∫ (25−5x−5e x ______ log (3) x)2((−5+3x) log(3)+e x ______ log (3) (2x2+3x log(3))) _________________________________________________________________________________ (−5+x) log(3)+e x _____

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Rubi [B] time = 0.88, antiderivative size = 278, normalized size of antiderivative = 13.24, number of steps used = 25, number of rules used = 6, integrand size = 69, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.087, Rules used = {6688, 12, 6742, 2196, 2176, 2194} \begin {gather*} 25 x^3+50 x^3 e^{\frac {x}{\log (3)}}+25 x^3 e^{\frac {2 x}{\log (3)}}-250 x^2+\frac {25}{2} x^2 \log (27) e^{\frac {2 x}{\log (3)}}-50 x^2 (5-\log (27)) e^{\frac {x}{\log (3)}}-150 x^2 \log (3) e^{\frac {x}{\log (3)}}-\frac {75}{2} x^2 \log (3) e^{\frac {2 x}{\log (3)}}+625 x-300 \log ^3(3) e^{\frac {x}{\log (3)}}-\frac {75}{4} \log ^3(3) e^{\frac {2 x}{\log (3)}}+300 x \log ^2(3) e^{\frac {x}{\log (3)}}+\frac {75}{2} x \log ^2(3) e^{\frac {2 x}{\log (3)}}+\frac {25}{4} \log ^2(3) \log (27) e^{\frac {2 x}{\log (3)}}-100 \log ^2(3) (5-\log (27)) e^{\frac {x}{\log (3)}}-125 x \log (81) e^{\frac {x}{\log (3)}}-\frac {25}{2} x \log (3) \log (27) e^{\frac {2 x}{\log (3)}}+100 x \log (3) (5-\log (27)) e^{\frac {x}{\log (3)}}+125 \log (3) \log (81) e^{\frac {x}{\log (3)}} \end {gather*}

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

625*x - 250*x^2 + 25*x^3 + 50*E^(x/Log[3])*x^3 + 25*E^((2*x)/Log[3])*x^3 - 150*E^(x/Log[3])*x^2*Log[3] - (75*E

^((2*x)/Log[3])*x^2*Log[3])/2 + 300*E^(x/Log[3])*x*Log[3]^2 + (75*E^((2*x)/Log[3])*x*Log[3]^2)/2 - 300*E^(x/Lo

g[3])*Log[3]^3 - (75*E^((2*x)/Log[3])*Log[3]^3)/4 - 50*E^(x/Log[3])*x^2*(5 - Log[27]) + 100*E^(x/Log[3])*x*Log

[3]*(5 - Log[27]) - 100*E^(x/Log[3])*Log[3]^2*(5 - Log[27]) + (25*E^((2*x)/Log[3])*x^2*Log[27])/2 - (25*E^((2*

x)/Log[3])*x*Log[3]*Log[27])/2 + (25*E^((2*x)/Log[3])*Log[3]^2*Log[27])/4 - 125*E^(x/Log[3])*x*Log[81] + 125*E

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[((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_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (5-x-e^{\frac {x}{\log (3)}} x\right ) \left (-((-5+3 x) \log (3))-e^{\frac {x}{\log (3)}} x (2 x+\log (27))\right )}{\log (3)} \, dx\\ &=\frac {25 \int \left (5-x-e^{\frac {x}{\log (3)}} x\right ) \left (-((-5+3 x) \log (3))-e^{\frac {x}{\log (3)}} x (2 x+\log (27))\right ) \, dx}{\log (3)}\\ &=\frac {25 \int \left (\left (25-20 x+3 x^2\right ) \log (3)+e^{\frac {2 x}{\log (3)}} x^2 (2 x+\log (27))+e^{\frac {x}{\log (3)}} x \left (2 x^2-2 x (5-\log (27))-5 \log (81)\right )\right ) \, dx}{\log (3)}\\ &=25 \int \left (25-20 x+3 x^2\right ) \, dx+\frac {25 \int e^{\frac {2 x}{\log (3)}} x^2 (2 x+\log (27)) \, dx}{\log (3)}+\frac {25 \int e^{\frac {x}{\log (3)}} x \left (2 x^2-2 x (5-\log (27))-5 \log (81)\right ) \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+\frac {25 \int \left (2 e^{\frac {2 x}{\log (3)}} x^3+e^{\frac {2 x}{\log (3)}} x^2 \log (27)\right ) \, dx}{\log (3)}+\frac {25 \int \left (2 e^{\frac {x}{\log (3)}} x^3+2 e^{\frac {x}{\log (3)}} x^2 (-5+\log (27))-5 e^{\frac {x}{\log (3)}} x \log (81)\right ) \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+\frac {50 \int e^{\frac {x}{\log (3)}} x^3 \, dx}{\log (3)}+\frac {50 \int e^{\frac {2 x}{\log (3)}} x^3 \, dx}{\log (3)}-\frac {(50 (5-\log (27))) \int e^{\frac {x}{\log (3)}} x^2 \, dx}{\log (3)}+\frac {(25 \log (27)) \int e^{\frac {2 x}{\log (3)}} x^2 \, dx}{\log (3)}-\frac {(125 \log (81)) \int e^{\frac {x}{\log (3)}} x \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)-75 \int e^{\frac {2 x}{\log (3)}} x^2 \, dx-150 \int e^{\frac {x}{\log (3)}} x^2 \, dx+(100 (5-\log (27))) \int e^{\frac {x}{\log (3)}} x \, dx-(25 \log (27)) \int e^{\frac {2 x}{\log (3)}} x \, dx+(125 \log (81)) \int e^{\frac {x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)+(75 \log (3)) \int e^{\frac {2 x}{\log (3)}} x \, dx+(300 \log (3)) \int e^{\frac {x}{\log (3)}} x \, dx-(100 \log (3) (5-\log (27))) \int e^{\frac {x}{\log (3)}} \, dx+\frac {1}{2} (25 \log (3) \log (27)) \int e^{\frac {2 x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)+300 e^{\frac {x}{\log (3)}} x \log ^2(3)+\frac {75}{2} e^{\frac {2 x}{\log (3)}} x \log ^2(3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))-100 e^{\frac {x}{\log (3)}} \log ^2(3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)+\frac {25}{4} e^{\frac {2 x}{\log (3)}} \log ^2(3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)-\frac {1}{2} \left (75 \log ^2(3)\right ) \int e^{\frac {2 x}{\log (3)}} \, dx-\left (300 \log ^2(3)\right ) \int e^{\frac {x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)+300 e^{\frac {x}{\log (3)}} x \log ^2(3)+\frac {75}{2} e^{\frac {2 x}{\log (3)}} x \log ^2(3)-300 e^{\frac {x}{\log (3)}} \log ^3(3)-\frac {75}{4} e^{\frac {2 x}{\log (3)}} \log ^3(3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))-100 e^{\frac {x}{\log (3)}} \log ^2(3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)+\frac {25}{4} e^{\frac {2 x}{\log (3)}} \log ^2(3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)\\ \end {aligned} \end {gather*}

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Mathematica [B] time = 0.20, size = 82, normalized size = 3.90 \begin {gather*} 25 \left (25 x-10 x^2+x^3+e^{\frac {2 x}{\log (3)}} x^3+e^{\frac {x}{\log (3)}} \left (-10 x^2+2 x^3-6 \log ^3(3)+x \left (6 \log ^2(3)-\log (9) \log (27)\right )+\log (27) \log (243)+\log ^2(3) (-15+\log (729))\right )\right ) \end {gather*}

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

25*(25*x - 10*x^2 + x^3 + E^((2*x)/Log[3])*x^3 + E^(x/Log[3])*(-10*x^2 + 2*x^3 - 6*Log[3]^3 + x*(6*Log[3]^2 -

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fricas [B] time = 0.68, size = 45, normalized size = 2.14 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 25 \, x^{3} - 250 \, x^{2} + 50 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\frac {x}{\log \relax (3)}} + 625 \, x \end {gather*}

integrate(((3*x*log(3)+2*x^2)*exp(x/log(3))+(3*x-5)*log(3))*(-5*x*exp(x/log(3))-5*x+25)^2/(x*log(3)*exp(x/log(

25*x^3*e^(2*x/log(3)) + 25*x^3 - 250*x^2 + 50*(x^3 - 5*x^2)*e^(x/log(3)) + 625*x

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giac [B] time = 0.14, size = 51, normalized size = 2.43 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 50 \, x^{3} e^{\frac {x}{\log \relax (3)}} + 25 \, x^{3} - 250 \, x^{2} e^{\frac {x}{\log \relax (3)}} - 250 \, x^{2} + 625 \, x \end {gather*}

integrate(((3*x*log(3)+2*x^2)*exp(x/log(3))+(3*x-5)*log(3))*(-5*x*exp(x/log(3))-5*x+25)^2/(x*log(3)*exp(x/log(

25*x^3*e^(2*x/log(3)) + 50*x^3*e^(x/log(3)) + 25*x^3 - 250*x^2*e^(x/log(3)) - 250*x^2 + 625*x

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method	result	size

risch	$25 \,{\mathrm e}^{\frac {2 x}{\ln \relax (3)}} x^{3}+25 x^{3}-250 x^{2}+625 x +\left (50 x^{3}-250 x^{2}\right ) {\mathrm e}^{\frac {x}{\ln \relax (3)}}$	$47$
norman	$625 x -250 x^{2}+25 x^{3}-250 x^{2} {\mathrm e}^{\frac {x}{\ln \relax (3)}}+50 \,{\mathrm e}^{\frac {x}{\ln \relax (3)}} x^{3}+25 \,{\mathrm e}^{\frac {2 x}{\ln \relax (3)}} x^{3}$	$53$

int(((3*x*ln(3)+2*x^2)*exp(x/ln(3))+(3*x-5)*ln(3))*(-5*x*exp(x/ln(3))-5*x+25)^2/(x*ln(3)*exp(x/ln(3))+(x-5)*ln

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maxima [B] time = 0.51, size = 45, normalized size = 2.14 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 25 \, x^{3} - 250 \, x^{2} + 50 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\frac {x}{\log \relax (3)}} + 625 \, x \end {gather*}

integrate(((3*x*log(3)+2*x^2)*exp(x/log(3))+(3*x-5)*log(3))*(-5*x*exp(x/log(3))-5*x+25)^2/(x*log(3)*exp(x/log(

25*x^3*e^(2*x/log(3)) + 25*x^3 - 250*x^2 + 50*(x^3 - 5*x^2)*e^(x/log(3)) + 625*x

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mupad [B] time = 4.56, size = 17, normalized size = 0.81 \begin {gather*} 25\,x\,{\left (x+x\,{\mathrm {e}}^{\frac {x}{\ln \relax (3)}}-5\right )}^2 \end {gather*}

int(((log(3)*(3*x - 5) + exp(x/log(3))*(3*x*log(3) + 2*x^2))*(5*x + 5*x*exp(x/log(3)) - 25)^2)/(log(3)*(x - 5)

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sympy [B] time = 0.22, size = 42, normalized size = 2.00 \begin {gather*} 25 x^{3} e^{\frac {2 x}{\log {\relax (3 )}}} + 25 x^{3} - 250 x^{2} + 625 x + \left (50 x^{3} - 250 x^{2}\right ) e^{\frac {x}{\log {\relax (3 )}}} \end {gather*}

integrate(((3*x*ln(3)+2*x**2)*exp(x/ln(3))+(3*x-5)*ln(3))*(-5*x*exp(x/ln(3))-5*x+25)**2/(x*ln(3)*exp(x/ln(3))+

25*x**3*exp(2*x/log(3)) + 25*x**3 - 250*x**2 + 625*x + (50*x**3 - 250*x**2)*exp(x/log(3))

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3.76.18 \(\int \frac {(25-5 x-5 e^{\frac {x}{\log (3)}} x)^2 ((-5+3 x) \log (3)+e^{\frac {x}{\log (3)}} (2 x^2+3 x \log (3)))}{(-5+x) \log (3)+e^{\frac {x}{\log (3)}} x \log (3)} \, dx\)