\(\int \frac {22500 x-7800 x^2+100 x^3+(22500 x-7800 x^2+100 x^3) \log (2)+(-22500+37500 x-15100 x^2+100 x^3+(-22500+37500 x-15100 x^2+100 x^3) \log (2)) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx\) [9463]

Optimal result

Integrand size = 85, antiderivative size = 25 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {4 (3-x) x (1+\log (2)) \log (-1+x)}{3-\frac {x}{25}} \]

[Out]

4*(-x+3)*x*ln(-1+x)*(1+ln(2))/(3-1/25*x)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6820, 12, 6874, 147, 2465, 2436, 2332, 2442, 36, 31} \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=7300 (1+\log (2)) \log (1-x)-100 (1-x) (1+\log (2)) \log (x-1)-\frac {540000 (1+\log (2)) \log (x-1)}{75-x} \]

[In]

Int[(22500*x - 7800*x^2 + 100*x^3 + (22500*x - 7800*x^2 + 100*x^3)*Log[2] + (-22500 + 37500*x - 15100*x^2 + 10
0*x^3 + (-22500 + 37500*x - 15100*x^2 + 100*x^3)*Log[2])*Log[-1 + x])/(-5625 + 5775*x - 151*x^2 + x^3),x]

[Out]

7300*(1 + Log[2])*Log[1 - x] - 100*(1 - x)*(1 + Log[2])*Log[-1 + x] - (540000*(1 + Log[2])*Log[-1 + x])/(75 -
x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && (IGtQ[m, 0] || IntegersQ[m, n])

Rule 2332

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

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {100 (1+\log (2)) \left (-x \left (225-78 x+x^2\right )-\left (-225+375 x-151 x^2+x^3\right ) \log (-1+x)\right )}{(1-x) (75-x)^2} \, dx \\ & = (100 (1+\log (2))) \int \frac {-x \left (225-78 x+x^2\right )-\left (-225+375 x-151 x^2+x^3\right ) \log (-1+x)}{(1-x) (75-x)^2} \, dx \\ & = (100 (1+\log (2))) \int \left (\frac {(-3+x) x}{(-75+x) (-1+x)}+\frac {\left (225-150 x+x^2\right ) \log (-1+x)}{(-75+x)^2}\right ) \, dx \\ & = (100 (1+\log (2))) \int \frac {(-3+x) x}{(-75+x) (-1+x)} \, dx+(100 (1+\log (2))) \int \frac {\left (225-150 x+x^2\right ) \log (-1+x)}{(-75+x)^2} \, dx \\ & = (100 (1+\log (2))) \int \left (1+\frac {2700}{37 (-75+x)}+\frac {1}{37 (-1+x)}\right ) \, dx+(100 (1+\log (2))) \int \left (\log (-1+x)-\frac {5400 \log (-1+x)}{(-75+x)^2}\right ) \, dx \\ & = 100 x (1+\log (2))+\frac {100}{37} (1+\log (2)) \log (1-x)+\frac {270000}{37} (1+\log (2)) \log (75-x)+(100 (1+\log (2))) \int \log (-1+x) \, dx-(540000 (1+\log (2))) \int \frac {\log (-1+x)}{(-75+x)^2} \, dx \\ & = 100 x (1+\log (2))+\frac {100}{37} (1+\log (2)) \log (1-x)+\frac {270000}{37} (1+\log (2)) \log (75-x)-\frac {540000 (1+\log (2)) \log (-1+x)}{75-x}+(100 (1+\log (2))) \text {Subst}(\int \log (x) \, dx,x,-1+x)-(540000 (1+\log (2))) \int \frac {1}{(-75+x) (-1+x)} \, dx \\ & = \frac {100}{37} (1+\log (2)) \log (1-x)+\frac {270000}{37} (1+\log (2)) \log (75-x)-100 (1-x) (1+\log (2)) \log (-1+x)-\frac {540000 (1+\log (2)) \log (-1+x)}{75-x}-\frac {1}{37} (270000 (1+\log (2))) \int \frac {1}{-75+x} \, dx+\frac {1}{37} (270000 (1+\log (2))) \int \frac {1}{-1+x} \, dx \\ & = 7300 (1+\log (2)) \log (1-x)-100 (1-x) (1+\log (2)) \log (-1+x)-\frac {540000 (1+\log (2)) \log (-1+x)}{75-x} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100}{37} (1+\log (2)) \left (5400 \text {arctanh}\left (\frac {1}{37} (-38+x)\right )+\log (1-x)+2700 \log (75-x)+\frac {199800 \log (-1+x)}{-75+x}+37 (-1+x) \log (-1+x)\right ) \]

[In]

Integrate[(22500*x - 7800*x^2 + 100*x^3 + (22500*x - 7800*x^2 + 100*x^3)*Log[2] + (-22500 + 37500*x - 15100*x^
2 + 100*x^3 + (-22500 + 37500*x - 15100*x^2 + 100*x^3)*Log[2])*Log[-1 + x])/(-5625 + 5775*x - 151*x^2 + x^3),x
]

[Out]

(100*(1 + Log[2])*(5400*ArcTanh[(-38 + x)/37] + Log[1 - x] + 2700*Log[75 - x] + (199800*Log[-1 + x])/(-75 + x)
+ 37*(-1 + x)*Log[-1 + x]))/37

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36

method	result	size
norman	\(\frac {\left (100 \ln \left (2\right )+100\right ) x^{2} \ln \left (-1+x \right )+\left (-300-300 \ln \left (2\right )\right ) x \ln \left (-1+x \right )}{x -75}\)	\(34\)
parallelrisch	\(\frac {100 \ln \left (2\right ) \ln \left (-1+x \right ) x^{2}-300 \ln \left (2\right ) \ln \left (-1+x \right ) x +100 \ln \left (-1+x \right ) x^{2}-300 \ln \left (-1+x \right ) x}{x -75}\)	\(44\)
risch	\(\frac {100 \left (x^{2} \ln \left (2\right )-75 x \ln \left (2\right )+x^{2}+5400 \ln \left (2\right )-75 x +5400\right ) \ln \left (-1+x \right )}{x -75}+7200 \ln \left (2\right ) \ln \left (-1+x \right )+7200 \ln \left (-1+x \right )\)	\(50\)
derivativedivides	\(100 \ln \left (2\right ) \left (\left (-1+x \right ) \ln \left (-1+x \right )+\frac {2700 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {\ln \left (-1+x \right )}{37}\right )+100 \left (-1+x \right ) \ln \left (-1+x \right )+\frac {270000 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {100 \ln \left (-1+x \right )}{37}\)	\(64\)
default	\(100 \ln \left (2\right ) \left (\left (-1+x \right ) \ln \left (-1+x \right )+\frac {2700 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {\ln \left (-1+x \right )}{37}\right )+100 \left (-1+x \right ) \ln \left (-1+x \right )+\frac {270000 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {100 \ln \left (-1+x \right )}{37}\)	\(64\)
parts	\(100 \left (1+\ln \left (2\right )\right ) \left (x +\frac {\ln \left (-1+x \right )}{37}+\frac {2700 \ln \left (x -75\right )}{37}\right )+100 \left (-1+x \right ) \ln \left (-1+x \right )-100 x +100+100 \ln \left (2\right ) \left (\left (-1+x \right ) \ln \left (-1+x \right )-x +1\right )+100 \left (-5400-5400 \ln \left (2\right )\right ) \left (\frac {\ln \left (x -75\right )}{74}-\frac {\ln \left (-1+x \right ) \left (-1+x \right )}{74 \left (x -75\right )}\right )\)	\(81\)

[In]

int((((100*x^3-15100*x^2+37500*x-22500)*ln(2)+100*x^3-15100*x^2+37500*x-22500)*ln(-1+x)+(100*x^3-7800*x^2+2250
0*x)*ln(2)+100*x^3-7800*x^2+22500*x)/(x^3-151*x^2+5775*x-5625),x,method=_RETURNVERBOSE)

[Out]

((100*ln(2)+100)*x^2*ln(-1+x)+(-300-300*ln(2))*x*ln(-1+x))/(x-75)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100 \, {\left (x^{2} + {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) - 3 \, x\right )} \log \left (x - 1\right )}{x - 75} \]

[In]

integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+37500*x-22500)*log(-1+x)+(100*x^3-7800*
x^2+22500*x)*log(2)+100*x^3-7800*x^2+22500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="fricas")

[Out]

100*(x^2 + (x^2 - 3*x)*log(2) - 3*x)*log(x - 1)/(x - 75)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\left (7200 \log {\left (2 \right )} + 7200\right ) \log {\left (x - 1 \right )} + \frac {\left (100 x^{2} \log {\left (2 \right )} + 100 x^{2} - 7500 x - 7500 x \log {\left (2 \right )} + 540000 \log {\left (2 \right )} + 540000\right ) \log {\left (x - 1 \right )}}{x - 75} \]

[In]

integrate((((100*x**3-15100*x**2+37500*x-22500)*ln(2)+100*x**3-15100*x**2+37500*x-22500)*ln(-1+x)+(100*x**3-78
00*x**2+22500*x)*ln(2)+100*x**3-7800*x**2+22500*x)/(x**3-151*x**2+5775*x-5625),x)

[Out]

(7200*log(2) + 7200)*log(x - 1) + (100*x**2*log(2) + 100*x**2 - 7500*x - 7500*x*log(2) + 540000*log(2) + 54000
0)*log(x - 1)/(x - 75)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100 \, {\left (x^{2} {\left (\log \left (2\right ) + 1\right )} - 3 \, x {\left (\log \left (2\right ) + 1\right )}\right )} \log \left (x - 1\right )}{x - 75} \]

[In]

integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+37500*x-22500)*log(-1+x)+(100*x^3-7800*
x^2+22500*x)*log(2)+100*x^3-7800*x^2+22500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="maxima")

[Out]

100*(x^2*(log(2) + 1) - 3*x*(log(2) + 1))*log(x - 1)/(x - 75)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=100 \, {\left (x {\left (\log \left (2\right ) + 1\right )} + \frac {5400 \, {\left (\log \left (2\right ) + 1\right )}}{x - 75}\right )} \log \left (x - 1\right ) + 7200 \, {\left (\log \left (2\right ) + 1\right )} \log \left (x - 1\right ) \]

[In]

integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+37500*x-22500)*log(-1+x)+(100*x^3-7800*
x^2+22500*x)*log(2)+100*x^3-7800*x^2+22500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="giac")

[Out]

100*(x*(log(2) + 1) + 5400*(log(2) + 1)/(x - 75))*log(x - 1) + 7200*(log(2) + 1)*log(x - 1)

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100\,x\,\ln \left (x-1\right )\,\left (\ln \left (2\right )+1\right )\,\left (x-3\right )}{x-75} \]

[In]

int((22500*x + log(2)*(22500*x - 7800*x^2 + 100*x^3) + log(x - 1)*(37500*x + log(2)*(37500*x - 15100*x^2 + 100
*x^3 - 22500) - 15100*x^2 + 100*x^3 - 22500) - 7800*x^2 + 100*x^3)/(5775*x - 151*x^2 + x^3 - 5625),x)

[Out]

(100*x*log(x - 1)*(log(2) + 1)*(x - 3))/(x - 75)