3.97.62 \(\int \frac {10240000+e^{2+x} (-73728000 x-36864000 x^2)+e^{4+2 x} (12441600 x^3+6220800 x^4)+e^{6+3 x} (-699840 x^5-349920 x^6)+e^{8+4 x} (13122 x^7+6561 x^8)}{10240000} \, dx\)

Optimal. Leaf size=18 \[ x+\left (2-\frac {9}{80} e^{2+x} x^2\right )^4 \]

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Rubi [B]  time = 0.69, antiderivative size = 56, normalized size of antiderivative = 3.11, number of steps used = 58, number of rules used = 5, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} \frac {6561 e^{4 x+8} x^8}{40960000}-\frac {729 e^{3 x+6} x^6}{64000}+\frac {243}{800} e^{2 x+4} x^4-\frac {18}{5} e^{x+2} x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10240000 + E^(2 + x)*(-73728000*x - 36864000*x^2) + E^(4 + 2*x)*(12441600*x^3 + 6220800*x^4) + E^(6 + 3*x
)*(-699840*x^5 - 349920*x^6) + E^(8 + 4*x)*(13122*x^7 + 6561*x^8))/10240000,x]

[Out]

x - (18*E^(2 + x)*x^2)/5 + (243*E^(4 + 2*x)*x^4)/800 - (729*E^(6 + 3*x)*x^6)/64000 + (6561*E^(8 + 4*x)*x^8)/40
960000

Rule 12

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 \left (10240000+e^{2+x} \left (-73728000 x-36864000 x^2\right )+e^{4+2 x} \left (12441600 x^3+6220800 x^4\right )+e^{6+3 x} \left (-699840 x^5-349920 x^6\right )+e^{8+4 x} \left (13122 x^7+6561 x^8\right )\right ) \, dx}{10240000}\\ &=x+\frac {\int e^{2+x} \left (-73728000 x-36864000 x^2\right ) \, dx}{10240000}+\frac {\int e^{4+2 x} \left (12441600 x^3+6220800 x^4\right ) \, dx}{10240000}+\frac {\int e^{6+3 x} \left (-699840 x^5-349920 x^6\right ) \, dx}{10240000}+\frac {\int e^{8+4 x} \left (13122 x^7+6561 x^8\right ) \, dx}{10240000}\\ &=x+\frac {\int e^{2+x} (-73728000-36864000 x) x \, dx}{10240000}+\frac {\int e^{6+3 x} (-699840-349920 x) x^5 \, dx}{10240000}+\frac {\int e^{8+4 x} x^7 (13122+6561 x) \, dx}{10240000}+\frac {\int e^{4+2 x} x^3 (12441600+6220800 x) \, dx}{10240000}\\ &=x+\frac {\int \left (-73728000 e^{2+x} x-36864000 e^{2+x} x^2\right ) \, dx}{10240000}+\frac {\int \left (12441600 e^{4+2 x} x^3+6220800 e^{4+2 x} x^4\right ) \, dx}{10240000}+\frac {\int \left (-699840 e^{6+3 x} x^5-349920 e^{6+3 x} x^6\right ) \, dx}{10240000}+\frac {\int \left (13122 e^{8+4 x} x^7+6561 e^{8+4 x} x^8\right ) \, dx}{10240000}\\ &=x+\frac {6561 \int e^{8+4 x} x^8 \, dx}{10240000}+\frac {6561 \int e^{8+4 x} x^7 \, dx}{5120000}-\frac {2187 \int e^{6+3 x} x^6 \, dx}{64000}-\frac {2187 \int e^{6+3 x} x^5 \, dx}{32000}+\frac {243}{400} \int e^{4+2 x} x^4 \, dx+\frac {243}{200} \int e^{4+2 x} x^3 \, dx-\frac {18}{5} \int e^{2+x} x^2 \, dx-\frac {36}{5} \int e^{2+x} x \, dx\\ &=x-\frac {36}{5} e^{2+x} x-\frac {18}{5} e^{2+x} x^2+\frac {243}{400} e^{4+2 x} x^3+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^5}{32000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^7}{20480000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {6561 \int e^{8+4 x} x^7 \, dx}{5120000}-\frac {45927 \int e^{8+4 x} x^6 \, dx}{20480000}+\frac {2187 \int e^{6+3 x} x^5 \, dx}{32000}+\frac {729 \int e^{6+3 x} x^4 \, dx}{6400}-\frac {243}{200} \int e^{4+2 x} x^3 \, dx-\frac {729}{400} \int e^{4+2 x} x^2 \, dx+\frac {36}{5} \int e^{2+x} \, dx+\frac {36}{5} \int e^{2+x} x \, dx\\ &=\frac {36 e^{2+x}}{5}+x-\frac {18}{5} e^{2+x} x^2-\frac {729}{800} e^{4+2 x} x^2+\frac {243}{800} e^{4+2 x} x^4+\frac {243 e^{6+3 x} x^4}{6400}-\frac {729 e^{6+3 x} x^6}{64000}-\frac {45927 e^{8+4 x} x^6}{81920000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {45927 \int e^{8+4 x} x^6 \, dx}{20480000}+\frac {137781 \int e^{8+4 x} x^5 \, dx}{40960000}-\frac {729 \int e^{6+3 x} x^4 \, dx}{6400}-\frac {243 \int e^{6+3 x} x^3 \, dx}{1600}+\frac {729}{400} \int e^{4+2 x} x \, dx+\frac {729}{400} \int e^{4+2 x} x^2 \, dx-\frac {36}{5} \int e^{2+x} \, dx\\ &=x+\frac {729}{800} e^{4+2 x} x-\frac {18}{5} e^{2+x} x^2-\frac {81 e^{6+3 x} x^3}{1600}+\frac {243}{800} e^{4+2 x} x^4+\frac {137781 e^{8+4 x} x^5}{163840000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {137781 \int e^{8+4 x} x^5 \, dx}{40960000}-\frac {137781 \int e^{8+4 x} x^4 \, dx}{32768000}+\frac {243 \int e^{6+3 x} x^2 \, dx}{1600}+\frac {243 \int e^{6+3 x} x^3 \, dx}{1600}-\frac {729}{800} \int e^{4+2 x} \, dx-\frac {729}{400} \int e^{4+2 x} x \, dx\\ &=-\frac {729 e^{4+2 x}}{1600}+x-\frac {18}{5} e^{2+x} x^2+\frac {81 e^{6+3 x} x^2}{1600}+\frac {243}{800} e^{4+2 x} x^4-\frac {137781 e^{8+4 x} x^4}{131072000}-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {137781 \int e^{8+4 x} x^3 \, dx}{32768000}+\frac {137781 \int e^{8+4 x} x^4 \, dx}{32768000}-\frac {81}{800} \int e^{6+3 x} x \, dx-\frac {243 \int e^{6+3 x} x^2 \, dx}{1600}+\frac {729}{800} \int e^{4+2 x} \, dx\\ &=x-\frac {27}{800} e^{6+3 x} x-\frac {18}{5} e^{2+x} x^2+\frac {137781 e^{8+4 x} x^3}{131072000}+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {413343 \int e^{8+4 x} x^2 \, dx}{131072000}-\frac {137781 \int e^{8+4 x} x^3 \, dx}{32768000}+\frac {27}{800} \int e^{6+3 x} \, dx+\frac {81}{800} \int e^{6+3 x} x \, dx\\ &=\frac {9}{800} e^{6+3 x}+x-\frac {18}{5} e^{2+x} x^2-\frac {413343 e^{8+4 x} x^2}{524288000}+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {413343 \int e^{8+4 x} x \, dx}{262144000}+\frac {413343 \int e^{8+4 x} x^2 \, dx}{131072000}-\frac {27}{800} \int e^{6+3 x} \, dx\\ &=x+\frac {413343 e^{8+4 x} x}{1048576000}-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}-\frac {413343 \int e^{8+4 x} \, dx}{1048576000}-\frac {413343 \int e^{8+4 x} x \, dx}{262144000}\\ &=-\frac {413343 e^{8+4 x}}{4194304000}+x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}+\frac {413343 \int e^{8+4 x} \, dx}{1048576000}\\ &=x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 56, normalized size = 3.11 \begin {gather*} x-\frac {18}{5} e^{2+x} x^2+\frac {243}{800} e^{4+2 x} x^4-\frac {729 e^{6+3 x} x^6}{64000}+\frac {6561 e^{8+4 x} x^8}{40960000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10240000 + E^(2 + x)*(-73728000*x - 36864000*x^2) + E^(4 + 2*x)*(12441600*x^3 + 6220800*x^4) + E^(6
+ 3*x)*(-699840*x^5 - 349920*x^6) + E^(8 + 4*x)*(13122*x^7 + 6561*x^8))/10240000,x]

[Out]

x - (18*E^(2 + x)*x^2)/5 + (243*E^(4 + 2*x)*x^4)/800 - (729*E^(6 + 3*x)*x^6)/64000 + (6561*E^(8 + 4*x)*x^8)/40
960000

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fricas [B]  time = 0.46, size = 44, normalized size = 2.44 \begin {gather*} \frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, x^{2} e^{\left (x + 2\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10240000*(6561*x^8+13122*x^7)*exp(2)^4*exp(x)^4+1/10240000*(-349920*x^6-699840*x^5)*exp(2)^3*exp(x
)^3+1/10240000*(6220800*x^4+12441600*x^3)*exp(2)^2*exp(x)^2+1/10240000*(-36864000*x^2-73728000*x)*exp(2)*exp(x
)+1,x, algorithm="fricas")

[Out]

6561/40960000*x^8*e^(4*x + 8) - 729/64000*x^6*e^(3*x + 6) + 243/800*x^4*e^(2*x + 4) - 18/5*x^2*e^(x + 2) + x

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giac [B]  time = 0.15, size = 44, normalized size = 2.44 \begin {gather*} \frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, x^{2} e^{\left (x + 2\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10240000*(6561*x^8+13122*x^7)*exp(2)^4*exp(x)^4+1/10240000*(-349920*x^6-699840*x^5)*exp(2)^3*exp(x
)^3+1/10240000*(6220800*x^4+12441600*x^3)*exp(2)^2*exp(x)^2+1/10240000*(-36864000*x^2-73728000*x)*exp(2)*exp(x
)+1,x, algorithm="giac")

[Out]

6561/40960000*x^8*e^(4*x + 8) - 729/64000*x^6*e^(3*x + 6) + 243/800*x^4*e^(2*x + 4) - 18/5*x^2*e^(x + 2) + x

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maple [B]  time = 0.04, size = 45, normalized size = 2.50

method	result	size
risch	\(x -\frac {18 x^{2} {\mathrm e}^{2+x}}{5}+\frac {243 x^{4} {\mathrm e}^{2 x +4}}{800}-\frac {729 x^{6} {\mathrm e}^{6+3 x}}{64000}+\frac {6561 x^{8} {\mathrm e}^{4 x +8}}{40960000}\)	\(45\)
default	\(x -\frac {18 x^{2} {\mathrm e}^{2} {\mathrm e}^{x}}{5}+\frac {243 \,{\mathrm e}^{4} {\mathrm e}^{2 x} x^{4}}{800}-\frac {729 \,{\mathrm e}^{6} {\mathrm e}^{3 x} x^{6}}{64000}+\frac {6561 \,{\mathrm e}^{8} {\mathrm e}^{4 x} x^{8}}{40960000}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10240000*(6561*x^8+13122*x^7)*exp(2)^4*exp(x)^4+1/10240000*(-349920*x^6-699840*x^5)*exp(2)^3*exp(x)^3+1/
10240000*(6220800*x^4+12441600*x^3)*exp(2)^2*exp(x)^2+1/10240000*(-36864000*x^2-73728000*x)*exp(2)*exp(x)+1,x,
method=_RETURNVERBOSE)

[Out]

x-18/5*x^2*exp(2+x)+243/800*x^4*exp(2*x+4)-729/64000*x^6*exp(6+3*x)+6561/40960000*x^8*exp(4*x+8)

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maxima [B]  time = 0.37, size = 68, normalized size = 3.78 \begin {gather*} \frac {6561}{40960000} \, x^{8} e^{\left (4 \, x + 8\right )} - \frac {729}{64000} \, x^{6} e^{\left (3 \, x + 6\right )} + \frac {243}{800} \, x^{4} e^{\left (2 \, x + 4\right )} - \frac {18}{5} \, {\left (x^{2} e^{2} - 2 \, x e^{2} + 2 \, e^{2}\right )} e^{x} - \frac {36}{5} \, {\left (x e^{2} - e^{2}\right )} e^{x} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10240000*(6561*x^8+13122*x^7)*exp(2)^4*exp(x)^4+1/10240000*(-349920*x^6-699840*x^5)*exp(2)^3*exp(x
)^3+1/10240000*(6220800*x^4+12441600*x^3)*exp(2)^2*exp(x)^2+1/10240000*(-36864000*x^2-73728000*x)*exp(2)*exp(x
)+1,x, algorithm="maxima")

[Out]

6561/40960000*x^8*e^(4*x + 8) - 729/64000*x^6*e^(3*x + 6) + 243/800*x^4*e^(2*x + 4) - 18/5*(x^2*e^2 - 2*x*e^2
+ 2*e^2)*e^x - 36/5*(x*e^2 - e^2)*e^x + x

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mupad [B]  time = 5.61, size = 44, normalized size = 2.44 \begin {gather*} x-\frac {18\,x^2\,{\mathrm {e}}^{x+2}}{5}+\frac {243\,x^4\,{\mathrm {e}}^{2\,x+4}}{800}-\frac {729\,x^6\,{\mathrm {e}}^{3\,x+6}}{64000}+\frac {6561\,x^8\,{\mathrm {e}}^{4\,x+8}}{40960000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*exp(8)*(13122*x^7 + 6561*x^8))/10240000 - (exp(2)*exp(x)*(73728000*x + 36864000*x^2))/10240000 -
(exp(3*x)*exp(6)*(699840*x^5 + 349920*x^6))/10240000 + (exp(2*x)*exp(4)*(12441600*x^3 + 6220800*x^4))/1024000
0 + 1,x)

[Out]

x - (18*x^2*exp(x + 2))/5 + (243*x^4*exp(2*x + 4))/800 - (729*x^6*exp(3*x + 6))/64000 + (6561*x^8*exp(4*x + 8)
)/40960000

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sympy [B]  time = 0.22, size = 60, normalized size = 3.33 \begin {gather*} \frac {6561 x^{8} e^{8} e^{4 x}}{40960000} - \frac {729 x^{6} e^{6} e^{3 x}}{64000} + \frac {243 x^{4} e^{4} e^{2 x}}{800} - \frac {18 x^{2} e^{2} e^{x}}{5} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10240000*(6561*x**8+13122*x**7)*exp(2)**4*exp(x)**4+1/10240000*(-349920*x**6-699840*x**5)*exp(2)**
3*exp(x)**3+1/10240000*(6220800*x**4+12441600*x**3)*exp(2)**2*exp(x)**2+1/10240000*(-36864000*x**2-73728000*x)
*exp(2)*exp(x)+1,x)

[Out]

6561*x**8*exp(8)*exp(4*x)/40960000 - 729*x**6*exp(6)*exp(3*x)/64000 + 243*x**4*exp(4)*exp(2*x)/800 - 18*x**2*e
xp(2)*exp(x)/5 + x

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