∫ e−2x(x2+e2x(5x2+50x3−3x4)+ex(5−x2−50x3+28x4−x5)+(x2−2x3+ex(−5−5x−x2+x3)) log(x))_____________________________________________________________________

Optimal. Leaf size=35 \[ \frac {1}{4} x \left (\frac {5}{x}-x+e^{-x} x\right ) \left (-25+x+\frac {e^{-x} \log (x)}{x}\right ) \]

________________________________________________________________________________________

Rubi [B] time = 1.98, antiderivative size = 82, normalized size of antiderivative = 2.34, number of steps used = 35, number of rules used = 8, integrand size = 93, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.086, Rules used = {12, 6742, 2194, 2554, 2178, 2176, 2199, 2177} \begin {gather*} \frac {1}{4} e^{-x} x^3-\frac {x^3}{4}-\frac {25}{4} e^{-x} x^2+\frac {25 x^2}{4}+\frac {5 x}{4}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)+\frac {5 e^{-x} \log (x)}{4 x} \end {gather*}

Int[(x^2 + E^(2*x)*(5*x^2 + 50*x^3 - 3*x^4) + E^x*(5 - x^2 - 50*x^3 + 28*x^4 - x^5) + (x^2 - 2*x^3 + E^x*(-5 -

(5*x)/4 + (25*x^2)/4 - (25*x^2)/(4*E^x) - x^3/4 + x^3/(4*E^x) + (5*Log[x])/(4*E^x*x) + (x*Log[x])/(4*E^(2*x))

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

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 ===

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

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

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]

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]

)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{-2 x} \left (x^2+e^{2 x} \left (5 x^2+50 x^3-3 x^4\right )+e^x \left (5-x^2-50 x^3+28 x^4-x^5\right )+\left (x^2-2 x^3+e^x \left (-5-5 x-x^2+x^3\right )\right ) \log (x)\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (5+e^{-2 x}+50 x-3 x^2+e^{-2 x} \log (x)-2 e^{-2 x} x \log (x)-\frac {e^{-x} \left (-5+x^2+50 x^3-28 x^4+x^5+5 \log (x)+5 x \log (x)+x^2 \log (x)-x^3 \log (x)\right )}{x^2}\right ) \, dx\\ &=\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {x^3}{4}+\frac {1}{4} \int e^{-2 x} \, dx+\frac {1}{4} \int e^{-2 x} \log (x) \, dx-\frac {1}{4} \int \frac {e^{-x} \left (-5+x^2+50 x^3-28 x^4+x^5+5 \log (x)+5 x \log (x)+x^2 \log (x)-x^3 \log (x)\right )}{x^2} \, dx-\frac {1}{2} \int e^{-2 x} x \log (x) \, dx\\ &=-\frac {1}{8} e^{-2 x}+\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {x^3}{4}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} \int -\frac {e^{-2 x}}{2 x} \, dx-\frac {1}{4} \int \left (\frac {e^{-x} \left (-5+x^2+50 x^3-28 x^4+x^5\right )}{x^2}-\frac {e^{-x} \left (-5-5 x-x^2+x^3\right ) \log (x)}{x^2}\right ) \, dx+\frac {1}{2} \int \frac {e^{-2 x} (-1-2 x)}{4 x} \, dx\\ &=-\frac {1}{8} e^{-2 x}+\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {x^3}{4}+\frac {1}{4} e^{-2 x} x \log (x)+\frac {1}{8} \int \frac {e^{-2 x}}{x} \, dx+\frac {1}{8} \int \frac {e^{-2 x} (-1-2 x)}{x} \, dx-\frac {1}{4} \int \frac {e^{-x} \left (-5+x^2+50 x^3-28 x^4+x^5\right )}{x^2} \, dx+\frac {1}{4} \int \frac {e^{-x} \left (-5-5 x-x^2+x^3\right ) \log (x)}{x^2} \, dx\\ &=-\frac {1}{8} e^{-2 x}+\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {x^3}{4}+\frac {\text {Ei}(-2 x)}{8}+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)+\frac {1}{8} \int \left (-2 e^{-2 x}-\frac {e^{-2 x}}{x}\right ) \, dx-\frac {1}{4} \int \frac {e^{-x} \left (5-x^2\right )}{x^2} \, dx-\frac {1}{4} \int \left (e^{-x}-\frac {5 e^{-x}}{x^2}+50 e^{-x} x-28 e^{-x} x^2+e^{-x} x^3\right ) \, dx\\ &=-\frac {1}{8} e^{-2 x}+\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {x^3}{4}+\frac {\text {Ei}(-2 x)}{8}+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)-\frac {1}{8} \int \frac {e^{-2 x}}{x} \, dx-\frac {1}{4} \int e^{-2 x} \, dx-\frac {1}{4} \int e^{-x} \, dx-\frac {1}{4} \int \left (-e^{-x}+\frac {5 e^{-x}}{x^2}\right ) \, dx-\frac {1}{4} \int e^{-x} x^3 \, dx+\frac {5}{4} \int \frac {e^{-x}}{x^2} \, dx+7 \int e^{-x} x^2 \, dx-\frac {25}{2} \int e^{-x} x \, dx\\ &=\frac {e^{-x}}{4}-\frac {5 e^{-x}}{4 x}+\frac {5 x}{4}+\frac {25 e^{-x} x}{2}+\frac {25 x^2}{4}-7 e^{-x} x^2-\frac {x^3}{4}+\frac {1}{4} e^{-x} x^3+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)+\frac {1}{4} \int e^{-x} \, dx-\frac {3}{4} \int e^{-x} x^2 \, dx-\frac {5}{4} \int \frac {e^{-x}}{x^2} \, dx-\frac {5}{4} \int \frac {e^{-x}}{x} \, dx-\frac {25}{2} \int e^{-x} \, dx+14 \int e^{-x} x \, dx\\ &=\frac {25 e^{-x}}{2}+\frac {5 x}{4}-\frac {3 e^{-x} x}{2}+\frac {25 x^2}{4}-\frac {25}{4} e^{-x} x^2-\frac {x^3}{4}+\frac {1}{4} e^{-x} x^3-\frac {5 \text {Ei}(-x)}{4}+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)+\frac {5}{4} \int \frac {e^{-x}}{x} \, dx-\frac {3}{2} \int e^{-x} x \, dx+14 \int e^{-x} \, dx\\ &=-\frac {3 e^{-x}}{2}+\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {25}{4} e^{-x} x^2-\frac {x^3}{4}+\frac {1}{4} e^{-x} x^3+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)-\frac {3}{2} \int e^{-x} \, dx\\ &=\frac {5 x}{4}+\frac {25 x^2}{4}-\frac {25}{4} e^{-x} x^2-\frac {x^3}{4}+\frac {1}{4} e^{-x} x^3+\frac {5 e^{-x} \log (x)}{4 x}+\frac {1}{4} e^{-2 x} x \log (x)-\frac {1}{4} e^{-x} x \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.13, size = 52, normalized size = 1.49 \begin {gather*} \frac {1}{4} \left (x \left (5+25 x+e^{-x} (-25+x) x-x^2\right )+\frac {e^{-2 x} \left (x^2-e^x \left (-5+x^2\right )\right ) \log (x)}{x}\right ) \end {gather*}

Integrate[(x^2 + E^(2*x)*(5*x^2 + 50*x^3 - 3*x^4) + E^x*(5 - x^2 - 50*x^3 + 28*x^4 - x^5) + (x^2 - 2*x^3 + E^x

(x*(5 + 25*x + ((-25 + x)*x)/E^x - x^2) + ((x^2 - E^x*(-5 + x^2))*Log[x])/(E^(2*x)*x))/4

________________________________________________________________________________________

fricas [A] time = 0.86, size = 59, normalized size = 1.69 \begin {gather*} -\frac {{\left ({\left (x^{4} - 25 \, x^{3} - 5 \, x^{2}\right )} e^{\left (2 \, x\right )} - {\left (x^{4} - 25 \, x^{3}\right )} e^{x} - {\left (x^{2} - {\left (x^{2} - 5\right )} e^{x}\right )} \log \relax (x)\right )} e^{\left (-2 \, x\right )}}{4 \, x} \end {gather*}

integrate(1/4*(((x^3-x^2-5*x-5)*exp(x)-2*x^3+x^2)*log(x)+(-3*x^4+50*x^3+5*x^2)*exp(x)^2+(-x^5+28*x^4-50*x^3-x^

-1/4*((x^4 - 25*x^3 - 5*x^2)*e^(2*x) - (x^4 - 25*x^3)*e^x - (x^2 - (x^2 - 5)*e^x)*log(x))*e^(-2*x)/x

________________________________________________________________________________________

giac [B] time = 0.25, size = 67, normalized size = 1.91 \begin {gather*} \frac {x^{4} e^{\left (-x\right )} - x^{4} - 25 \, x^{3} e^{\left (-x\right )} - x^{2} e^{\left (-x\right )} \log \relax (x) + x^{2} e^{\left (-2 \, x\right )} \log \relax (x) + 25 \, x^{3} + 5 \, x^{2} + 5 \, e^{\left (-x\right )} \log \relax (x)}{4 \, x} \end {gather*}

integrate(1/4*(((x^3-x^2-5*x-5)*exp(x)-2*x^3+x^2)*log(x)+(-3*x^4+50*x^3+5*x^2)*exp(x)^2+(-x^5+28*x^4-50*x^3-x^

1/4*(x^4*e^(-x) - x^4 - 25*x^3*e^(-x) - x^2*e^(-x)*log(x) + x^2*e^(-2*x)*log(x) + 25*x^3 + 5*x^2 + 5*e^(-x)*lo

________________________________________________________________________________________


method	result	size

risch	$-\frac {\left ({\mathrm e}^{x} x^{2}-x^{2}-5 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-2 x} \ln \relax (x )}{4 x}-\frac {x \left ({\mathrm e}^{x} x^{2}-x^{2}-25 \,{\mathrm e}^{x} x +25 x -5 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{4}$	$60$

int(1/4*(((x^3-x^2-5*x-5)*exp(x)-2*x^3+x^2)*ln(x)+(-3*x^4+50*x^3+5*x^2)*exp(x)^2+(-x^5+28*x^4-50*x^3-x^2+5)*ex

-1/4*(exp(x)*x^2-x^2-5*exp(x))/x*exp(-2*x)*ln(x)-1/4*x*(exp(x)*x^2-x^2-25*exp(x)*x+25*x-5*exp(x))*exp(-x)

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{4} \, x^{3} + \frac {25}{4} \, x^{2} + \frac {1}{4} \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} - 7 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac {25}{2} \, {\left (x + 1\right )} e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \log \relax (x) + \frac {5}{4} \, x - \frac {2 \, {\left (x^{2} - 5\right )} e^{\left (-x\right )} \log \relax (x) - {\left (2 \, x^{2} + x\right )} e^{\left (-2 \, x\right )} \log \relax (x)}{8 \, x} + \frac {1}{8} \, {\rm Ei}\left (-2 \, x\right ) + \frac {1}{4} \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {5}{4} \, \Gamma \left (-1, x\right ) - \frac {1}{8} \, \int \frac {{\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )}}{x}\,{d x} + \frac {1}{4} \, \int \frac {{\left (x^{2} - 5\right )} e^{\left (-x\right )}}{x^{2}}\,{d x} \end {gather*}

integrate(1/4*(((x^3-x^2-5*x-5)*exp(x)-2*x^3+x^2)*log(x)+(-3*x^4+50*x^3+5*x^2)*exp(x)^2+(-x^5+28*x^4-50*x^3-x^

-1/4*x^3 + 25/4*x^2 + 1/4*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) - 7*(x^2 + 2*x + 2)*e^(-x) + 25/2*(x + 1)*e^(-x) - 1/

8*e^(-2*x)*log(x) + 5/4*x - 1/8*(2*(x^2 - 5)*e^(-x)*log(x) - (2*x^2 + x)*e^(-2*x)*log(x))/x + 1/8*Ei(-2*x) + 1

/4*e^(-x) - 1/8*e^(-2*x) - 5/4*gamma(-1, x) - 1/8*integrate((2*x + 1)*e^(-2*x)/x, x) + 1/4*integrate((x^2 - 5)

________________________________________________________________________________________

mupad [B] time = 5.79, size = 140, normalized size = 4.00 \begin {gather*} \frac {x\,\left (-x^2+25\,x+5\right )}{4}-\frac {5\,{\mathrm {e}}^{-x}-x\,{\mathrm {e}}^{-x}+25\,x^3\,{\mathrm {e}}^{-x}-x^4\,{\mathrm {e}}^{-x}}{4\,x}-\frac {{\mathrm {e}}^{-2\,x}}{8}+\frac {5\,x\,{\mathrm {e}}^{-x}-x^2\,{\mathrm {e}}^{-x}-x^3\,{\mathrm {e}}^{-x}\,\ln \relax (x)+5\,x\,{\mathrm {e}}^{-x}\,\ln \relax (x)}{4\,x^2}-\frac {{\mathrm {e}}^{-2\,x}\,\ln \relax (x)}{8}+\frac {\frac {x\,{\mathrm {e}}^{-2\,x}}{4}+\frac {x^2\,{\mathrm {e}}^{-2\,x}\,\ln \relax (x)}{2}+\frac {x\,{\mathrm {e}}^{-2\,x}\,\ln \relax (x)}{4}}{2\,x} \end {gather*}

int(-(exp(-2*x)*((log(x)*(exp(x)*(5*x + x^2 - x^3 + 5) - x^2 + 2*x^3))/4 - (exp(2*x)*(5*x^2 + 50*x^3 - 3*x^4))

/4 + (exp(x)*(x^2 + 50*x^3 - 28*x^4 + x^5 - 5))/4 - x^2/4))/x^2,x)

(x*(25*x - x^2 + 5))/4 - (5*exp(-x) - x*exp(-x) + 25*x^3*exp(-x) - x^4*exp(-x))/(4*x) - exp(-2*x)/8 + (5*x*exp

(-x) - x^2*exp(-x) - x^3*exp(-x)*log(x) + 5*x*exp(-x)*log(x))/(4*x^2) - (exp(-2*x)*log(x))/8 + ((x*exp(-2*x))/

4 + (x^2*exp(-2*x)*log(x))/2 + (x*exp(-2*x)*log(x))/4)/(2*x)

________________________________________________________________________________________

sympy [B] time = 0.38, size = 60, normalized size = 1.71 \begin {gather*} - \frac {x^{3}}{4} + \frac {25 x^{2}}{4} + \frac {5 x}{4} + \frac {4 x^{2} e^{- 2 x} \log {\relax (x )} + \left (4 x^{4} - 100 x^{3} - 4 x^{2} \log {\relax (x )} + 20 \log {\relax (x )}\right ) e^{- x}}{16 x} \end {gather*}

integrate(1/4*(((x**3-x**2-5*x-5)*exp(x)-2*x**3+x**2)*ln(x)+(-3*x**4+50*x**3+5*x**2)*exp(x)**2+(-x**5+28*x**4-

-x**3/4 + 25*x**2/4 + 5*x/4 + (4*x**2*exp(-2*x)*log(x) + (4*x**4 - 100*x**3 - 4*x**2*log(x) + 20*log(x))*exp(-

________________________________________________________________________________________

3.84.75 \(\int \frac {e^{-2 x} (x^2+e^{2 x} (5 x^2+50 x^3-3 x^4)+e^x (5-x^2-50 x^3+28 x^4-x^5)+(x^2-2 x^3+e^x (-5-5 x-x^2+x^3)) \log (x))}{4 x^2} \, dx\)