3.10.8 \(\int \frac {-125+x+4000 x^2-32000 e^4 x^2+4000 x^3-32000 x^4-64000 x^5-32000 x^6+e^2 (-4000 x+64000 x^3+64000 x^4)+(8000 x^2-64000 e^4 x^2+12000 x^3-128000 x^4-320000 x^5-192000 x^6+e^2 (-4000 x+192000 x^3+256000 x^4)) \log (x)}{x} \, dx\)

Optimal. Leaf size=25 \[ x-5 \left (5+80 x \left (e^2-x-x^2\right )\right )^2 \log (x) \]

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Rubi [B]  time = 0.22, antiderivative size = 84, normalized size of antiderivative = 3.36, number of steps used = 13, number of rules used = 5, integrand size = 113, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6, 14, 2356, 2295, 2304} \begin {gather*} -32000 x^6 \log (x)-64000 x^5 \log (x)-32000 \left (1-2 e^2\right ) x^4 \log (x)+4000 \left (1+16 e^2\right ) x^3 \log (x)+4000 \left (1-8 e^4\right ) x^2 \log (x)+\left (1-4000 e^2\right ) x+4000 e^2 x-4000 e^2 x \log (x)-125 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-125 + x + 4000*x^2 - 32000*E^4*x^2 + 4000*x^3 - 32000*x^4 - 64000*x^5 - 32000*x^6 + E^2*(-4000*x + 64000
*x^3 + 64000*x^4) + (8000*x^2 - 64000*E^4*x^2 + 12000*x^3 - 128000*x^4 - 320000*x^5 - 192000*x^6 + E^2*(-4000*
x + 192000*x^3 + 256000*x^4))*Log[x])/x,x]

[Out]

4000*E^2*x + (1 - 4000*E^2)*x - 125*Log[x] - 4000*E^2*x*Log[x] + 4000*(1 - 8*E^4)*x^2*Log[x] + 4000*(1 + 16*E^
2)*x^3*Log[x] - 32000*(1 - 2*E^2)*x^4*Log[x] - 64000*x^5*Log[x] - 32000*x^6*Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 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} &=\int \frac {-125+x+\left (4000-32000 e^4\right ) x^2+4000 x^3-32000 x^4-64000 x^5-32000 x^6+e^2 \left (-4000 x+64000 x^3+64000 x^4\right )+\left (8000 x^2-64000 e^4 x^2+12000 x^3-128000 x^4-320000 x^5-192000 x^6+e^2 \left (-4000 x+192000 x^3+256000 x^4\right )\right ) \log (x)}{x} \, dx\\ &=\int \left (\frac {-125+\left (1-4000 e^2\right ) x+4000 \left (1-8 e^4\right ) x^2+4000 \left (1+16 e^2\right ) x^3-32000 \left (1-2 e^2\right ) x^4-64000 x^5-32000 x^6}{x}-4000 \left (e^2-2 x-3 x^2\right ) \left (1+16 e^2 x-16 x^2-16 x^3\right ) \log (x)\right ) \, dx\\ &=-\left (4000 \int \left (e^2-2 x-3 x^2\right ) \left (1+16 e^2 x-16 x^2-16 x^3\right ) \log (x) \, dx\right )+\int \frac {-125+\left (1-4000 e^2\right ) x+4000 \left (1-8 e^4\right ) x^2+4000 \left (1+16 e^2\right ) x^3-32000 \left (1-2 e^2\right ) x^4-64000 x^5-32000 x^6}{x} \, dx\\ &=-\left (4000 \int \left (e^2 \log (x)+2 \left (-1+8 e^4\right ) x \log (x)-3 \left (1+16 e^2\right ) x^2 \log (x)-32 \left (-1+2 e^2\right ) x^3 \log (x)+80 x^4 \log (x)+48 x^5 \log (x)\right ) \, dx\right )+\int \left (1-4000 e^2-\frac {125}{x}+4000 \left (1-8 e^4\right ) x+4000 \left (1+16 e^2\right ) x^2-32000 \left (1-2 e^2\right ) x^3-64000 x^4-32000 x^5\right ) \, dx\\ &=\left (1-4000 e^2\right ) x+2000 \left (1-8 e^4\right ) x^2+\frac {4000}{3} \left (1+16 e^2\right ) x^3-8000 \left (1-2 e^2\right ) x^4-12800 x^5-\frac {16000 x^6}{3}-125 \log (x)-192000 \int x^5 \log (x) \, dx-320000 \int x^4 \log (x) \, dx-\left (4000 e^2\right ) \int \log (x) \, dx-\left (128000 \left (1-2 e^2\right )\right ) \int x^3 \log (x) \, dx+\left (12000 \left (1+16 e^2\right )\right ) \int x^2 \log (x) \, dx+\left (8000 \left (1-8 e^4\right )\right ) \int x \log (x) \, dx\\ &=4000 e^2 x+\left (1-4000 e^2\right ) x-125 \log (x)-4000 e^2 x \log (x)+4000 \left (1-8 e^4\right ) x^2 \log (x)+4000 \left (1+16 e^2\right ) x^3 \log (x)-32000 \left (1-2 e^2\right ) x^4 \log (x)-64000 x^5 \log (x)-32000 x^6 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 79, normalized size = 3.16 \begin {gather*} x-125 \log (x)-4000 e^2 x \log (x)+4000 x^2 \log (x)-32000 e^4 x^2 \log (x)+4000 x^3 \log (x)+64000 e^2 x^3 \log (x)-32000 x^4 \log (x)+64000 e^2 x^4 \log (x)-64000 x^5 \log (x)-32000 x^6 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-125 + x + 4000*x^2 - 32000*E^4*x^2 + 4000*x^3 - 32000*x^4 - 64000*x^5 - 32000*x^6 + E^2*(-4000*x +
64000*x^3 + 64000*x^4) + (8000*x^2 - 64000*E^4*x^2 + 12000*x^3 - 128000*x^4 - 320000*x^5 - 192000*x^6 + E^2*(
-4000*x + 192000*x^3 + 256000*x^4))*Log[x])/x,x]

[Out]

x - 125*Log[x] - 4000*E^2*x*Log[x] + 4000*x^2*Log[x] - 32000*E^4*x^2*Log[x] + 4000*x^3*Log[x] + 64000*E^2*x^3*
Log[x] - 32000*x^4*Log[x] + 64000*E^2*x^4*Log[x] - 64000*x^5*Log[x] - 32000*x^6*Log[x]

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fricas [B]  time = 0.46, size = 58, normalized size = 2.32 \begin {gather*} -125 \, {\left (256 \, x^{6} + 512 \, x^{5} + 256 \, x^{4} - 32 \, x^{3} + 256 \, x^{2} e^{4} - 32 \, x^{2} - 32 \, {\left (16 \, x^{4} + 16 \, x^{3} - x\right )} e^{2} + 1\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64000*x^2*exp(2)^2+(256000*x^4+192000*x^3-4000*x)*exp(2)-192000*x^6-320000*x^5-128000*x^4+12000*x
^3+8000*x^2)*log(x)-32000*x^2*exp(2)^2+(64000*x^4+64000*x^3-4000*x)*exp(2)-32000*x^6-64000*x^5-32000*x^4+4000*
x^3+4000*x^2+x-125)/x,x, algorithm="fricas")

[Out]

-125*(256*x^6 + 512*x^5 + 256*x^4 - 32*x^3 + 256*x^2*e^4 - 32*x^2 - 32*(16*x^4 + 16*x^3 - x)*e^2 + 1)*log(x) +
x

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giac [B]  time = 0.64, size = 75, normalized size = 3.00 \begin {gather*} -32000 \, x^{6} \log \relax (x) - 64000 \, x^{5} \log \relax (x) + 64000 \, x^{4} e^{2} \log \relax (x) - 32000 \, x^{4} \log \relax (x) + 64000 \, x^{3} e^{2} \log \relax (x) + 4000 \, x^{3} \log \relax (x) - 32000 \, x^{2} e^{4} \log \relax (x) + 4000 \, x^{2} \log \relax (x) - 4000 \, x e^{2} \log \relax (x) + x - 125 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64000*x^2*exp(2)^2+(256000*x^4+192000*x^3-4000*x)*exp(2)-192000*x^6-320000*x^5-128000*x^4+12000*x
^3+8000*x^2)*log(x)-32000*x^2*exp(2)^2+(64000*x^4+64000*x^3-4000*x)*exp(2)-32000*x^6-64000*x^5-32000*x^4+4000*
x^3+4000*x^2+x-125)/x,x, algorithm="giac")

[Out]

-32000*x^6*log(x) - 64000*x^5*log(x) + 64000*x^4*e^2*log(x) - 32000*x^4*log(x) + 64000*x^3*e^2*log(x) + 4000*x
^3*log(x) - 32000*x^2*e^4*log(x) + 4000*x^2*log(x) - 4000*x*e^2*log(x) + x - 125*log(x)

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maple [B]  time = 0.04, size = 62, normalized size = 2.48

method	result	size
risch	\(\left (-32000 x^{6}+64000 x^{4} {\mathrm e}^{2}-64000 x^{5}-32000 x^{2} {\mathrm e}^{4}+64000 x^{3} {\mathrm e}^{2}-32000 x^{4}+4000 x^{3}-4000 \,{\mathrm e}^{2} x +4000 x^{2}\right ) \ln \relax (x )+x -125 \ln \relax (x )\)	\(62\)
norman	\(x -125 \ln \relax (x )+\left (-32000 \,{\mathrm e}^{4}+4000\right ) x^{2} \ln \relax (x )+\left (64000 \,{\mathrm e}^{2}-32000\right ) x^{4} \ln \relax (x )+\left (64000 \,{\mathrm e}^{2}+4000\right ) x^{3} \ln \relax (x )-64000 x^{5} \ln \relax (x )-32000 x^{6} \ln \relax (x )-4000 x \,{\mathrm e}^{2} \ln \relax (x )\)	\(66\)
default	\(x -4000 \,{\mathrm e}^{2} x +4000 x^{2} \ln \relax (x )-125 \ln \relax (x )+4000 x^{3} \ln \relax (x )-32000 x^{6} \ln \relax (x )-64000 x^{5} \ln \relax (x )+\frac {64000 x^{3} {\mathrm e}^{2}}{3}-16000 x^{2} {\mathrm e}^{4}-32000 x^{4} \ln \relax (x )+16000 x^{4} {\mathrm e}^{2}-4000 \,{\mathrm e}^{2} \left (x \ln \relax (x )-x \right )+192000 \,{\mathrm e}^{2} \left (\frac {x^{3} \ln \relax (x )}{3}-\frac {x^{3}}{9}\right )+256000 \,{\mathrm e}^{2} \left (\frac {x^{4} \ln \relax (x )}{4}-\frac {x^{4}}{16}\right )-64000 \,{\mathrm e}^{4} \left (\frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4}\right )\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-64000*x^2*exp(2)^2+(256000*x^4+192000*x^3-4000*x)*exp(2)-192000*x^6-320000*x^5-128000*x^4+12000*x^3+800
0*x^2)*ln(x)-32000*x^2*exp(2)^2+(64000*x^4+64000*x^3-4000*x)*exp(2)-32000*x^6-64000*x^5-32000*x^4+4000*x^3+400
0*x^2+x-125)/x,x,method=_RETURNVERBOSE)

[Out]

(-32000*x^6+64000*x^4*exp(2)-64000*x^5-32000*x^2*exp(4)+64000*x^3*exp(2)-32000*x^4+4000*x^3-4000*exp(2)*x+4000
*x^2)*ln(x)+x-125*ln(x)

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maxima [B]  time = 0.34, size = 130, normalized size = 5.20 \begin {gather*} -32000 \, x^{6} \log \relax (x) - 64000 \, x^{5} \log \relax (x) + 16000 \, x^{4} e^{2} - 32000 \, x^{4} \log \relax (x) + \frac {64000}{3} \, x^{3} e^{2} + 4000 \, x^{3} \log \relax (x) - 16000 \, x^{2} e^{4} + 4000 \, x^{2} \log \relax (x) - 16000 \, {\left (2 \, x^{2} \log \relax (x) - x^{2}\right )} e^{4} + 16000 \, {\left (4 \, x^{4} \log \relax (x) - x^{4}\right )} e^{2} + \frac {64000}{3} \, {\left (3 \, x^{3} \log \relax (x) - x^{3}\right )} e^{2} - 4000 \, {\left (x \log \relax (x) - x\right )} e^{2} - 4000 \, x e^{2} + x - 125 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64000*x^2*exp(2)^2+(256000*x^4+192000*x^3-4000*x)*exp(2)-192000*x^6-320000*x^5-128000*x^4+12000*x
^3+8000*x^2)*log(x)-32000*x^2*exp(2)^2+(64000*x^4+64000*x^3-4000*x)*exp(2)-32000*x^6-64000*x^5-32000*x^4+4000*
x^3+4000*x^2+x-125)/x,x, algorithm="maxima")

[Out]

-32000*x^6*log(x) - 64000*x^5*log(x) + 16000*x^4*e^2 - 32000*x^4*log(x) + 64000/3*x^3*e^2 + 4000*x^3*log(x) -
16000*x^2*e^4 + 4000*x^2*log(x) - 16000*(2*x^2*log(x) - x^2)*e^4 + 16000*(4*x^4*log(x) - x^4)*e^2 + 64000/3*(3
*x^3*log(x) - x^3)*e^2 - 4000*(x*log(x) - x)*e^2 - 4000*x*e^2 + x - 125*log(x)

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mupad [B]  time = 0.88, size = 67, normalized size = 2.68 \begin {gather*} x^3\,\ln \relax (x)\,\left (64000\,{\mathrm {e}}^2+4000\right )-64000\,x^5\,\ln \relax (x)-32000\,x^6\,\ln \relax (x)-x\,\left (4000\,{\mathrm {e}}^2\,\ln \relax (x)-1\right )-x^2\,\ln \relax (x)\,\left (32000\,{\mathrm {e}}^4-4000\right )-125\,\ln \relax (x)+x^4\,\ln \relax (x)\,\left (64000\,{\mathrm {e}}^2-32000\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32000*x^2*exp(4) - exp(2)*(64000*x^3 - 4000*x + 64000*x^4) - x + log(x)*(64000*x^2*exp(4) - exp(2)*(1920
00*x^3 - 4000*x + 256000*x^4) - 8000*x^2 - 12000*x^3 + 128000*x^4 + 320000*x^5 + 192000*x^6) - 4000*x^2 - 4000
*x^3 + 32000*x^4 + 64000*x^5 + 32000*x^6 + 125)/x,x)

[Out]

x^3*log(x)*(64000*exp(2) + 4000) - 64000*x^5*log(x) - 32000*x^6*log(x) - x*(4000*exp(2)*log(x) - 1) - x^2*log(
x)*(32000*exp(4) - 4000) - 125*log(x) + x^4*log(x)*(64000*exp(2) - 32000)

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sympy [B]  time = 0.22, size = 66, normalized size = 2.64 \begin {gather*} x + \left (- 32000 x^{6} - 64000 x^{5} - 32000 x^{4} + 64000 x^{4} e^{2} + 4000 x^{3} + 64000 x^{3} e^{2} - 32000 x^{2} e^{4} + 4000 x^{2} - 4000 x e^{2}\right ) \log {\relax (x )} - 125 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64000*x**2*exp(2)**2+(256000*x**4+192000*x**3-4000*x)*exp(2)-192000*x**6-320000*x**5-128000*x**4+
12000*x**3+8000*x**2)*ln(x)-32000*x**2*exp(2)**2+(64000*x**4+64000*x**3-4000*x)*exp(2)-32000*x**6-64000*x**5-3
2000*x**4+4000*x**3+4000*x**2+x-125)/x,x)

[Out]

x + (-32000*x**6 - 64000*x**5 - 32000*x**4 + 64000*x**4*exp(2) + 4000*x**3 + 64000*x**3*exp(2) - 32000*x**2*ex
p(4) + 4000*x**2 - 4000*x*exp(2))*log(x) - 125*log(x)

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