3.93.64 \(\int \frac {e^{-2 x} (e^{2 x} (-2000-400 x-3000 x^3+449700 x^4+135000 x^5+13500 x^6+450 x^7)+e^x (30000 x+39000 x^2-443400 x^3+315300 x^4+121500 x^5+13050 x^6+450 x^7) \log (4)+(-450000 x^3-135000 x^4-13500 x^5-450 x^6) \log ^2(4))}{1000 x^3+300 x^4+30 x^5+x^6} \, dx\)

Optimal. Leaf size=28 \[ 25 \left (\frac {2}{x (10+x)}+3 \left (x-e^{-x} \log (4)\right )\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 2.07, antiderivative size = 77, normalized size of antiderivative = 2.75, number of steps used = 17, number of rules used = 7, integrand size = 130, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6741, 6742, 1620, 2194, 2177, 2178, 2176} \begin {gather*} 225 x^2+\frac {1}{x^2}+\frac {1501}{5 (x+10)}+\frac {1}{(x+10)^2}-\frac {1}{5 x}+225 e^{-2 x} \log ^2(4)-450 e^{-x} x \log (4)+\frac {30 e^{-x} \log (4)}{x+10}-\frac {30 e^{-x} \log (4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(-2000 - 400*x - 3000*x^3 + 449700*x^4 + 135000*x^5 + 13500*x^6 + 450*x^7) + E^x*(30000*x + 39000
*x^2 - 443400*x^3 + 315300*x^4 + 121500*x^5 + 13050*x^6 + 450*x^7)*Log[4] + (-450000*x^3 - 135000*x^4 - 13500*
x^5 - 450*x^6)*Log[4]^2)/(E^(2*x)*(1000*x^3 + 300*x^4 + 30*x^5 + x^6)),x]

[Out]

x^(-2) - 1/(5*x) + 225*x^2 + (10 + x)^(-2) + 1501/(5*(10 + x)) - (30*Log[4])/(E^x*x) - (450*x*Log[4])/E^x + (3
0*Log[4])/(E^x*(10 + x)) + (225*Log[4]^2)/E^(2*x)

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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 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} &=\int \frac {e^{-2 x} \left (e^{2 x} \left (-2000-400 x-3000 x^3+449700 x^4+135000 x^5+13500 x^6+450 x^7\right )+e^x \left (30000 x+39000 x^2-443400 x^3+315300 x^4+121500 x^5+13050 x^6+450 x^7\right ) \log (4)+\left (-450000 x^3-135000 x^4-13500 x^5-450 x^6\right ) \log ^2(4)\right )}{x^3 (10+x)^3} \, dx\\ &=\int \left (\frac {50 \left (2+30 x^2+3 x^3\right ) \left (-20-4 x+300 x^2+60 x^3+3 x^4\right )}{x^3 (10+x)^3}+\frac {150 e^{-x} \left (20+24 x-298 x^2+240 x^3+57 x^4+3 x^5\right ) \log (4)}{x^2 (10+x)^2}-450 e^{-2 x} \log ^2(4)\right ) \, dx\\ &=50 \int \frac {\left (2+30 x^2+3 x^3\right ) \left (-20-4 x+300 x^2+60 x^3+3 x^4\right )}{x^3 (10+x)^3} \, dx+(150 \log (4)) \int \frac {e^{-x} \left (20+24 x-298 x^2+240 x^3+57 x^4+3 x^5\right )}{x^2 (10+x)^2} \, dx-\left (450 \log ^2(4)\right ) \int e^{-2 x} \, dx\\ &=225 e^{-2 x} \log ^2(4)+50 \int \left (-\frac {1}{25 x^3}+\frac {1}{250 x^2}+9 x-\frac {1}{25 (10+x)^3}-\frac {1501}{250 (10+x)^2}\right ) \, dx+(150 \log (4)) \int \left (-3 e^{-x}+\frac {e^{-x}}{5 x^2}+\frac {e^{-x}}{5 x}+3 e^{-x} x-\frac {e^{-x}}{5 (10+x)^2}-\frac {e^{-x}}{5 (10+x)}\right ) \, dx\\ &=\frac {1}{x^2}-\frac {1}{5 x}+225 x^2+\frac {1}{(10+x)^2}+\frac {1501}{5 (10+x)}+225 e^{-2 x} \log ^2(4)+(30 \log (4)) \int \frac {e^{-x}}{x^2} \, dx+(30 \log (4)) \int \frac {e^{-x}}{x} \, dx-(30 \log (4)) \int \frac {e^{-x}}{(10+x)^2} \, dx-(30 \log (4)) \int \frac {e^{-x}}{10+x} \, dx-(450 \log (4)) \int e^{-x} \, dx+(450 \log (4)) \int e^{-x} x \, dx\\ &=\frac {1}{x^2}-\frac {1}{5 x}+225 x^2+\frac {1}{(10+x)^2}+\frac {1501}{5 (10+x)}+450 e^{-x} \log (4)-\frac {30 e^{-x} \log (4)}{x}-450 e^{-x} x \log (4)+\frac {30 e^{-x} \log (4)}{10+x}-30 e^{10} \text {Ei}(-10-x) \log (4)+30 \text {Ei}(-x) \log (4)+225 e^{-2 x} \log ^2(4)-(30 \log (4)) \int \frac {e^{-x}}{x} \, dx+(30 \log (4)) \int \frac {e^{-x}}{10+x} \, dx+(450 \log (4)) \int e^{-x} \, dx\\ &=\frac {1}{x^2}-\frac {1}{5 x}+225 x^2+\frac {1}{(10+x)^2}+\frac {1501}{5 (10+x)}-\frac {30 e^{-x} \log (4)}{x}-450 e^{-x} x \log (4)+\frac {30 e^{-x} \log (4)}{10+x}+225 e^{-2 x} \log ^2(4)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 42, normalized size = 1.50 \begin {gather*} \frac {25 e^{-2 x} \left (e^x \left (2+30 x^2+3 x^3\right )-3 x (10+x) \log (4)\right )^2}{x^2 (10+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-2000 - 400*x - 3000*x^3 + 449700*x^4 + 135000*x^5 + 13500*x^6 + 450*x^7) + E^x*(30000*x +
39000*x^2 - 443400*x^3 + 315300*x^4 + 121500*x^5 + 13050*x^6 + 450*x^7)*Log[4] + (-450000*x^3 - 135000*x^4 -
13500*x^5 - 450*x^6)*Log[4]^2)/(E^(2*x)*(1000*x^3 + 300*x^4 + 30*x^5 + x^6)),x]

[Out]

(25*(E^x*(2 + 30*x^2 + 3*x^3) - 3*x*(10 + x)*Log[4])^2)/(E^(2*x)*x^2*(10 + x)^2)

________________________________________________________________________________________

fricas [B]  time = 0.53, size = 106, normalized size = 3.79 \begin {gather*} -\frac {25 \, {\left (12 \, {\left (3 \, x^{5} + 60 \, x^{4} + 300 \, x^{3} + 2 \, x^{2} + 20 \, x\right )} e^{x} \log \relax (2) - 36 \, {\left (x^{4} + 20 \, x^{3} + 100 \, x^{2}\right )} \log \relax (2)^{2} - {\left (9 \, x^{6} + 180 \, x^{5} + 900 \, x^{4} + 12 \, x^{3} + 120 \, x^{2} + 4\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x^{4} + 20 \, x^{3} + 100 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((450*x^7+13500*x^6+135000*x^5+449700*x^4-3000*x^3-400*x-2000)*exp(x)^2+2*(450*x^7+13050*x^6+121500*
x^5+315300*x^4-443400*x^3+39000*x^2+30000*x)*log(2)*exp(x)+4*(-450*x^6-13500*x^5-135000*x^4-450000*x^3)*log(2)
^2)/(x^6+30*x^5+300*x^4+1000*x^3)/exp(x)^2,x, algorithm="fricas")

[Out]

-25*(12*(3*x^5 + 60*x^4 + 300*x^3 + 2*x^2 + 20*x)*e^x*log(2) - 36*(x^4 + 20*x^3 + 100*x^2)*log(2)^2 - (9*x^6 +
180*x^5 + 900*x^4 + 12*x^3 + 120*x^2 + 4)*e^(2*x))*e^(-2*x)/(x^4 + 20*x^3 + 100*x^2)

________________________________________________________________________________________

giac [B]  time = 0.17, size = 137, normalized size = 4.89 \begin {gather*} -\frac {25 \, {\left (36 \, x^{5} e^{\left (-x\right )} \log \relax (2) - 36 \, x^{4} e^{\left (-2 \, x\right )} \log \relax (2)^{2} - 9 \, x^{6} + 720 \, x^{4} e^{\left (-x\right )} \log \relax (2) - 720 \, x^{3} e^{\left (-2 \, x\right )} \log \relax (2)^{2} - 180 \, x^{5} + 3600 \, x^{3} e^{\left (-x\right )} \log \relax (2) - 3600 \, x^{2} e^{\left (-2 \, x\right )} \log \relax (2)^{2} - 900 \, x^{4} + 24 \, x^{2} e^{\left (-x\right )} \log \relax (2) - 12 \, x^{3} + 240 \, x e^{\left (-x\right )} \log \relax (2) - 120 \, x^{2} - 4\right )}}{x^{4} + 20 \, x^{3} + 100 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((450*x^7+13500*x^6+135000*x^5+449700*x^4-3000*x^3-400*x-2000)*exp(x)^2+2*(450*x^7+13050*x^6+121500*
x^5+315300*x^4-443400*x^3+39000*x^2+30000*x)*log(2)*exp(x)+4*(-450*x^6-13500*x^5-135000*x^4-450000*x^3)*log(2)
^2)/(x^6+30*x^5+300*x^4+1000*x^3)/exp(x)^2,x, algorithm="giac")

[Out]

-25*(36*x^5*e^(-x)*log(2) - 36*x^4*e^(-2*x)*log(2)^2 - 9*x^6 + 720*x^4*e^(-x)*log(2) - 720*x^3*e^(-2*x)*log(2)
^2 - 180*x^5 + 3600*x^3*e^(-x)*log(2) - 3600*x^2*e^(-2*x)*log(2)^2 - 900*x^4 + 24*x^2*e^(-x)*log(2) - 12*x^3 +
240*x*e^(-x)*log(2) - 120*x^2 - 4)/(x^4 + 20*x^3 + 100*x^2)

________________________________________________________________________________________

maple [A]  time = 0.22, size = 71, normalized size = 2.54

method	result	size
risch	\(225 x^{2}+\frac {300 x^{3}+3000 x^{2}+100}{x^{2} \left (x^{2}+20 x +100\right )}-\frac {300 \ln \relax (2) \left (3 x^{3}+30 x^{2}+2\right ) {\mathrm e}^{-x}}{x \left (x +10\right )}+900 \ln \relax (2)^{2} {\mathrm e}^{-2 x}\)	\(71\)
default	\(\frac {1}{x^{2}}-\frac {1}{5 x}+\frac {1}{\left (x +10\right )^{2}}+\frac {1501}{5 \left (x +10\right )}+225 x^{2}+900 \ln \relax (2)^{2} {\mathrm e}^{-2 x}-\frac {600 \ln \relax (2) {\mathrm e}^{-x}}{x^{2}+20 x +100}-\frac {6000 \ln \relax (2) {\mathrm e}^{-x}}{x \left (x^{2}+20 x +100\right )}-900 \ln \relax (2) {\mathrm e}^{-x} x\)	\(85\)
norman	\(\frac {\left (-2247000 \,{\mathrm e}^{2 x} x^{2}-449700 \,{\mathrm e}^{2 x} x^{3}+100 \,{\mathrm e}^{2 x}+90000 x^{2} \ln \relax (2)^{2}+18000 x^{3} \ln \relax (2)^{2}+900 x^{4} \ln \relax (2)^{2}+4500 x^{5} {\mathrm e}^{2 x}+225 x^{6} {\mathrm e}^{2 x}-6000 x \ln \relax (2) {\mathrm e}^{x}-600 x^{2} \ln \relax (2) {\mathrm e}^{x}-90000 x^{3} \ln \relax (2) {\mathrm e}^{x}-18000 \,{\mathrm e}^{x} \ln \relax (2) x^{4}-900 \,{\mathrm e}^{x} \ln \relax (2) x^{5}\right ) {\mathrm e}^{-2 x}}{x^{2} \left (x +10\right )^{2}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((450*x^7+13500*x^6+135000*x^5+449700*x^4-3000*x^3-400*x-2000)*exp(x)^2+2*(450*x^7+13050*x^6+121500*x^5+31
5300*x^4-443400*x^3+39000*x^2+30000*x)*ln(2)*exp(x)+4*(-450*x^6-13500*x^5-135000*x^4-450000*x^3)*ln(2)^2)/(x^6
+30*x^5+300*x^4+1000*x^3)/exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

225*x^2+(300*x^3+3000*x^2+100)/x^2/(x^2+20*x+100)-300*ln(2)*(3*x^3+30*x^2+2)/x/(x+10)*exp(-x)+900*ln(2)^2*exp(
-2*x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2700000 \, \int \frac {e^{\left (-2 \, x\right )}}{x^{4} + 40 \, x^{3} + 600 \, x^{2} + 4000 \, x + 10000}\,{d x} \log \relax (2)^{2} + \frac {1800000 \, e^{20} E_{3}\left (2 \, x + 20\right ) \log \relax (2)^{2}}{{\left (x + 10\right )}^{2}} + \frac {25 \, {\left (9 \, x^{7} + 270 \, x^{6} + 2700 \, x^{5} + 9012 \, x^{4} + 240 \, x^{3} + 1200 \, x^{2} - 12 \, {\left (3 \, x^{6} \log \relax (2) + 90 \, x^{5} \log \relax (2) + 900 \, x^{4} \log \relax (2) + 3002 \, x^{3} \log \relax (2) + 40 \, x^{2} \log \relax (2) + 200 \, x \log \relax (2)\right )} e^{\left (-x\right )} + 36 \, {\left (x^{5} \log \relax (2)^{2} + 30 \, x^{4} \log \relax (2)^{2} + 300 \, x^{3} \log \relax (2)^{2}\right )} e^{\left (-2 \, x\right )} + 4 \, x + 40\right )}}{x^{5} + 30 \, x^{4} + 300 \, x^{3} + 1000 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((450*x^7+13500*x^6+135000*x^5+449700*x^4-3000*x^3-400*x-2000)*exp(x)^2+2*(450*x^7+13050*x^6+121500*
x^5+315300*x^4-443400*x^3+39000*x^2+30000*x)*log(2)*exp(x)+4*(-450*x^6-13500*x^5-135000*x^4-450000*x^3)*log(2)
^2)/(x^6+30*x^5+300*x^4+1000*x^3)/exp(x)^2,x, algorithm="maxima")

[Out]

-2700000*integrate(e^(-2*x)/(x^4 + 40*x^3 + 600*x^2 + 4000*x + 10000), x)*log(2)^2 + 1800000*e^20*exp_integral
_e(3, 2*x + 20)*log(2)^2/(x + 10)^2 + 25*(9*x^7 + 270*x^6 + 2700*x^5 + 9012*x^4 + 240*x^3 + 1200*x^2 - 12*(3*x
^6*log(2) + 90*x^5*log(2) + 900*x^4*log(2) + 3002*x^3*log(2) + 40*x^2*log(2) + 200*x*log(2))*e^(-x) + 36*(x^5*
log(2)^2 + 30*x^4*log(2)^2 + 300*x^3*log(2)^2)*e^(-2*x) + 4*x + 40)/(x^5 + 30*x^4 + 300*x^3 + 1000*x^2)

________________________________________________________________________________________

mupad [B]  time = 7.07, size = 79, normalized size = 2.82 \begin {gather*} \frac {300\,x^3+3000\,x^2+100}{x^4+20\,x^3+100\,x^2}+900\,{\mathrm {e}}^{-2\,x}\,{\ln \relax (2)}^2+225\,x^2-\frac {{\mathrm {e}}^{-x}\,\left (900\,\ln \relax (2)\,x^3+9000\,\ln \relax (2)\,x^2+600\,\ln \relax (2)\right )}{x^2+10\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*x)*(exp(2*x)*(449700*x^4 - 3000*x^3 - 400*x + 135000*x^5 + 13500*x^6 + 450*x^7 - 2000) - 4*log(2)^
2*(450000*x^3 + 135000*x^4 + 13500*x^5 + 450*x^6) + 2*exp(x)*log(2)*(30000*x + 39000*x^2 - 443400*x^3 + 315300
*x^4 + 121500*x^5 + 13050*x^6 + 450*x^7)))/(1000*x^3 + 300*x^4 + 30*x^5 + x^6),x)

[Out]

(3000*x^2 + 300*x^3 + 100)/(100*x^2 + 20*x^3 + x^4) + 900*exp(-2*x)*log(2)^2 + 225*x^2 - (exp(-x)*(600*log(2)
+ 9000*x^2*log(2) + 900*x^3*log(2)))/(10*x + x^2)

________________________________________________________________________________________

sympy [B]  time = 0.34, size = 87, normalized size = 3.11 \begin {gather*} 225 x^{2} + \frac {300 x^{3} + 3000 x^{2} + 100}{x^{4} + 20 x^{3} + 100 x^{2}} + \frac {\left (900 x^{2} \log {\relax (2 )}^{2} + 9000 x \log {\relax (2 )}^{2}\right ) e^{- 2 x} + \left (- 900 x^{3} \log {\relax (2 )} - 9000 x^{2} \log {\relax (2 )} - 600 \log {\relax (2 )}\right ) e^{- x}}{x^{2} + 10 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((450*x**7+13500*x**6+135000*x**5+449700*x**4-3000*x**3-400*x-2000)*exp(x)**2+2*(450*x**7+13050*x**6
+121500*x**5+315300*x**4-443400*x**3+39000*x**2+30000*x)*ln(2)*exp(x)+4*(-450*x**6-13500*x**5-135000*x**4-4500
00*x**3)*ln(2)**2)/(x**6+30*x**5+300*x**4+1000*x**3)/exp(x)**2,x)

[Out]

225*x**2 + (300*x**3 + 3000*x**2 + 100)/(x**4 + 20*x**3 + 100*x**2) + ((900*x**2*log(2)**2 + 9000*x*log(2)**2)
*exp(-2*x) + (-900*x**3*log(2) - 9000*x**2*log(2) - 600*log(2))*exp(-x))/(x**2 + 10*x)

________________________________________________________________________________________