3.66.22 \(\int \frac {1}{625} e^{-2 x} (-920-200 e^2+258 x+1250 e^{2 x} x-18 x^2+e (-860+120 x)+e^x (1000+e (500-500 x)-1300 x+150 x^2)) \, dx\)

Optimal. Leaf size=23 \[ \left (-\frac {2}{5} e^{-x} \left (-2-e+\frac {3 x}{10}\right )+x\right )^2 \]

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Rubi [B]  time = 0.91, antiderivative size = 150, normalized size of antiderivative = 6.52, number of steps used = 20, number of rules used = 6, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 6741, 6742, 2194, 2176, 2196} \begin {gather*} \frac {9}{625} e^{-2 x} x^2-\frac {6}{25} e^{-x} x^2+x^2-\frac {24}{125} e^{-2 x} x-\frac {12 e^{-x} x}{25}+\frac {4}{25} (13+5 e) e^{-x} x-\frac {6}{125} e^{1-2 x}-\frac {12 e^{-2 x}}{125}-\frac {12 e^{-x}}{25}+\frac {2}{125} e^{1-2 x} (43-6 x)+\frac {4}{125} \left (23+5 e^2\right ) e^{-2 x}+\frac {4}{25} (13+5 e) e^{-x}-\frac {4}{5} (2+e) e^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-920 - 200*E^2 + 258*x + 1250*E^(2*x)*x - 18*x^2 + E*(-860 + 120*x) + E^x*(1000 + E*(500 - 500*x) - 1300*
x + 150*x^2))/(625*E^(2*x)),x]

[Out]

(-6*E^(1 - 2*x))/125 - 12/(125*E^(2*x)) - 12/(25*E^x) - (4*(2 + E))/(5*E^x) + (4*(13 + 5*E))/(25*E^x) + (4*(23
+ 5*E^2))/(125*E^(2*x)) + (2*E^(1 - 2*x)*(43 - 6*x))/125 - (24*x)/(125*E^(2*x)) - (12*x)/(25*E^x) + (4*(13 +
5*E)*x)/(25*E^x) + x^2 + (9*x^2)/(625*E^(2*x)) - (6*x^2)/(25*E^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 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 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 2196

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

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int e^{-2 x} \left (-920-200 e^2+258 x+1250 e^{2 x} x-18 x^2+e (-860+120 x)+e^x \left (1000+e (500-500 x)-1300 x+150 x^2\right )\right ) \, dx\\ &=\frac {1}{625} \int e^{-2 x} \left (-920 \left (1+\frac {5 e^2}{23}\right )+258 x+1250 e^{2 x} x-18 x^2+e (-860+120 x)+e^x \left (1000+e (500-500 x)-1300 x+150 x^2\right )\right ) \, dx\\ &=\frac {1}{625} \int \left (-40 e^{-2 x} \left (23+5 e^2\right )+1250 x+258 e^{-2 x} x-18 e^{-2 x} x^2+20 e^{1-2 x} (-43+6 x)+50 e^{-x} \left (10 (2+e)-2 (13+5 e) x+3 x^2\right )\right ) \, dx\\ &=x^2-\frac {18}{625} \int e^{-2 x} x^2 \, dx+\frac {4}{125} \int e^{1-2 x} (-43+6 x) \, dx+\frac {2}{25} \int e^{-x} \left (10 (2+e)-2 (13+5 e) x+3 x^2\right ) \, dx+\frac {258}{625} \int e^{-2 x} x \, dx-\frac {1}{125} \left (8 \left (23+5 e^2\right )\right ) \int e^{-2 x} \, dx\\ &=\frac {4}{125} e^{-2 x} \left (23+5 e^2\right )+\frac {2}{125} e^{1-2 x} (43-6 x)-\frac {129}{625} e^{-2 x} x+x^2+\frac {9}{625} e^{-2 x} x^2-\frac {18}{625} \int e^{-2 x} x \, dx+\frac {2}{25} \int \left (10 e^{-x} (2+e)-2 e^{-x} (13+5 e) x+3 e^{-x} x^2\right ) \, dx+\frac {12}{125} \int e^{1-2 x} \, dx+\frac {129}{625} \int e^{-2 x} \, dx\\ &=-\frac {6}{125} e^{1-2 x}-\frac {129 e^{-2 x}}{1250}+\frac {4}{125} e^{-2 x} \left (23+5 e^2\right )+\frac {2}{125} e^{1-2 x} (43-6 x)-\frac {24}{125} e^{-2 x} x+x^2+\frac {9}{625} e^{-2 x} x^2-\frac {9}{625} \int e^{-2 x} \, dx+\frac {6}{25} \int e^{-x} x^2 \, dx+\frac {1}{5} (4 (2+e)) \int e^{-x} \, dx-\frac {1}{25} (4 (13+5 e)) \int e^{-x} x \, dx\\ &=-\frac {6}{125} e^{1-2 x}-\frac {12 e^{-2 x}}{125}-\frac {4}{5} e^{-x} (2+e)+\frac {4}{125} e^{-2 x} \left (23+5 e^2\right )+\frac {2}{125} e^{1-2 x} (43-6 x)-\frac {24}{125} e^{-2 x} x+\frac {4}{25} e^{-x} (13+5 e) x+x^2+\frac {9}{625} e^{-2 x} x^2-\frac {6}{25} e^{-x} x^2+\frac {12}{25} \int e^{-x} x \, dx-\frac {1}{25} (4 (13+5 e)) \int e^{-x} \, dx\\ &=-\frac {6}{125} e^{1-2 x}-\frac {12 e^{-2 x}}{125}-\frac {4}{5} e^{-x} (2+e)+\frac {4}{25} e^{-x} (13+5 e)+\frac {4}{125} e^{-2 x} \left (23+5 e^2\right )+\frac {2}{125} e^{1-2 x} (43-6 x)-\frac {24}{125} e^{-2 x} x-\frac {12 e^{-x} x}{25}+\frac {4}{25} e^{-x} (13+5 e) x+x^2+\frac {9}{625} e^{-2 x} x^2-\frac {6}{25} e^{-x} x^2+\frac {12}{25} \int e^{-x} \, dx\\ &=-\frac {6}{125} e^{1-2 x}-\frac {12 e^{-2 x}}{125}-\frac {12 e^{-x}}{25}-\frac {4}{5} e^{-x} (2+e)+\frac {4}{25} e^{-x} (13+5 e)+\frac {4}{125} e^{-2 x} \left (23+5 e^2\right )+\frac {2}{125} e^{1-2 x} (43-6 x)-\frac {24}{125} e^{-2 x} x-\frac {12 e^{-x} x}{25}+\frac {4}{25} e^{-x} (13+5 e) x+x^2+\frac {9}{625} e^{-2 x} x^2-\frac {6}{25} e^{-x} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.56, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{625} e^{-2 x} \left (20+10 e-3 x+25 e^x x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-920 - 200*E^2 + 258*x + 1250*E^(2*x)*x - 18*x^2 + E*(-860 + 120*x) + E^x*(1000 + E*(500 - 500*x) -
1300*x + 150*x^2))/(625*E^(2*x)),x]

[Out]

(20 + 10*E - 3*x + 25*E^x*x)^2/(625*E^(2*x))

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fricas [B]  time = 0.65, size = 56, normalized size = 2.43 \begin {gather*} \frac {1}{625} \, {\left (625 \, x^{2} e^{\left (2 \, x\right )} + 9 \, x^{2} - 20 \, {\left (3 \, x - 20\right )} e - 50 \, {\left (3 \, x^{2} - 10 \, x e - 20 \, x\right )} e^{x} - 120 \, x + 100 \, e^{2} + 400\right )} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(1250*x*exp(x)^2+((-500*x+500)*exp(1)+150*x^2-1300*x+1000)*exp(x)-200*exp(1)^2+(120*x-860)*exp
(1)-18*x^2+258*x-920)/exp(x)^2,x, algorithm="fricas")

[Out]

1/625*(625*x^2*e^(2*x) + 9*x^2 - 20*(3*x - 20)*e - 50*(3*x^2 - 10*x*e - 20*x)*e^x - 120*x + 100*e^2 + 400)*e^(
-2*x)

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giac [B]  time = 0.19, size = 76, normalized size = 3.30 \begin {gather*} -\frac {6}{25} \, x^{2} e^{\left (-x\right )} + \frac {9}{625} \, x^{2} e^{\left (-2 \, x\right )} + x^{2} + \frac {8}{5} \, x e^{\left (-x\right )} - \frac {24}{125} \, x e^{\left (-2 \, x\right )} + \frac {4}{5} \, x e^{\left (-x + 1\right )} - \frac {12}{125} \, x e^{\left (-2 \, x + 1\right )} + \frac {16}{25} \, e^{\left (-2 \, x\right )} + \frac {4}{25} \, e^{\left (-2 \, x + 2\right )} + \frac {16}{25} \, e^{\left (-2 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(1250*x*exp(x)^2+((-500*x+500)*exp(1)+150*x^2-1300*x+1000)*exp(x)-200*exp(1)^2+(120*x-860)*exp
(1)-18*x^2+258*x-920)/exp(x)^2,x, algorithm="giac")

[Out]

-6/25*x^2*e^(-x) + 9/625*x^2*e^(-2*x) + x^2 + 8/5*x*e^(-x) - 24/125*x*e^(-2*x) + 4/5*x*e^(-x + 1) - 12/125*x*e
^(-2*x + 1) + 16/25*e^(-2*x) + 4/25*e^(-2*x + 2) + 16/25*e^(-2*x + 1)

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maple [B]  time = 0.07, size = 54, normalized size = 2.35

method	result	size
risch	\(x^{2}+\frac {\left (500 x \,{\mathrm e}-150 x^{2}+1000 x \right ) {\mathrm e}^{-x}}{625}+\frac {\left (100 \,{\mathrm e}^{2}-60 x \,{\mathrm e}+9 x^{2}+400 \,{\mathrm e}-120 x +400\right ) {\mathrm e}^{-2 x}}{625}\)	\(54\)
norman	\(\left (\left (-\frac {24}{125}-\frac {12 \,{\mathrm e}}{125}\right ) x +{\mathrm e}^{2 x} x^{2}+\left (\frac {4 \,{\mathrm e}}{5}+\frac {8}{5}\right ) x \,{\mathrm e}^{x}+\frac {9 x^{2}}{625}-\frac {6 \,{\mathrm e}^{x} x^{2}}{25}+\frac {16}{25}+\frac {16 \,{\mathrm e}}{25}+\frac {4 \,{\mathrm e}^{2}}{25}\right ) {\mathrm e}^{-2 x}\)	\(56\)
default	\(x^{2}+\frac {16 \,{\mathrm e}^{-2 x}}{25}-\frac {24 x \,{\mathrm e}^{-2 x}}{125}+\frac {8 x \,{\mathrm e}^{-x}}{5}+\frac {9 x^{2} {\mathrm e}^{-2 x}}{625}-\frac {6 x^{2} {\mathrm e}^{-x}}{25}-\frac {4 \,{\mathrm e} \,{\mathrm e}^{-x}}{5}+\frac {86 \,{\mathrm e}^{-2 x} {\mathrm e}}{125}+\frac {4 \,{\mathrm e}^{-2 x} {\mathrm e}^{2}}{25}+\frac {24 \,{\mathrm e} \left (-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )}{125}-\frac {4 \,{\mathrm e} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )}{5}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*(1250*x*exp(x)^2+((-500*x+500)*exp(1)+150*x^2-1300*x+1000)*exp(x)-200*exp(1)^2+(120*x-860)*exp(1)-18
*x^2+258*x-920)/exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

x^2+1/625*(500*x*exp(1)-150*x^2+1000*x)*exp(-x)+1/625*(100*exp(2)-60*x*exp(1)+9*x^2+400*exp(1)-120*x+400)*exp(
-2*x)

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maxima [B]  time = 0.37, size = 117, normalized size = 5.09 \begin {gather*} x^{2} - \frac {6}{25} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac {4}{5} \, {\left (x e + e\right )} e^{\left (-x\right )} + \frac {52}{25} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {9}{1250} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {6}{125} \, {\left (2 \, x e + e\right )} e^{\left (-2 \, x\right )} - \frac {129}{1250} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {8}{5} \, e^{\left (-x\right )} + \frac {92}{125} \, e^{\left (-2 \, x\right )} - \frac {4}{5} \, e^{\left (-x + 1\right )} + \frac {4}{25} \, e^{\left (-2 \, x + 2\right )} + \frac {86}{125} \, e^{\left (-2 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(1250*x*exp(x)^2+((-500*x+500)*exp(1)+150*x^2-1300*x+1000)*exp(x)-200*exp(1)^2+(120*x-860)*exp
(1)-18*x^2+258*x-920)/exp(x)^2,x, algorithm="maxima")

[Out]

x^2 - 6/25*(x^2 + 2*x + 2)*e^(-x) + 4/5*(x*e + e)*e^(-x) + 52/25*(x + 1)*e^(-x) + 9/1250*(2*x^2 + 2*x + 1)*e^(
-2*x) - 6/125*(2*x*e + e)*e^(-2*x) - 129/1250*(2*x + 1)*e^(-2*x) - 8/5*e^(-x) + 92/125*e^(-2*x) - 4/5*e^(-x +
1) + 4/25*e^(-2*x + 2) + 86/125*e^(-2*x + 1)

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mupad [B]  time = 0.24, size = 22, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^{-2\,x}\,{\left (10\,\mathrm {e}-3\,x+25\,x\,{\mathrm {e}}^x+20\right )}^2}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*x)*((8*exp(2))/25 - (258*x)/625 - 2*x*exp(2*x) + (exp(x)*(1300*x - 150*x^2 + exp(1)*(500*x - 500)
- 1000))/625 + (18*x^2)/625 - (exp(1)*(120*x - 860))/625 + 184/125),x)

[Out]

(exp(-2*x)*(10*exp(1) - 3*x + 25*x*exp(x) + 20)^2)/625

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sympy [B]  time = 0.21, size = 56, normalized size = 2.43 \begin {gather*} x^{2} + \frac {\left (- 3750 x^{2} + 25000 x + 12500 e x\right ) e^{- x}}{15625} + \frac {\left (225 x^{2} - 1500 e x - 3000 x + 10000 + 2500 e^{2} + 10000 e\right ) e^{- 2 x}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(1250*x*exp(x)**2+((-500*x+500)*exp(1)+150*x**2-1300*x+1000)*exp(x)-200*exp(1)**2+(120*x-860)*
exp(1)-18*x**2+258*x-920)/exp(x)**2,x)

[Out]

x**2 + (-3750*x**2 + 25000*x + 12500*E*x)*exp(-x)/15625 + (225*x**2 - 1500*E*x - 3000*x + 10000 + 2500*exp(2)
+ 10000*E)*exp(-2*x)/15625

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