3.63.28 \(\int \frac {e^{-\frac {4 x^2}{25}} (e^{20+4 x} (-9375+12500 x-1000 x^2)+e^{15+3 x+\frac {26 x^2}{25}} (5000 x-7500 x^2-4400 x^3)+e^{10+2 x+\frac {52 x^2}{25}} (-750 x^2+1500 x^3+2880 x^4)+e^{5+x+\frac {78 x^2}{25}} (-100 x^4-592 x^5)+e^{\frac {104 x^2}{25}} (5 x^4+40 x^6))}{5 x^4} \, dx\)

Optimal. Leaf size=29 \[ \left (-e^{x^2}+\frac {5 e^{5+x-\frac {x^2}{25}}}{x}\right )^4 x \]

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Rubi [B]  time = 2.01, antiderivative size = 129, normalized size of antiderivative = 4.45, number of steps used = 13, number of rules used = 8, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 6688, 6742, 2236, 2226, 2204, 2212, 2288} \begin {gather*} e^{4 x^2} x-20 e^{\frac {74 x^2}{25}+x+5}+\frac {150 e^{\frac {48 x^2}{25}+2 x+10} \left (48 x^2+25 x\right )}{(48 x+25) x^2}+\frac {625 e^{-\frac {4 x^2}{25}+4 x+20} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {500 e^{\frac {22 x^2}{25}+3 x+15} \left (44 x^2+75 x\right )}{(44 x+75) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(20 + 4*x)*(-9375 + 12500*x - 1000*x^2) + E^(15 + 3*x + (26*x^2)/25)*(5000*x - 7500*x^2 - 4400*x^3) + E
^(10 + 2*x + (52*x^2)/25)*(-750*x^2 + 1500*x^3 + 2880*x^4) + E^(5 + x + (78*x^2)/25)*(-100*x^4 - 592*x^5) + E^
((104*x^2)/25)*(5*x^4 + 40*x^6))/(5*E^((4*x^2)/25)*x^4),x]

[Out]

-20*E^(5 + x + (74*x^2)/25) + E^(4*x^2)*x + (625*E^(20 + 4*x - (4*x^2)/25)*(25*x - 2*x^2))/((25 - 2*x)*x^4) -
(500*E^(15 + 3*x + (22*x^2)/25)*(75*x + 44*x^2))/(x^3*(75 + 44*x)) + (150*E^(10 + 2*x + (48*x^2)/25)*(25*x + 4
8*x^2))/(x^2*(25 + 48*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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{x^4} \, dx\\ &=\frac {1}{5} \int \frac {e^{-\frac {4 x^2}{25}} \left (5 e^{5+x}-e^{\frac {26 x^2}{25}} x\right )^3 \left (-5 e^{\frac {26 x^2}{25}} x \left (1+8 x^2\right )-e^{5+x} \left (75-100 x+8 x^2\right )\right )}{x^4} \, dx\\ &=\frac {1}{5} \int \left (-4 e^{5+x+\frac {74 x^2}{25}} (25+148 x)+5 e^{4 x^2} \left (1+8 x^2\right )-\frac {125 e^{20+4 x-\frac {4 x^2}{25}} \left (75-100 x+8 x^2\right )}{x^4}-\frac {100 e^{15+3 x+\frac {22 x^2}{25}} \left (-50+75 x+44 x^2\right )}{x^3}+\frac {30 e^{10+2 x+\frac {48 x^2}{25}} \left (-25+50 x+96 x^2\right )}{x^2}\right ) \, dx\\ &=-\left (\frac {4}{5} \int e^{5+x+\frac {74 x^2}{25}} (25+148 x) \, dx\right )+6 \int \frac {e^{10+2 x+\frac {48 x^2}{25}} \left (-25+50 x+96 x^2\right )}{x^2} \, dx-20 \int \frac {e^{15+3 x+\frac {22 x^2}{25}} \left (-50+75 x+44 x^2\right )}{x^3} \, dx-25 \int \frac {e^{20+4 x-\frac {4 x^2}{25}} \left (75-100 x+8 x^2\right )}{x^4} \, dx+\int e^{4 x^2} \left (1+8 x^2\right ) \, dx\\ &=-20 e^{5+x+\frac {74 x^2}{25}}+\frac {625 e^{20+4 x-\frac {4 x^2}{25}} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {500 e^{15+3 x+\frac {22 x^2}{25}} \left (75 x+44 x^2\right )}{x^3 (75+44 x)}+\frac {150 e^{10+2 x+\frac {48 x^2}{25}} \left (25 x+48 x^2\right )}{x^2 (25+48 x)}+\int \left (e^{4 x^2}+8 e^{4 x^2} x^2\right ) \, dx\\ &=-20 e^{5+x+\frac {74 x^2}{25}}+\frac {625 e^{20+4 x-\frac {4 x^2}{25}} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {500 e^{15+3 x+\frac {22 x^2}{25}} \left (75 x+44 x^2\right )}{x^3 (75+44 x)}+\frac {150 e^{10+2 x+\frac {48 x^2}{25}} \left (25 x+48 x^2\right )}{x^2 (25+48 x)}+8 \int e^{4 x^2} x^2 \, dx+\int e^{4 x^2} \, dx\\ &=-20 e^{5+x+\frac {74 x^2}{25}}+e^{4 x^2} x+\frac {625 e^{20+4 x-\frac {4 x^2}{25}} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {500 e^{15+3 x+\frac {22 x^2}{25}} \left (75 x+44 x^2\right )}{x^3 (75+44 x)}+\frac {150 e^{10+2 x+\frac {48 x^2}{25}} \left (25 x+48 x^2\right )}{x^2 (25+48 x)}+\frac {1}{4} \sqrt {\pi } \text {erfi}(2 x)-\int e^{4 x^2} \, dx\\ &=-20 e^{5+x+\frac {74 x^2}{25}}+e^{4 x^2} x+\frac {625 e^{20+4 x-\frac {4 x^2}{25}} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {500 e^{15+3 x+\frac {22 x^2}{25}} \left (75 x+44 x^2\right )}{x^3 (75+44 x)}+\frac {150 e^{10+2 x+\frac {48 x^2}{25}} \left (25 x+48 x^2\right )}{x^2 (25+48 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 34, normalized size = 1.17 \begin {gather*} \frac {e^{-\frac {4 x^2}{25}} \left (-5 e^{5+x}+e^{\frac {26 x^2}{25}} x\right )^4}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(20 + 4*x)*(-9375 + 12500*x - 1000*x^2) + E^(15 + 3*x + (26*x^2)/25)*(5000*x - 7500*x^2 - 4400*x^
3) + E^(10 + 2*x + (52*x^2)/25)*(-750*x^2 + 1500*x^3 + 2880*x^4) + E^(5 + x + (78*x^2)/25)*(-100*x^4 - 592*x^5
) + E^((104*x^2)/25)*(5*x^4 + 40*x^6))/(5*E^((4*x^2)/25)*x^4),x]

[Out]

(-5*E^(5 + x) + E^((26*x^2)/25)*x)^4/(E^((4*x^2)/25)*x^3)

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fricas [B]  time = 0.60, size = 71, normalized size = 2.45 \begin {gather*} \frac {{\left (x^{4} - 20 \, x^{3} e^{\left (-\frac {26}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (-\frac {52}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (-\frac {78}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {104}{25} \, x^{2} + 4 \, x + 20\right )}\right )} e^{\left (4 \, x^{2}\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4)*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^
3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x)^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp
(1/25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4,x, algorithm="fricas")

[Out]

(x^4 - 20*x^3*e^(-26/25*x^2 + x + 5) + 150*x^2*e^(-52/25*x^2 + 2*x + 10) - 500*x*e^(-78/25*x^2 + 3*x + 15) + 6
25*e^(-104/25*x^2 + 4*x + 20))*e^(4*x^2)/x^3

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giac [B]  time = 0.20, size = 72, normalized size = 2.48 \begin {gather*} \frac {x^{4} e^{\left (4 \, x^{2}\right )} - 20 \, x^{3} e^{\left (\frac {74}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (\frac {48}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (\frac {22}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {4}{25} \, x^{2} + 4 \, x + 20\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4)*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^
3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x)^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp
(1/25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4,x, algorithm="giac")

[Out]

(x^4*e^(4*x^2) - 20*x^3*e^(74/25*x^2 + x + 5) + 150*x^2*e^(48/25*x^2 + 2*x + 10) - 500*x*e^(22/25*x^2 + 3*x +
15) + 625*e^(-4/25*x^2 + 4*x + 20))/x^3

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maple [B]  time = 0.12, size = 69, normalized size = 2.38

method	result	size
risch	\(x \,{\mathrm e}^{4 x^{2}}-20 \,{\mathrm e}^{5+x +\frac {74}{25} x^{2}}+\frac {150 \,{\mathrm e}^{2 x +10+\frac {48}{25} x^{2}}}{x}-\frac {500 \,{\mathrm e}^{15+3 x +\frac {22}{25} x^{2}}}{x^{2}}+\frac {625 \,{\mathrm e}^{20+4 x -\frac {4}{25} x^{2}}}{x^{3}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4)*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^3+(288
0*x^4+1500*x^3-750*x^2)*exp(5+x)^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp(1/25*
x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

x*exp(4*x^2)-20*exp(5+x+74/25*x^2)+150/x*exp(2*x+10+48/25*x^2)-500/x^2*exp(15+3*x+22/25*x^2)+625/x^3*exp(20+4*
x-4/25*x^2)

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maxima [B]  time = 0.42, size = 72, normalized size = 2.48 \begin {gather*} \frac {x^{4} e^{\left (4 \, x^{2}\right )} - 20 \, x^{3} e^{\left (\frac {74}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (\frac {48}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (\frac {22}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {4}{25} \, x^{2} + 4 \, x + 20\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4)*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^
3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x)^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp
(1/25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4,x, algorithm="maxima")

[Out]

(x^4*e^(4*x^2) - 20*x^3*e^(74/25*x^2 + x + 5) + 150*x^2*e^(48/25*x^2 + 2*x + 10) - 500*x*e^(22/25*x^2 + 3*x +
15) + 625*e^(-4/25*x^2 + 4*x + 20))/x^3

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mupad [B]  time = 5.24, size = 68, normalized size = 2.34 \begin {gather*} \frac {625\,{\mathrm {e}}^{-\frac {4\,x^2}{25}+4\,x+20}}{x^3}-20\,{\mathrm {e}}^{\frac {74\,x^2}{25}+x+5}-\frac {500\,{\mathrm {e}}^{\frac {22\,x^2}{25}+3\,x+15}}{x^2}+\frac {150\,{\mathrm {e}}^{\frac {48\,x^2}{25}+2\,x+10}}{x}+x\,{\mathrm {e}}^{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(4*x^2)/25)*((exp(4*x + 20)*(1000*x^2 - 12500*x + 9375))/5 - (exp(4*x^2)*exp((4*x^2)/25)*(5*x^4 + 4
0*x^6))/5 - (exp(2*x + 10)*exp(2*x^2)*exp((2*x^2)/25)*(1500*x^3 - 750*x^2 + 2880*x^4))/5 + (exp(x^2)*exp(3*x +
15)*exp(x^2/25)*(7500*x^2 - 5000*x + 4400*x^3))/5 + (exp(x + 5)*exp(3*x^2)*exp((3*x^2)/25)*(100*x^4 + 592*x^5
))/5))/x^4,x)

[Out]

(625*exp(4*x - (4*x^2)/25 + 20))/x^3 - 20*exp(x + (74*x^2)/25 + 5) - (500*exp(3*x + (22*x^2)/25 + 15))/x^2 + (
150*exp(2*x + (48*x^2)/25 + 10))/x + x*exp(4*x^2)

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sympy [B]  time = 0.86, size = 90, normalized size = 3.10 \begin {gather*} x e^{4 x^{2}} + \frac {\left (- 20 x^{6} e^{\frac {78 x^{2}}{25}} e^{x + 5} + 150 x^{5} e^{\frac {52 x^{2}}{25}} e^{2 x + 10} - 500 x^{4} e^{\frac {26 x^{2}}{25}} e^{3 x + 15} + 625 x^{3} e^{4 x + 20}\right ) e^{- \frac {4 x^{2}}{25}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((40*x**6+5*x**4)*exp(1/25*x**2)**4*exp(x**2)**4+(-592*x**5-100*x**4)*exp(5+x)*exp(1/25*x**2)**3
*exp(x**2)**3+(2880*x**4+1500*x**3-750*x**2)*exp(5+x)**2*exp(1/25*x**2)**2*exp(x**2)**2+(-4400*x**3-7500*x**2+
5000*x)*exp(5+x)**3*exp(1/25*x**2)*exp(x**2)+(-1000*x**2+12500*x-9375)*exp(5+x)**4)/x**4/exp(1/25*x**2)**4,x)

[Out]

x*exp(4*x**2) + (-20*x**6*exp(78*x**2/25)*exp(x + 5) + 150*x**5*exp(52*x**2/25)*exp(2*x + 10) - 500*x**4*exp(2
6*x**2/25)*exp(3*x + 15) + 625*x**3*exp(4*x + 20))*exp(-4*x**2/25)/x**6

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