3.48.65 \(\int \frac {e^{-4 x} (288 x-576 x^2+(-2880 x+5760 x^2-1152 x^3+1152 x^4) \log (5)+(7200 x-14400 x^2+5760 x^3-5760 x^4+864 x^5-576 x^6) \log ^2(5))}{25 \log ^2(5)} \, dx\)

Optimal. Leaf size=25 \[ \frac {144}{25} e^{-4 x} x^2 \left (5+x^2-\frac {1}{\log (5)}\right )^2 \]

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Rubi [B]  time = 0.53, antiderivative size = 83, normalized size of antiderivative = 3.32, number of steps used = 53, number of rules used = 4, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 2196, 2176, 2194} \begin {gather*} \frac {144}{25} e^{-4 x} x^6+\frac {288}{5} e^{-4 x} x^4-\frac {288 e^{-4 x} x^4}{25 \log (5)}+144 e^{-4 x} x^2+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(288*x - 576*x^2 + (-2880*x + 5760*x^2 - 1152*x^3 + 1152*x^4)*Log[5] + (7200*x - 14400*x^2 + 5760*x^3 - 57
60*x^4 + 864*x^5 - 576*x^6)*Log[5]^2)/(25*E^(4*x)*Log[5]^2),x]

[Out]

(144*x^2)/E^(4*x) + (288*x^4)/(5*E^(4*x)) + (144*x^6)/(25*E^(4*x)) + (144*x^2)/(25*E^(4*x)*Log[5]^2) - (288*x^
2)/(5*E^(4*x)*Log[5]) - (288*x^4)/(25*E^(4*x)*Log[5])

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-4 x} \left (288 x-576 x^2+\left (-2880 x+5760 x^2-1152 x^3+1152 x^4\right ) \log (5)+\left (7200 x-14400 x^2+5760 x^3-5760 x^4+864 x^5-576 x^6\right ) \log ^2(5)\right ) \, dx}{25 \log ^2(5)}\\ &=\frac {\int \left (288 e^{-4 x} x-576 e^{-4 x} x^2+576 e^{-4 x} x \left (-5+10 x-2 x^2+2 x^3\right ) \log (5)-288 e^{-4 x} x \left (-25+50 x-20 x^2+20 x^3-3 x^4+2 x^5\right ) \log ^2(5)\right ) \, dx}{25 \log ^2(5)}\\ &=-\left (\frac {288}{25} \int e^{-4 x} x \left (-25+50 x-20 x^2+20 x^3-3 x^4+2 x^5\right ) \, dx\right )+\frac {288 \int e^{-4 x} x \, dx}{25 \log ^2(5)}-\frac {576 \int e^{-4 x} x^2 \, dx}{25 \log ^2(5)}+\frac {576 \int e^{-4 x} x \left (-5+10 x-2 x^2+2 x^3\right ) \, dx}{25 \log (5)}\\ &=-\frac {72 e^{-4 x} x}{25 \log ^2(5)}+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {288}{25} \int \left (-25 e^{-4 x} x+50 e^{-4 x} x^2-20 e^{-4 x} x^3+20 e^{-4 x} x^4-3 e^{-4 x} x^5+2 e^{-4 x} x^6\right ) \, dx+\frac {72 \int e^{-4 x} \, dx}{25 \log ^2(5)}-\frac {288 \int e^{-4 x} x \, dx}{25 \log ^2(5)}+\frac {576 \int \left (-5 e^{-4 x} x+10 e^{-4 x} x^2-2 e^{-4 x} x^3+2 e^{-4 x} x^4\right ) \, dx}{25 \log (5)}\\ &=-\frac {18 e^{-4 x}}{25 \log ^2(5)}+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {576}{25} \int e^{-4 x} x^6 \, dx+\frac {864}{25} \int e^{-4 x} x^5 \, dx+\frac {1152}{5} \int e^{-4 x} x^3 \, dx-\frac {1152}{5} \int e^{-4 x} x^4 \, dx+288 \int e^{-4 x} x \, dx-576 \int e^{-4 x} x^2 \, dx-\frac {72 \int e^{-4 x} \, dx}{25 \log ^2(5)}-\frac {1152 \int e^{-4 x} x^3 \, dx}{25 \log (5)}+\frac {1152 \int e^{-4 x} x^4 \, dx}{25 \log (5)}-\frac {576 \int e^{-4 x} x \, dx}{5 \log (5)}+\frac {1152 \int e^{-4 x} x^2 \, dx}{5 \log (5)}\\ &=-72 e^{-4 x} x+144 e^{-4 x} x^2-\frac {288}{5} e^{-4 x} x^3+\frac {288}{5} e^{-4 x} x^4-\frac {216}{25} e^{-4 x} x^5+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}+\frac {144 e^{-4 x} x}{5 \log (5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}+\frac {288 e^{-4 x} x^3}{25 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}-\frac {864}{25} \int e^{-4 x} x^5 \, dx+\frac {216}{5} \int e^{-4 x} x^4 \, dx+72 \int e^{-4 x} \, dx+\frac {864}{5} \int e^{-4 x} x^2 \, dx-\frac {1152}{5} \int e^{-4 x} x^3 \, dx-288 \int e^{-4 x} x \, dx-\frac {144 \int e^{-4 x} \, dx}{5 \log (5)}-\frac {864 \int e^{-4 x} x^2 \, dx}{25 \log (5)}+\frac {1152 \int e^{-4 x} x^3 \, dx}{25 \log (5)}+\frac {576 \int e^{-4 x} x \, dx}{5 \log (5)}\\ &=-18 e^{-4 x}+\frac {504}{5} e^{-4 x} x^2+\frac {234}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}+\frac {36 e^{-4 x}}{5 \log (5)}-\frac {1224 e^{-4 x} x^2}{25 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}+\frac {216}{5} \int e^{-4 x} x^3 \, dx-\frac {216}{5} \int e^{-4 x} x^4 \, dx-72 \int e^{-4 x} \, dx+\frac {432}{5} \int e^{-4 x} x \, dx-\frac {864}{5} \int e^{-4 x} x^2 \, dx-\frac {432 \int e^{-4 x} x \, dx}{25 \log (5)}+\frac {144 \int e^{-4 x} \, dx}{5 \log (5)}+\frac {864 \int e^{-4 x} x^2 \, dx}{25 \log (5)}\\ &=-\frac {108}{5} e^{-4 x} x+144 e^{-4 x} x^2-\frac {54}{5} e^{-4 x} x^3+\frac {288}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}+\frac {108 e^{-4 x} x}{25 \log (5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}+\frac {108}{5} \int e^{-4 x} \, dx+\frac {162}{5} \int e^{-4 x} x^2 \, dx-\frac {216}{5} \int e^{-4 x} x^3 \, dx-\frac {432}{5} \int e^{-4 x} x \, dx-\frac {108 \int e^{-4 x} \, dx}{25 \log (5)}+\frac {432 \int e^{-4 x} x \, dx}{25 \log (5)}\\ &=-\frac {27}{5} e^{-4 x}+\frac {1359}{10} e^{-4 x} x^2+\frac {288}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}+\frac {27 e^{-4 x}}{25 \log (5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}+\frac {81}{5} \int e^{-4 x} x \, dx-\frac {108}{5} \int e^{-4 x} \, dx-\frac {162}{5} \int e^{-4 x} x^2 \, dx+\frac {108 \int e^{-4 x} \, dx}{25 \log (5)}\\ &=-\frac {81}{20} e^{-4 x} x+144 e^{-4 x} x^2+\frac {288}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}+\frac {81}{20} \int e^{-4 x} \, dx-\frac {81}{5} \int e^{-4 x} x \, dx\\ &=-\frac {81}{80} e^{-4 x}+144 e^{-4 x} x^2+\frac {288}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}-\frac {81}{20} \int e^{-4 x} \, dx\\ &=144 e^{-4 x} x^2+\frac {288}{5} e^{-4 x} x^4+\frac {144}{25} e^{-4 x} x^6+\frac {144 e^{-4 x} x^2}{25 \log ^2(5)}-\frac {288 e^{-4 x} x^2}{5 \log (5)}-\frac {288 e^{-4 x} x^4}{25 \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 30, normalized size = 1.20 \begin {gather*} \frac {144 e^{-4 x} x^2 \left (-1+5 \log (5)+x^2 \log (5)\right )^2}{25 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(288*x - 576*x^2 + (-2880*x + 5760*x^2 - 1152*x^3 + 1152*x^4)*Log[5] + (7200*x - 14400*x^2 + 5760*x^
3 - 5760*x^4 + 864*x^5 - 576*x^6)*Log[5]^2)/(25*E^(4*x)*Log[5]^2),x]

[Out]

(144*x^2*(-1 + 5*Log[5] + x^2*Log[5])^2)/(25*E^(4*x)*Log[5]^2)

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fricas [B]  time = 0.78, size = 46, normalized size = 1.84 \begin {gather*} \frac {144 \, {\left ({\left (x^{6} + 10 \, x^{4} + 25 \, x^{2}\right )} \log \relax (5)^{2} + x^{2} - 2 \, {\left (x^{4} + 5 \, x^{2}\right )} \log \relax (5)\right )} e^{\left (-4 \, x\right )}}{25 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-576*x^6+864*x^5-5760*x^4+5760*x^3-14400*x^2+7200*x)*log(5)^2+(1152*x^4-1152*x^3+5760*x^2-288
0*x)*log(5)-576*x^2+288*x)/log(5)^2/exp(x)^4,x, algorithm="fricas")

[Out]

144/25*((x^6 + 10*x^4 + 25*x^2)*log(5)^2 + x^2 - 2*(x^4 + 5*x^2)*log(5))*e^(-4*x)/log(5)^2

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giac [B]  time = 0.20, size = 54, normalized size = 2.16 \begin {gather*} \frac {144 \, {\left (x^{6} \log \relax (5)^{2} + 10 \, x^{4} \log \relax (5)^{2} - 2 \, x^{4} \log \relax (5) + 25 \, x^{2} \log \relax (5)^{2} - 10 \, x^{2} \log \relax (5) + x^{2}\right )} e^{\left (-4 \, x\right )}}{25 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-576*x^6+864*x^5-5760*x^4+5760*x^3-14400*x^2+7200*x)*log(5)^2+(1152*x^4-1152*x^3+5760*x^2-288
0*x)*log(5)-576*x^2+288*x)/log(5)^2/exp(x)^4,x, algorithm="giac")

[Out]

144/25*(x^6*log(5)^2 + 10*x^4*log(5)^2 - 2*x^4*log(5) + 25*x^2*log(5)^2 - 10*x^2*log(5) + x^2)*e^(-4*x)/log(5)
^2

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maple [A]  time = 0.06, size = 28, normalized size = 1.12

method	result	size
gosper	\(\frac {144 x^{2} \left (x^{2} \ln \relax (5)+5 \ln \relax (5)-1\right )^{2} {\mathrm e}^{-4 x}}{25 \ln \relax (5)^{2}}\)	\(28\)
risch	\(\frac {\left (144 x^{6} \ln \relax (5)^{2}+1440 x^{4} \ln \relax (5)^{2}-288 x^{4} \ln \relax (5)+3600 x^{2} \ln \relax (5)^{2}-1440 x^{2} \ln \relax (5)+144 x^{2}\right ) {\mathrm e}^{-4 x}}{25 \ln \relax (5)^{2}}\)	\(58\)
default	\(\frac {144 \,{\mathrm e}^{-4 x} x^{2}-1440 \,{\mathrm e}^{-4 x} x^{2} \ln \relax (5)+3600 \,{\mathrm e}^{-4 x} x^{2} \ln \relax (5)^{2}-288 \,{\mathrm e}^{-4 x} x^{4} \ln \relax (5)+1440 \,{\mathrm e}^{-4 x} x^{4} \ln \relax (5)^{2}+144 \,{\mathrm e}^{-4 x} x^{6} \ln \relax (5)^{2}}{25 \ln \relax (5)^{2}}\)	\(78\)
meijerg	\(\frac {9 \left (28672 x^{6}+43008 x^{5}+53760 x^{4}+53760 x^{3}+40320 x^{2}+20160 x +5040\right ) {\mathrm e}^{-4 x}}{44800}-\frac {9 \left (6144 x^{5}+7680 x^{4}+7680 x^{3}+5760 x^{2}+2880 x +720\right ) {\mathrm e}^{-4 x}}{6400}+\frac {\left (-\frac {1152 \ln \relax (5)^{2}}{5}+\frac {1152 \ln \relax (5)}{25}\right ) \left (24-\frac {\left (1280 x^{4}+1280 x^{3}+960 x^{2}+480 x +120\right ) {\mathrm e}^{-4 x}}{5}\right )}{1024 \ln \relax (5)^{2}}+\frac {\left (\frac {1152 \ln \relax (5)^{2}}{5}-\frac {1152 \ln \relax (5)}{25}\right ) \left (6-\frac {\left (256 x^{3}+192 x^{2}+96 x +24\right ) {\mathrm e}^{-4 x}}{4}\right )}{256 \ln \relax (5)^{2}}+\frac {\left (-576 \ln \relax (5)^{2}+\frac {1152 \ln \relax (5)}{5}-\frac {576}{25}\right ) \left (2-\frac {\left (48 x^{2}+24 x +6\right ) {\mathrm e}^{-4 x}}{3}\right )}{64 \ln \relax (5)^{2}}+\frac {\left (288 \ln \relax (5)^{2}-\frac {576 \ln \relax (5)}{5}+\frac {288}{25}\right ) \left (1-\frac {\left (8 x +2\right ) {\mathrm e}^{-4 x}}{2}\right )}{16 \ln \relax (5)^{2}}\)	\(221\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*((-576*x^6+864*x^5-5760*x^4+5760*x^3-14400*x^2+7200*x)*ln(5)^2+(1152*x^4-1152*x^3+5760*x^2-2880*x)*ln
(5)-576*x^2+288*x)/ln(5)^2/exp(x)^4,x,method=_RETURNVERBOSE)

[Out]

144/25*x^2*(x^2*ln(5)+5*ln(5)-1)^2/ln(5)^2/exp(x)^4

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maxima [B]  time = 0.37, size = 280, normalized size = 11.20 \begin {gather*} \frac {9 \, {\left ({\left (256 \, x^{6} + 384 \, x^{5} + 480 \, x^{4} + 480 \, x^{3} + 360 \, x^{2} + 180 \, x + 45\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} - 3 \, {\left (128 \, x^{5} + 160 \, x^{4} + 160 \, x^{3} + 120 \, x^{2} + 60 \, x + 15\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} + 80 \, {\left (32 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 12 \, x + 3\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} - 80 \, {\left (32 \, x^{3} + 24 \, x^{2} + 12 \, x + 3\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} + 800 \, {\left (8 \, x^{2} + 4 \, x + 1\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} - 800 \, {\left (4 \, x + 1\right )} e^{\left (-4 \, x\right )} \log \relax (5)^{2} - 16 \, {\left (32 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 12 \, x + 3\right )} e^{\left (-4 \, x\right )} \log \relax (5) + 16 \, {\left (32 \, x^{3} + 24 \, x^{2} + 12 \, x + 3\right )} e^{\left (-4 \, x\right )} \log \relax (5) - 320 \, {\left (8 \, x^{2} + 4 \, x + 1\right )} e^{\left (-4 \, x\right )} \log \relax (5) + 320 \, {\left (4 \, x + 1\right )} e^{\left (-4 \, x\right )} \log \relax (5) + 32 \, {\left (8 \, x^{2} + 4 \, x + 1\right )} e^{\left (-4 \, x\right )} - 32 \, {\left (4 \, x + 1\right )} e^{\left (-4 \, x\right )}\right )}}{400 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-576*x^6+864*x^5-5760*x^4+5760*x^3-14400*x^2+7200*x)*log(5)^2+(1152*x^4-1152*x^3+5760*x^2-288
0*x)*log(5)-576*x^2+288*x)/log(5)^2/exp(x)^4,x, algorithm="maxima")

[Out]

9/400*((256*x^6 + 384*x^5 + 480*x^4 + 480*x^3 + 360*x^2 + 180*x + 45)*e^(-4*x)*log(5)^2 - 3*(128*x^5 + 160*x^4
+ 160*x^3 + 120*x^2 + 60*x + 15)*e^(-4*x)*log(5)^2 + 80*(32*x^4 + 32*x^3 + 24*x^2 + 12*x + 3)*e^(-4*x)*log(5)
^2 - 80*(32*x^3 + 24*x^2 + 12*x + 3)*e^(-4*x)*log(5)^2 + 800*(8*x^2 + 4*x + 1)*e^(-4*x)*log(5)^2 - 800*(4*x +
1)*e^(-4*x)*log(5)^2 - 16*(32*x^4 + 32*x^3 + 24*x^2 + 12*x + 3)*e^(-4*x)*log(5) + 16*(32*x^3 + 24*x^2 + 12*x +
3)*e^(-4*x)*log(5) - 320*(8*x^2 + 4*x + 1)*e^(-4*x)*log(5) + 320*(4*x + 1)*e^(-4*x)*log(5) + 32*(8*x^2 + 4*x
+ 1)*e^(-4*x) - 32*(4*x + 1)*e^(-4*x))/log(5)^2

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mupad [B]  time = 3.57, size = 50, normalized size = 2.00 \begin {gather*} \frac {144\,x^6\,{\mathrm {e}}^{-4\,x}}{25}+\frac {288\,x^4\,{\mathrm {e}}^{-4\,x}\,\left (5\,\ln \relax (5)-1\right )}{25\,\ln \relax (5)}+\frac {144\,x^2\,{\mathrm {e}}^{-4\,x}\,{\left (5\,\ln \relax (5)-1\right )}^2}{25\,{\ln \relax (5)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4*x)*((288*x)/25 + (log(5)^2*(7200*x - 14400*x^2 + 5760*x^3 - 5760*x^4 + 864*x^5 - 576*x^6))/25 - (l
og(5)*(2880*x - 5760*x^2 + 1152*x^3 - 1152*x^4))/25 - (576*x^2)/25))/log(5)^2,x)

[Out]

(144*x^6*exp(-4*x))/25 + (288*x^4*exp(-4*x)*(5*log(5) - 1))/(25*log(5)) + (144*x^2*exp(-4*x)*(5*log(5) - 1)^2)
/(25*log(5)^2)

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sympy [B]  time = 0.20, size = 63, normalized size = 2.52 \begin {gather*} \frac {\left (144 x^{6} \log {\relax (5 )}^{2} - 288 x^{4} \log {\relax (5 )} + 1440 x^{4} \log {\relax (5 )}^{2} - 1440 x^{2} \log {\relax (5 )} + 144 x^{2} + 3600 x^{2} \log {\relax (5 )}^{2}\right ) e^{- 4 x}}{25 \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-576*x**6+864*x**5-5760*x**4+5760*x**3-14400*x**2+7200*x)*ln(5)**2+(1152*x**4-1152*x**3+5760*
x**2-2880*x)*ln(5)-576*x**2+288*x)/ln(5)**2/exp(x)**4,x)

[Out]

(144*x**6*log(5)**2 - 288*x**4*log(5) + 1440*x**4*log(5)**2 - 1440*x**2*log(5) + 144*x**2 + 3600*x**2*log(5)**
2)*exp(-4*x)/(25*log(5)**2)

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