3.7.99 \(\int (4 x+24 x^2+26 x^3-30 x^4+6 x^5+e^{4-2 x} (2 x-2 x^2)+e^{2-x} (6 x+16 x^2-14 x^3+2 x^4)+(6 x+18 x^2-8 x^3+e^{2-x} (4 x-2 x^2)) \log (x)+2 x \log ^2(x)) \, dx\)

Optimal. Leaf size=24 \[ x^2 \left (1+e^{2-x}+x-(-2+x) x+\log (x)\right )^2 \]

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Rubi [B]  time = 0.43, antiderivative size = 112, normalized size of antiderivative = 4.67, number of steps used = 34, number of rules used = 8, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {1593, 2196, 2176, 2194, 2554, 14, 2305, 2304} \begin {gather*} x^6-6 x^5-2 e^{2-x} x^4+7 x^4-2 x^4 \log (x)+6 e^{2-x} x^3+6 x^3+6 x^3 \log (x)+e^{4-2 x} x^2+2 e^{2-x} x^2+x^2+x^2 \log ^2(x)+2 e^{2-x} x^2 \log (x)+2 x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4*x + 24*x^2 + 26*x^3 - 30*x^4 + 6*x^5 + E^(4 - 2*x)*(2*x - 2*x^2) + E^(2 - x)*(6*x + 16*x^2 - 14*x^3 + 2*
x^4) + (6*x + 18*x^2 - 8*x^3 + E^(2 - x)*(4*x - 2*x^2))*Log[x] + 2*x*Log[x]^2,x]

[Out]

x^2 + E^(4 - 2*x)*x^2 + 2*E^(2 - x)*x^2 + 6*x^3 + 6*E^(2 - x)*x^3 + 7*x^4 - 2*E^(2 - x)*x^4 - 6*x^5 + x^6 + 2*
x^2*Log[x] + 2*E^(2 - x)*x^2*Log[x] + 6*x^3*Log[x] - 2*x^4*Log[x] + x^2*Log[x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2554

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x^2+8 x^3+\frac {13 x^4}{2}-6 x^5+x^6+2 \int x \log ^2(x) \, dx+\int e^{4-2 x} \left (2 x-2 x^2\right ) \, dx+\int e^{2-x} \left (6 x+16 x^2-14 x^3+2 x^4\right ) \, dx+\int \left (6 x+18 x^2-8 x^3+e^{2-x} \left (4 x-2 x^2\right )\right ) \log (x) \, dx\\ &=2 x^2+8 x^3+\frac {13 x^4}{2}-6 x^5+x^6+3 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)-2 \int x \log (x) \, dx+\int e^{4-2 x} (2-2 x) x \, dx-\int x \left (3+2 e^{2-x}+6 x-2 x^2\right ) \, dx+\int \left (6 e^{2-x} x+16 e^{2-x} x^2-14 e^{2-x} x^3+2 e^{2-x} x^4\right ) \, dx\\ &=\frac {5 x^2}{2}+8 x^3+\frac {13 x^4}{2}-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)+2 \int e^{2-x} x^4 \, dx+6 \int e^{2-x} x \, dx-14 \int e^{2-x} x^3 \, dx+16 \int e^{2-x} x^2 \, dx+\int \left (2 e^{4-2 x} x-2 e^{4-2 x} x^2\right ) \, dx-\int \left (2 e^{2-x} x-x \left (-3-6 x+2 x^2\right )\right ) \, dx\\ &=-6 e^{2-x} x+\frac {5 x^2}{2}-16 e^{2-x} x^2+8 x^3+14 e^{2-x} x^3+\frac {13 x^4}{2}-2 e^{2-x} x^4-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)+2 \int e^{4-2 x} x \, dx-2 \int e^{2-x} x \, dx-2 \int e^{4-2 x} x^2 \, dx+6 \int e^{2-x} \, dx+8 \int e^{2-x} x^3 \, dx+32 \int e^{2-x} x \, dx-42 \int e^{2-x} x^2 \, dx+\int x \left (-3-6 x+2 x^2\right ) \, dx\\ &=-6 e^{2-x}-e^{4-2 x} x-36 e^{2-x} x+\frac {5 x^2}{2}+e^{4-2 x} x^2+26 e^{2-x} x^2+8 x^3+6 e^{2-x} x^3+\frac {13 x^4}{2}-2 e^{2-x} x^4-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)-2 \int e^{2-x} \, dx-2 \int e^{4-2 x} x \, dx+24 \int e^{2-x} x^2 \, dx+32 \int e^{2-x} \, dx-84 \int e^{2-x} x \, dx+\int e^{4-2 x} \, dx+\int \left (-3 x-6 x^2+2 x^3\right ) \, dx\\ &=-\frac {1}{2} e^{4-2 x}-36 e^{2-x}+48 e^{2-x} x+x^2+e^{4-2 x} x^2+2 e^{2-x} x^2+6 x^3+6 e^{2-x} x^3+7 x^4-2 e^{2-x} x^4-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)+48 \int e^{2-x} x \, dx-84 \int e^{2-x} \, dx-\int e^{4-2 x} \, dx\\ &=48 e^{2-x}+x^2+e^{4-2 x} x^2+2 e^{2-x} x^2+6 x^3+6 e^{2-x} x^3+7 x^4-2 e^{2-x} x^4-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)+48 \int e^{2-x} \, dx\\ &=x^2+e^{4-2 x} x^2+2 e^{2-x} x^2+6 x^3+6 e^{2-x} x^3+7 x^4-2 e^{2-x} x^4-6 x^5+x^6+2 x^2 \log (x)+2 e^{2-x} x^2 \log (x)+6 x^3 \log (x)-2 x^4 \log (x)+x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 35, normalized size = 1.46 \begin {gather*} e^{-2 x} x^2 \left (e^2+e^x \left (1+3 x-x^2\right )+e^x \log (x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4*x + 24*x^2 + 26*x^3 - 30*x^4 + 6*x^5 + E^(4 - 2*x)*(2*x - 2*x^2) + E^(2 - x)*(6*x + 16*x^2 - 14*x^
3 + 2*x^4) + (6*x + 18*x^2 - 8*x^3 + E^(2 - x)*(4*x - 2*x^2))*Log[x] + 2*x*Log[x]^2,x]

[Out]

(x^2*(E^2 + E^x*(1 + 3*x - x^2) + E^x*Log[x])^2)/E^(2*x)

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fricas [B]  time = 0.92, size = 91, normalized size = 3.79 \begin {gather*} x^{6} - 6 \, x^{5} + 7 \, x^{4} + x^{2} \log \relax (x)^{2} + 6 \, x^{3} + x^{2} e^{\left (-2 \, x + 4\right )} + x^{2} - 2 \, {\left (x^{4} - 3 \, x^{3} - x^{2}\right )} e^{\left (-x + 2\right )} - 2 \, {\left (x^{4} - 3 \, x^{3} - x^{2} e^{\left (-x + 2\right )} - x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^2+((-2*x^2+4*x)*exp(2-x)-8*x^3+18*x^2+6*x)*log(x)+(-2*x^2+2*x)*exp(2-x)^2+(2*x^4-14*x^3+1
6*x^2+6*x)*exp(2-x)+6*x^5-30*x^4+26*x^3+24*x^2+4*x,x, algorithm="fricas")

[Out]

x^6 - 6*x^5 + 7*x^4 + x^2*log(x)^2 + 6*x^3 + x^2*e^(-2*x + 4) + x^2 - 2*(x^4 - 3*x^3 - x^2)*e^(-x + 2) - 2*(x^
4 - 3*x^3 - x^2*e^(-x + 2) - x^2)*log(x)

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giac [B]  time = 0.22, size = 113, normalized size = 4.71 \begin {gather*} x^{6} - 6 \, x^{5} + 7 \, x^{4} + x^{2} \log \relax (x)^{2} + 6 \, x^{3} + x^{2} e^{\left (-2 \, x + 4\right )} - x^{2} \log \relax (x) + x^{2} - 2 \, {\left (x^{4} - 3 \, x^{3} - x^{2} + x + 1\right )} e^{\left (-x + 2\right )} + 2 \, {\left (x + 1\right )} e^{\left (-x + 2\right )} - {\left (2 \, x^{4} - 6 \, x^{3} - 2 \, x^{2} e^{\left (-x + 2\right )} - 3 \, x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^2+((-2*x^2+4*x)*exp(2-x)-8*x^3+18*x^2+6*x)*log(x)+(-2*x^2+2*x)*exp(2-x)^2+(2*x^4-14*x^3+1
6*x^2+6*x)*exp(2-x)+6*x^5-30*x^4+26*x^3+24*x^2+4*x,x, algorithm="giac")

[Out]

x^6 - 6*x^5 + 7*x^4 + x^2*log(x)^2 + 6*x^3 + x^2*e^(-2*x + 4) - x^2*log(x) + x^2 - 2*(x^4 - 3*x^3 - x^2 + x +
1)*e^(-x + 2) + 2*(x + 1)*e^(-x + 2) - (2*x^4 - 6*x^3 - 2*x^2*e^(-x + 2) - 3*x^2)*log(x)

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maple [B]  time = 0.06, size = 108, normalized size = 4.50

method	result	size
risch	\(x^{6}-2 x^{4} \ln \relax (x )-2 \,{\mathrm e}^{2-x} x^{4}-6 x^{5}+x^{2} \ln \relax (x )^{2}+2 \ln \relax (x ) {\mathrm e}^{2-x} x^{2}+6 x^{3} \ln \relax (x )+{\mathrm e}^{4-2 x} x^{2}+6 x^{3} {\mathrm e}^{2-x}+7 x^{4}+2 x^{2} \ln \relax (x )+2 x^{2} {\mathrm e}^{2-x}+6 x^{3}+x^{2}\)	\(108\)
default	\(-2 x^{4} \ln \relax (x )+7 x^{4}+6 x^{3} \ln \relax (x )+6 x^{3}+2 x^{2} \ln \relax (x )+x^{2}+2 x \,{\mathrm e}^{2-x}+2 \ln \relax (x ) {\mathrm e}^{2-x} x^{2}+20 \,{\mathrm e}^{2-x}+4 \,{\mathrm e}^{4-2 x}-4 \left (2-x \right ) {\mathrm e}^{4-2 x}+{\mathrm e}^{4-2 x} \left (2-x \right )^{2}-2 \,{\mathrm e}^{2-x} \left (2-x \right )^{4}+10 \left (2-x \right )^{3} {\mathrm e}^{2-x}-10 \left (2-x \right )^{2} {\mathrm e}^{2-x}-14 \,{\mathrm e}^{2-x} \left (2-x \right )-6 x^{5}+x^{6}+x^{2} \ln \relax (x )^{2}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*ln(x)^2+((-2*x^2+4*x)*exp(2-x)-8*x^3+18*x^2+6*x)*ln(x)+(-2*x^2+2*x)*exp(2-x)^2+(2*x^4-14*x^3+16*x^2+6*
x)*exp(2-x)+6*x^5-30*x^4+26*x^3+24*x^2+4*x,x,method=_RETURNVERBOSE)

[Out]

x^6-2*x^4*ln(x)-2*exp(2-x)*x^4-6*x^5+x^2*ln(x)^2+2*ln(x)*exp(2-x)*x^2+6*x^3*ln(x)+exp(4-2*x)*x^2+6*x^3*exp(2-x
)+7*x^4+2*x^2*ln(x)+2*x^2*exp(2-x)+6*x^3+x^2

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maxima [B]  time = 0.45, size = 128, normalized size = 5.33 \begin {gather*} x^{6} - 6 \, x^{5} + 7 \, x^{4} + \frac {1}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + 6 \, x^{3} + x^{2} e^{\left (-2 \, x + 4\right )} + \frac {1}{2} \, x^{2} - 2 \, {\left (x^{4} e^{2} - 3 \, x^{3} e^{2} - x^{2} e^{2} + x e^{2} + e^{2}\right )} e^{\left (-x\right )} + 2 \, {\left (x e^{2} + e^{2}\right )} e^{\left (-x\right )} - {\left (2 \, x^{4} - 6 \, x^{3} - 2 \, x^{2} e^{\left (-x + 2\right )} - 3 \, x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x)^2+((-2*x^2+4*x)*exp(2-x)-8*x^3+18*x^2+6*x)*log(x)+(-2*x^2+2*x)*exp(2-x)^2+(2*x^4-14*x^3+1
6*x^2+6*x)*exp(2-x)+6*x^5-30*x^4+26*x^3+24*x^2+4*x,x, algorithm="maxima")

[Out]

x^6 - 6*x^5 + 7*x^4 + 1/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 6*x^3 + x^2*e^(-2*x + 4) + 1/2*x^2 - 2*(x^4*e^2 -
3*x^3*e^2 - x^2*e^2 + x*e^2 + e^2)*e^(-x) + 2*(x*e^2 + e^2)*e^(-x) - (2*x^4 - 6*x^3 - 2*x^2*e^(-x + 2) - 3*x^2
)*log(x)

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mupad [B]  time = 0.74, size = 107, normalized size = 4.46 \begin {gather*} 2\,x^2\,\ln \relax (x)+6\,x^3\,\ln \relax (x)-2\,x^4\,\ln \relax (x)+x^2\,{\ln \relax (x)}^2+2\,x^2\,{\mathrm {e}}^{2-x}+6\,x^3\,{\mathrm {e}}^{2-x}-2\,x^4\,{\mathrm {e}}^{2-x}+x^2\,{\mathrm {e}}^{4-2\,x}+x^2+6\,x^3+7\,x^4-6\,x^5+x^6+2\,x^2\,{\mathrm {e}}^{2-x}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x + 2*x*log(x)^2 + log(x)*(6*x + exp(2 - x)*(4*x - 2*x^2) + 18*x^2 - 8*x^3) + exp(4 - 2*x)*(2*x - 2*x^2)
+ exp(2 - x)*(6*x + 16*x^2 - 14*x^3 + 2*x^4) + 24*x^2 + 26*x^3 - 30*x^4 + 6*x^5,x)

[Out]

2*x^2*log(x) + 6*x^3*log(x) - 2*x^4*log(x) + x^2*log(x)^2 + 2*x^2*exp(2 - x) + 6*x^3*exp(2 - x) - 2*x^4*exp(2
- x) + x^2*exp(4 - 2*x) + x^2 + 6*x^3 + 7*x^4 - 6*x^5 + x^6 + 2*x^2*exp(2 - x)*log(x)

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sympy [B]  time = 0.41, size = 87, normalized size = 3.62 \begin {gather*} x^{6} - 6 x^{5} + 7 x^{4} + 6 x^{3} + x^{2} e^{4 - 2 x} + x^{2} \log {\relax (x )}^{2} + x^{2} + \left (- 2 x^{4} + 6 x^{3} + 2 x^{2}\right ) \log {\relax (x )} + \left (- 2 x^{4} + 6 x^{3} + 2 x^{2} \log {\relax (x )} + 2 x^{2}\right ) e^{2 - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*ln(x)**2+((-2*x**2+4*x)*exp(2-x)-8*x**3+18*x**2+6*x)*ln(x)+(-2*x**2+2*x)*exp(2-x)**2+(2*x**4-14*
x**3+16*x**2+6*x)*exp(2-x)+6*x**5-30*x**4+26*x**3+24*x**2+4*x,x)

[Out]

x**6 - 6*x**5 + 7*x**4 + 6*x**3 + x**2*exp(4 - 2*x) + x**2*log(x)**2 + x**2 + (-2*x**4 + 6*x**3 + 2*x**2)*log(
x) + (-2*x**4 + 6*x**3 + 2*x**2*log(x) + 2*x**2)*exp(2 - x)

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