\(\int \frac {-10267500+5476000 x-547600 x^2+(8556250-4107000 x+273800 x^2) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx\) [7656]

Optimal result

Integrand size = 37, antiderivative size = 22 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=5+\frac {34225 (5-2 x)^2 (2-\log (x))^2}{x^2} \]

[Out]

5+34225*(5-2*x)^2*(-ln(x)+2)^2/x^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(22)=44\).

Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {14, 2404, 2341, 2338, 2395, 2342} \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {3422500}{x^2}+\frac {855625 \log ^2(x)}{x^2}-\frac {3422500 \log (x)}{x^2}-\frac {2738000}{x}-\frac {684500 \log ^2(x)}{x}+136900 \log ^2(x)+\frac {2738000 \log (x)}{x}-547600 \log (x) \]

[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 2338

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 2341

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 2342

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 2395

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 2404

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {34225 (5-2 x)^2 (-2+\log (x))^2}{x^2} \]

[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]

(34225*(5 - 2*x)^2*(-2 + Log[x])^2)/x^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {34225 \, {\left ({\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right )^{2} - 4 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right ) - 80 \, x + 100\right )}}{x^{2}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).

Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=- 547600 \log {\left (x \right )} + \frac {\left (2738000 x - 3422500\right ) \log {\left (x \right )}}{x^{2}} - \frac {2738000 x - 3422500}{x^{2}} + \frac {\left (136900 x^{2} - 684500 x + 855625\right ) \log {\left (x \right )}^{2}}{x^{2}} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (20) = 40\).

Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=136900 \, \log \left (x\right )^{2} - \frac {684500 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )}}{x} + \frac {4107000 \, \log \left (x\right )}{x} + \frac {855625 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{2 \, x^{2}} - \frac {1369000}{x} - \frac {4278125 \, \log \left (x\right )}{x^{2}} + \frac {5989375}{2 \, x^{2}} - 547600 \, \log \left (x\right ) \]

[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)

Giac [F]

\[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\int { \frac {68450 \, {\left (5 \, {\left (2 \, x - 5\right )} \log \left (x\right )^{2} - 8 \, x^{2} + {\left (4 \, x^{2} - 60 \, x + 125\right )} \log \left (x\right ) + 80 \, x - 150\right )}}{x^{3}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 12.70 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-10267500+5476000 x-547600 x^2+\left (8556250-4107000 x+273800 x^2\right ) \log (x)+(-1711250+684500 x) \log ^2(x)}{x^3} \, dx=\frac {x\,\left (855625\,{\ln \left (x\right )}^2-3422500\,\ln \left (x\right )+3422500\right )-x^2\,\left (684500\,{\ln \left (x\right )}^2-2738000\,\ln \left (x\right )+2738000\right )}{x^3}-547600\,\ln \left (x\right )+136900\,{\ln \left (x\right )}^2 \]

[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