrm{e}}^{-x}}{10x^2+9x-3}$	$19$
risch	$\frac{x{\mathrm{e}}^{-x}}{10x^2+9x-3}$	$19$
default	$\frac{{\mathrm{e}}^{-x}(20x+9)}{670x^2+603x-201}+\frac{3{\mathrm{e}}^{-x}(3x-2)}{67(10x^2+9x-3)}+\frac{19{\mathrm{e}}^{-x}(47x-9)}{670(10x^2+9x-3)}-\frac{3{\mathrm{e}}^{-x}(171x-47)}{670(10x^2+9x-3)}$	$94$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x,method=_RETURNVERBOSE)

[Out]

x/exp(x)/(10*x^2+9*x-3)

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maxima [A]  time = 0.44, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{x{e}^{(-x)}}{10x^2+9x-3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="maxima")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

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mupad [B]  time = 0.21, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{x{\mathrm{e}}^{-x}}{10x^2+9x-3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(19*x^2 - 3*x + 10*x^3 + 3))/(21*x^2 - 54*x + 180*x^3 + 100*x^4 + 9),x)

[Out]

(x*exp(-x))/(9*x + 10*x^2 - 3)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.64 $$\begin{array}{c}\frac{x{e}^{-x}}{10x^2+9x-3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x**3-19*x**2+3*x-3)/(100*x**4+180*x**3+21*x**2-54*x+9)/exp(x),x)

[Out]

x*exp(-x)/(10*x**2 + 9*x - 3)

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