3.56.58 \(\int (8 x-12 x^2+24 x^3-12 x^4+8 x^5+e^{e^3} (4-4 x+8 x^2)+(6 e^{e^3} x^2+16 x^3-10 x^4+14 x^5) \log (x)+6 x^5 \log ^2(x)) \, dx\)

Optimal. Leaf size=24 \[ \left (e^{e^3}+2 x-x^2+x^3 (1+\log (x))\right )^2 \]

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Rubi [B]  time = 0.11, antiderivative size = 93, normalized size of antiderivative = 3.88, number of steps used = 10, number of rules used = 3, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2356, 2304, 2305} \begin {gather*} x^6+x^6 \log ^2(x)+2 x^6 \log (x)-2 x^5-2 x^5 \log (x)+5 x^4+4 x^4 \log (x)+2 e^{e^3} x^3-4 x^3+2 e^{e^3} x^3 \log (x)-2 e^{e^3} x^2+4 x^2+4 e^{e^3} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8*x - 12*x^2 + 24*x^3 - 12*x^4 + 8*x^5 + E^E^3*(4 - 4*x + 8*x^2) + (6*E^E^3*x^2 + 16*x^3 - 10*x^4 + 14*x^5
)*Log[x] + 6*x^5*Log[x]^2,x]

[Out]

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

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 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x^2-4 x^3+6 x^4-\frac {12 x^5}{5}+\frac {4 x^6}{3}+6 \int x^5 \log ^2(x) \, dx+e^{e^3} \int \left (4-4 x+8 x^2\right ) \, dx+\int \left (6 e^{e^3} x^2+16 x^3-10 x^4+14 x^5\right ) \log (x) \, dx\\ &=4 e^{e^3} x+4 x^2-2 e^{e^3} x^2-4 x^3+\frac {8}{3} e^{e^3} x^3+6 x^4-\frac {12 x^5}{5}+\frac {4 x^6}{3}+x^6 \log ^2(x)-2 \int x^5 \log (x) \, dx+\int \left (6 e^{e^3} x^2 \log (x)+16 x^3 \log (x)-10 x^4 \log (x)+14 x^5 \log (x)\right ) \, dx\\ &=4 e^{e^3} x+4 x^2-2 e^{e^3} x^2-4 x^3+\frac {8}{3} e^{e^3} x^3+6 x^4-\frac {12 x^5}{5}+\frac {25 x^6}{18}-\frac {1}{3} x^6 \log (x)+x^6 \log ^2(x)-10 \int x^4 \log (x) \, dx+14 \int x^5 \log (x) \, dx+16 \int x^3 \log (x) \, dx+\left (6 e^{e^3}\right ) \int x^2 \log (x) \, dx\\ &=4 e^{e^3} x+4 x^2-2 e^{e^3} x^2-4 x^3+2 e^{e^3} x^3+5 x^4-2 x^5+x^6+2 e^{e^3} x^3 \log (x)+4 x^4 \log (x)-2 x^5 \log (x)+2 x^6 \log (x)+x^6 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 24, normalized size = 1.00 \begin {gather*} \left (e^{e^3}+x \left (2-x+x^2\right )+x^3 \log (x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8*x - 12*x^2 + 24*x^3 - 12*x^4 + 8*x^5 + E^E^3*(4 - 4*x + 8*x^2) + (6*E^E^3*x^2 + 16*x^3 - 10*x^4 +
14*x^5)*Log[x] + 6*x^5*Log[x]^2,x]

[Out]

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

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fricas [B]  time = 0.53, size = 74, normalized size = 3.08 \begin {gather*} x^{6} \log \relax (x)^{2} + x^{6} - 2 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{3} - x^{2} + 2 \, x\right )} e^{\left (e^{3}\right )} + 2 \, {\left (x^{6} - x^{5} + 2 \, x^{4} + x^{3} e^{\left (e^{3}\right )}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*x^5*log(x)^2+(6*x^2*exp(exp(3))+14*x^5-10*x^4+16*x^3)*log(x)+(8*x^2-4*x+4)*exp(exp(3))+8*x^5-12*x^
4+24*x^3-12*x^2+8*x,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.16, size = 90, normalized size = 3.75 \begin {gather*} x^{6} \log \relax (x)^{2} + 2 \, x^{6} \log \relax (x) + x^{6} - 2 \, x^{5} \log \relax (x) - 2 \, x^{5} + 4 \, x^{4} \log \relax (x) + 2 \, x^{3} e^{\left (e^{3}\right )} \log \relax (x) + 5 \, x^{4} - \frac {2}{3} \, x^{3} e^{\left (e^{3}\right )} - 4 \, x^{3} + 4 \, x^{2} + \frac {2}{3} \, {\left (4 \, x^{3} - 3 \, x^{2} + 6 \, x\right )} e^{\left (e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*x^5*log(x)^2+(6*x^2*exp(exp(3))+14*x^5-10*x^4+16*x^3)*log(x)+(8*x^2-4*x+4)*exp(exp(3))+8*x^5-12*x^
4+24*x^3-12*x^2+8*x,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.07, size = 82, normalized size = 3.42

method	result	size
norman	\(x^{6}+x^{6} \ln \relax (x )^{2}+\left (-2 \,{\mathrm e}^{{\mathrm e}^{3}}+4\right ) x^{2}+\left (2 \,{\mathrm e}^{{\mathrm e}^{3}}-4\right ) x^{3}+5 x^{4}-2 x^{5}+4 x \,{\mathrm e}^{{\mathrm e}^{3}}+4 x^{4} \ln \relax (x )-2 x^{5} \ln \relax (x )+2 x^{6} \ln \relax (x )+2 \,{\mathrm e}^{{\mathrm e}^{3}} \ln \relax (x ) x^{3}\)	\(82\)
risch	\(x^{6} \ln \relax (x )^{2}+2 x^{6} \ln \relax (x )-2 x^{5} \ln \relax (x )+x^{6}+2 \,{\mathrm e}^{{\mathrm e}^{3}} \ln \relax (x ) x^{3}+4 x^{4} \ln \relax (x )-2 x^{5}+2 x^{3} {\mathrm e}^{{\mathrm e}^{3}}+5 x^{4}-2 x^{2} {\mathrm e}^{{\mathrm e}^{3}}-4 x^{3}+4 x \,{\mathrm e}^{{\mathrm e}^{3}}+4 x^{2}\)	\(86\)
default	\(2 x^{6} \ln \relax (x )+x^{6}-2 x^{5} \ln \relax (x )-2 x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{3}} \ln \relax (x ) x^{3}-\frac {2 x^{3} {\mathrm e}^{{\mathrm e}^{3}}}{3}+4 x^{4} \ln \relax (x )+5 x^{4}+{\mathrm e}^{{\mathrm e}^{3}} \left (\frac {8}{3} x^{3}-2 x^{2}+4 x \right )+4 x^{2}-4 x^{3}+x^{6} \ln \relax (x )^{2}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x^5*ln(x)^2+(6*x^2*exp(exp(3))+14*x^5-10*x^4+16*x^3)*ln(x)+(8*x^2-4*x+4)*exp(exp(3))+8*x^5-12*x^4+24*x^3
-12*x^2+8*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.37, size = 98, normalized size = 4.08 \begin {gather*} \frac {1}{18} \, {\left (18 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 1\right )} x^{6} + \frac {17}{18} \, x^{6} - 2 \, x^{5} + 5 \, x^{4} - \frac {2}{3} \, x^{3} e^{\left (e^{3}\right )} - 4 \, x^{3} + 4 \, x^{2} + \frac {2}{3} \, {\left (4 \, x^{3} - 3 \, x^{2} + 6 \, x\right )} e^{\left (e^{3}\right )} + \frac {1}{3} \, {\left (7 \, x^{6} - 6 \, x^{5} + 12 \, x^{4} + 6 \, x^{3} e^{\left (e^{3}\right )}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*x^5*log(x)^2+(6*x^2*exp(exp(3))+14*x^5-10*x^4+16*x^3)*log(x)+(8*x^2-4*x+4)*exp(exp(3))+8*x^5-12*x^
4+24*x^3-12*x^2+8*x,x, algorithm="maxima")

[Out]

1/18*(18*log(x)^2 - 6*log(x) + 1)*x^6 + 17/18*x^6 - 2*x^5 + 5*x^4 - 2/3*x^3*e^(e^3) - 4*x^3 + 4*x^2 + 2/3*(4*x
^3 - 3*x^2 + 6*x)*e^(e^3) + 1/3*(7*x^6 - 6*x^5 + 12*x^4 + 6*x^3*e^(e^3))*log(x)

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mupad [B]  time = 3.67, size = 39, normalized size = 1.62 \begin {gather*} x\,\left (x^2\,\ln \relax (x)-x+x^2+2\right )\,\left (2\,x+2\,{\mathrm {e}}^{{\mathrm {e}}^3}+x^3\,\ln \relax (x)-x^2+x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x + log(x)*(6*x^2*exp(exp(3)) + 16*x^3 - 10*x^4 + 14*x^5) + 6*x^5*log(x)^2 + exp(exp(3))*(8*x^2 - 4*x +
4) - 12*x^2 + 24*x^3 - 12*x^4 + 8*x^5,x)

[Out]

x*(x^2*log(x) - x + x^2 + 2)*(2*x + 2*exp(exp(3)) + x^3*log(x) - x^2 + x^3)

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sympy [B]  time = 0.16, size = 82, normalized size = 3.42 \begin {gather*} x^{6} \log {\relax (x )}^{2} + x^{6} - 2 x^{5} + 5 x^{4} + x^{3} \left (-4 + 2 e^{e^{3}}\right ) + x^{2} \left (4 - 2 e^{e^{3}}\right ) + 4 x e^{e^{3}} + \left (2 x^{6} - 2 x^{5} + 4 x^{4} + 2 x^{3} e^{e^{3}}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*x**5*ln(x)**2+(6*x**2*exp(exp(3))+14*x**5-10*x**4+16*x**3)*ln(x)+(8*x**2-4*x+4)*exp(exp(3))+8*x**5
-12*x**4+24*x**3-12*x**2+8*x,x)

[Out]

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

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