3.77.56 \(\int \frac {-10267500+5476000 x-547600 x^2+(8556250-4107000 x+273800 x^2) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ 5+\frac {34225 (5-2 x)^2 (2-\log (x))^2}{x^2} \]

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Rubi [B]  time = 0.17, antiderivative size = 53, normalized size of antiderivative = 2.41, number of steps used = 15, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {14, 2357, 2304, 2301, 2353, 2305} \begin {gather*} \frac {3422500}{x^2}+\frac {855625 \log ^2(x)}{x^2}-\frac {3422500 \log (x)}{x^2}-\frac {2738000}{x}-\frac {684500 \log ^2(x)}{x}+136900 \log ^2(x)+\frac {2738000 \log (x)}{x}-547600 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10267500 + 5476000*x - 547600*x^2 + (8556250 - 4107000*x + 273800*x^2)*Log[x] + (-1711250 + 684500*x)*Lo
g[x]^2)/x^3,x]

[Out]

3422500/x^2 - 2738000/x - 547600*Log[x] - (3422500*Log[x])/x^2 + (2738000*Log[x])/x + 136900*Log[x]^2 + (85562
5*Log[x]^2)/x^2 - (684500*Log[x]^2)/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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, 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 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 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {136900 \left (75-40 x+4 x^2\right )}{x^3}+\frac {68450 (-25+2 x) (-5+2 x) \log (x)}{x^3}+\frac {342250 (-5+2 x) \log ^2(x)}{x^3}\right ) \, dx\\ &=68450 \int \frac {(-25+2 x) (-5+2 x) \log (x)}{x^3} \, dx-136900 \int \frac {75-40 x+4 x^2}{x^3} \, dx+342250 \int \frac {(-5+2 x) \log ^2(x)}{x^3} \, dx\\ &=68450 \int \left (\frac {125 \log (x)}{x^3}-\frac {60 \log (x)}{x^2}+\frac {4 \log (x)}{x}\right ) \, dx-136900 \int \left (\frac {75}{x^3}-\frac {40}{x^2}+\frac {4}{x}\right ) \, dx+342250 \int \left (-\frac {5 \log ^2(x)}{x^3}+\frac {2 \log ^2(x)}{x^2}\right ) \, dx\\ &=\frac {5133750}{x^2}-\frac {5476000}{x}-547600 \log (x)+273800 \int \frac {\log (x)}{x} \, dx+684500 \int \frac {\log ^2(x)}{x^2} \, dx-1711250 \int \frac {\log ^2(x)}{x^3} \, dx-4107000 \int \frac {\log (x)}{x^2} \, dx+8556250 \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {5989375}{2 x^2}-\frac {1369000}{x}-547600 \log (x)-\frac {4278125 \log (x)}{x^2}+\frac {4107000 \log (x)}{x}+136900 \log ^2(x)+\frac {855625 \log ^2(x)}{x^2}-\frac {684500 \log ^2(x)}{x}+1369000 \int \frac {\log (x)}{x^2} \, dx-1711250 \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {3422500}{x^2}-\frac {2738000}{x}-547600 \log (x)-\frac {3422500 \log (x)}{x^2}+\frac {2738000 \log (x)}{x}+136900 \log ^2(x)+\frac {855625 \log ^2(x)}{x^2}-\frac {684500 \log ^2(x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 57, normalized size = 2.59 \begin {gather*} 68450 \left (\frac {50}{x^2}-\frac {40}{x}-8 \log (x)-\frac {50 \log (x)}{x^2}+\frac {40 \log (x)}{x}+2 \log ^2(x)+\frac {25 \log ^2(x)}{2 x^2}-\frac {10 \log ^2(x)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10267500 + 5476000*x - 547600*x^2 + (8556250 - 4107000*x + 273800*x^2)*Log[x] + (-1711250 + 684500
*x)*Log[x]^2)/x^3,x]

[Out]

68450*(50/x^2 - 40/x - 8*Log[x] - (50*Log[x])/x^2 + (40*Log[x])/x + 2*Log[x]^2 + (25*Log[x]^2)/(2*x^2) - (10*L
og[x]^2)/x)

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fricas [A]  time = 0.72, size = 39, normalized size = 1.77 \begin {gather*} \frac {34225 \, {\left ({\left (4 \, x^{2} - 20 \, x + 25\right )} \log \relax (x)^{2} - 4 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \relax (x) - 80 \, x + 100\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log(x)-547600*x^2+5476000*x-10267500)/x^
3,x, algorithm="fricas")

[Out]

34225*((4*x^2 - 20*x + 25)*log(x)^2 - 4*(4*x^2 - 20*x + 25)*log(x) - 80*x + 100)/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {68450 \, {\left (5 \, {\left (2 \, x - 5\right )} \log \relax (x)^{2} - 8 \, x^{2} + {\left (4 \, x^{2} - 60 \, x + 125\right )} \log \relax (x) + 80 \, x - 150\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log(x)-547600*x^2+5476000*x-10267500)/x^
3,x, algorithm="giac")

[Out]

integrate(68450*(5*(2*x - 5)*log(x)^2 - 8*x^2 + (4*x^2 - 60*x + 125)*log(x) + 80*x - 150)/x^3, x)

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maple [B]  time = 0.02, size = 48, normalized size = 2.18

method	result	size
norman	\(\frac {3422500-547600 x^{2} \ln \relax (x )-2738000 x +855625 \ln \relax (x )^{2}+2738000 x \ln \relax (x )-684500 x \ln \relax (x )^{2}+136900 x^{2} \ln \relax (x )^{2}-3422500 \ln \relax (x )}{x^{2}}\)	\(48\)
risch	\(\frac {34225 \left (4 x^{2}-20 x +25\right ) \ln \relax (x )^{2}}{x^{2}}+\frac {684500 \left (4 x -5\right ) \ln \relax (x )}{x^{2}}-\frac {136900 \left (4 x^{2} \ln \relax (x )+20 x -25\right )}{x^{2}}\)	\(50\)
default	\(-\frac {684500 \ln \relax (x )^{2}}{x}+\frac {2738000 \ln \relax (x )}{x}-\frac {2738000}{x}+136900 \ln \relax (x )^{2}+\frac {855625 \ln \relax (x )^{2}}{x^{2}}-\frac {3422500 \ln \relax (x )}{x^{2}}+\frac {3422500}{x^{2}}-547600 \ln \relax (x )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((684500*x-1711250)*ln(x)^2+(273800*x^2-4107000*x+8556250)*ln(x)-547600*x^2+5476000*x-10267500)/x^3,x,meth
od=_RETURNVERBOSE)

[Out]

(3422500-547600*x^2*ln(x)-2738000*x+855625*ln(x)^2+2738000*x*ln(x)-684500*x*ln(x)^2+136900*x^2*ln(x)^2-3422500
*ln(x))/x^2

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maxima [B]  time = 0.36, size = 67, normalized size = 3.05 \begin {gather*} 136900 \, \log \relax (x)^{2} - \frac {684500 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 2\right )}}{x} + \frac {4107000 \, \log \relax (x)}{x} + \frac {855625 \, {\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )}}{2 \, x^{2}} - \frac {1369000}{x} - \frac {4278125 \, \log \relax (x)}{x^{2}} + \frac {5989375}{2 \, x^{2}} - 547600 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((684500*x-1711250)*log(x)^2+(273800*x^2-4107000*x+8556250)*log(x)-547600*x^2+5476000*x-10267500)/x^
3,x, algorithm="maxima")

[Out]

136900*log(x)^2 - 684500*(log(x)^2 + 2*log(x) + 2)/x + 4107000*log(x)/x + 855625/2*(2*log(x)^2 + 2*log(x) + 1)
/x^2 - 1369000/x - 4278125*log(x)/x^2 + 5989375/2/x^2 - 547600*log(x)

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mupad [B]  time = 5.16, size = 47, normalized size = 2.14 \begin {gather*} \frac {x\,\left (855625\,{\ln \relax (x)}^2-3422500\,\ln \relax (x)+3422500\right )-x^2\,\left (684500\,{\ln \relax (x)}^2-2738000\,\ln \relax (x)+2738000\right )}{x^3}-547600\,\ln \relax (x)+136900\,{\ln \relax (x)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5476000*x + log(x)*(273800*x^2 - 4107000*x + 8556250) - 547600*x^2 + log(x)^2*(684500*x - 1711250) - 1026
7500)/x^3,x)

[Out]

(x*(855625*log(x)^2 - 3422500*log(x) + 3422500) - x^2*(684500*log(x)^2 - 2738000*log(x) + 2738000))/x^3 - 5476
00*log(x) + 136900*log(x)^2

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sympy [B]  time = 0.21, size = 42, normalized size = 1.91 \begin {gather*} - 547600 \log {\relax (x )} + \frac {\left (2738000 x - 3422500\right ) \log {\relax (x )}}{x^{2}} - \frac {2738000 x - 3422500}{x^{2}} + \frac {\left (136900 x^{2} - 684500 x + 855625\right ) \log {\relax (x )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((684500*x-1711250)*ln(x)**2+(273800*x**2-4107000*x+8556250)*ln(x)-547600*x**2+5476000*x-10267500)/x
**3,x)

[Out]

-547600*log(x) + (2738000*x - 3422500)*log(x)/x**2 - (2738000*x - 3422500)/x**2 + (136900*x**2 - 684500*x + 85
5625)*log(x)**2/x**2

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