\(\int \frac {-162+460 x-75 x^2+(54-153 x+24 x^2) \log (6-x)}{-6 x^{10}+x^{11}} \, dx\) [6286]

Optimal result

Integrand size = 39, antiderivative size = 19 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {(-1+3 x) (3-\log (6-x))}{x^9} \]

[Out]

(-1+3*x)/x^9*(3-ln(6-x))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1607, 6874, 907, 45, 2461, 12, 78} \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {3}{x^9}+\frac {\log (6-x)}{x^9}+\frac {9}{x^8}-\frac {3 \log (6-x)}{x^8} \]

[In]

Int[(-162 + 460*x - 75*x^2 + (54 - 153*x + 24*x^2)*Log[6 - x])/(-6*x^10 + x^11),x]

[Out]

-3/x^9 + 9/x^8 + Log[6 - x]/x^9 - (3*Log[6 - x])/x^8

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2461

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[a + b*Log[c*(d + e*x)^n], u, x] - Dist[b*e*n, Int[SimplifyI
ntegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x]
&& IntegerQ[m] && IntegerQ[q] && IntegerQ[r]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{(-6+x) x^{10}} \, dx \\ & = \int \left (\frac {-162+460 x-75 x^2}{(-6+x) x^{10}}+\frac {3 (-3+8 x) \log (6-x)}{x^{10}}\right ) \, dx \\ & = 3 \int \frac {(-3+8 x) \log (6-x)}{x^{10}} \, dx+\int \frac {-162+460 x-75 x^2}{(-6+x) x^{10}} \, dx \\ & = \frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8}+3 \int \frac {1-3 x}{3 (6-x) x^9} \, dx+\int \left (-\frac {17}{10077696 (-6+x)}+\frac {27}{x^{10}}-\frac {433}{6 x^9}+\frac {17}{36 x^8}+\frac {17}{216 x^7}+\frac {17}{1296 x^6}+\frac {17}{7776 x^5}+\frac {17}{46656 x^4}+\frac {17}{279936 x^3}+\frac {17}{1679616 x^2}+\frac {17}{10077696 x}\right ) \, dx \\ & = -\frac {3}{x^9}+\frac {433}{48 x^8}-\frac {17}{252 x^7}-\frac {17}{1296 x^6}-\frac {17}{6480 x^5}-\frac {17}{31104 x^4}-\frac {17}{139968 x^3}-\frac {17}{559872 x^2}-\frac {17}{1679616 x}-\frac {17 \log (6-x)}{10077696}+\frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8}+\frac {17 \log (x)}{10077696}+\int \frac {1-3 x}{(6-x) x^9} \, dx \\ & = -\frac {3}{x^9}+\frac {433}{48 x^8}-\frac {17}{252 x^7}-\frac {17}{1296 x^6}-\frac {17}{6480 x^5}-\frac {17}{31104 x^4}-\frac {17}{139968 x^3}-\frac {17}{559872 x^2}-\frac {17}{1679616 x}-\frac {17 \log (6-x)}{10077696}+\frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8}+\frac {17 \log (x)}{10077696}+\int \left (\frac {17}{10077696 (-6+x)}+\frac {1}{6 x^9}-\frac {17}{36 x^8}-\frac {17}{216 x^7}-\frac {17}{1296 x^6}-\frac {17}{7776 x^5}-\frac {17}{46656 x^4}-\frac {17}{279936 x^3}-\frac {17}{1679616 x^2}-\frac {17}{10077696 x}\right ) \, dx \\ & = -\frac {3}{x^9}+\frac {9}{x^8}+\frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {3}{x^9}+\frac {9}{x^8}+\frac {\log (6-x)}{x^9}-\frac {3 \log (6-x)}{x^8} \]

[In]

Integrate[(-162 + 460*x - 75*x^2 + (54 - 153*x + 24*x^2)*Log[6 - x])/(-6*x^10 + x^11),x]

[Out]

-3/x^9 + 9/x^8 + Log[6 - x]/x^9 - (3*Log[6 - x])/x^8

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32

method	result	size
norman	\(\frac {-3+9 x -3 \ln \left (-x +6\right ) x +\ln \left (-x +6\right )}{x^{9}}\)	\(25\)
risch	\(-\frac {\left (-1+3 x \right ) \ln \left (-x +6\right )}{x^{9}}+\frac {9 x -3}{x^{9}}\)	\(28\)
parallelrisch	\(-\frac {36+36 \ln \left (-x +6\right ) x -108 x -12 \ln \left (-x +6\right )}{12 x^{9}}\)	\(28\)
derivativedivides	\(-\frac {3}{x^{9}}+\frac {9}{x^{8}}-\frac {17 \ln \left (-x +6\right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}[phone] x -45349632\right )}{10077696 x^{9}}\)	\(175\)
default	\(-\frac {3}{x^{9}}+\frac {9}{x^{8}}-\frac {17 \ln \left (-x +6\right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}[phone] x -45349632\right )}{10077696 x^{9}}\)	\(175\)
parts	\(-\frac {17 \ln \left (-x \right )}{10077696}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{7}-48 \left (-x +6\right )^{6}+1008 \left (-x +6\right )^{5}-12096 \left (-x +6\right )^{4}+90720 \left (-x +6\right )^{3}-435456 \left (-x +6\right )^{2}-1306368 x +5598720\right )}{559872 x^{8}}+\frac {9}{x^{8}}+\frac {\ln \left (-x +6\right ) \left (-x +6\right ) \left (\left (-x +6\right )^{8}-54 \left (-x +6\right )^{7}+1296 \left (-x +6\right )^{6}-18144 \left (-x +6\right )^{5}+163296 \left (-x +6\right )^{4}-979776 \left (-x +6\right )^{3}+3919104 \left (-x +6\right )^{2}[phone] x -45349632\right )}{10077696 x^{9}}-\frac {3}{x^{9}}+\frac {17 \ln \left (x \right )}{10077696}-\frac {17 \ln \left (-6+x \right )}{10077696}\)	\(183\)

[In]

int(((24*x^2-153*x+54)*ln(-x+6)-75*x^2+460*x-162)/(x^11-6*x^10),x,method=_RETURNVERBOSE)

[Out]

(-3+9*x-3*ln(-x+6)*x+ln(-x+6))/x^9

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (-x + 6\right ) - 9 \, x + 3}{x^{9}} \]

[In]

integrate(((24*x^2-153*x+54)*log(-x+6)-75*x^2+460*x-162)/(x^11-6*x^10),x, algorithm="fricas")

[Out]

-((3*x - 1)*log(-x + 6) - 9*x + 3)/x^9

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {\left (1 - 3 x\right ) \log {\left (6 - x \right )}}{x^{9}} - \frac {3 - 9 x}{x^{9}} \]

[In]

integrate(((24*x**2-153*x+54)*ln(-x+6)-75*x**2+460*x-162)/(x**11-6*x**10),x)

[Out]

(1 - 3*x)*log(6 - x)/x**9 - (3 - 9*x)/x**9

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.89 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=\frac {210 \, x^{8} + 630 \, x^{7} + 2520 \, x^{6} + 11340 \, x^{5} + 54432 \, x^{4} + 272160 \, x^{3} + 1399680 \, x^{2} + 35 \, {\left (x^{9} - 1119744 \, x + 373248\right )} \log \left (-x + 6\right ) + 124921440 \, x}{13063680 \, x^{9}} - \frac {35 \, x^{8} + 105 \, x^{7} + 420 \, x^{6} + 1890 \, x^{5} + 9072 \, x^{4} + 45360 \, x^{3} + 233280 \, x^{2} + 1224720 \, x + 6531840}{2177280 \, x^{9}} - \frac {1}{373248} \, \log \left (x - 6\right ) \]

[In]

integrate(((24*x^2-153*x+54)*log(-x+6)-75*x^2+460*x-162)/(x^11-6*x^10),x, algorithm="maxima")

[Out]

1/13063680*(210*x^8 + 630*x^7 + 2520*x^6 + 11340*x^5 + 54432*x^4 + 272160*x^3 + 1399680*x^2 + 35*(x^9 - 111974
4*x + 373248)*log(-x + 6) + 124921440*x)/x^9 - 1/2177280*(35*x^8 + 105*x^7 + 420*x^6 + 1890*x^5 + 9072*x^4 + 4
5360*x^3 + 233280*x^2 + 1224720*x + 6531840)/x^9 - 1/373248*log(x - 6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 7.53 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (-x + 6\right )}{{\left (x - 6\right )}^{9} + 54 \, {\left (x - 6\right )}^{8} + 1296 \, {\left (x - 6\right )}^{7} + 18144 \, {\left (x - 6\right )}^{6} + 163296 \, {\left (x - 6\right )}^{5} + 979776 \, {\left (x - 6\right )}^{4} + 3919104 \, {\left (x - 6\right )}^{3} + 10077696 \, {\left (x - 6\right )}^{2} + 15116544 \, x - 80621568} + \frac {3 \, {\left (3 \, x - 1\right )}}{{\left (x - 6\right )}^{9} + 54 \, {\left (x - 6\right )}^{8} + 1296 \, {\left (x - 6\right )}^{7} + 18144 \, {\left (x - 6\right )}^{6} + 163296 \, {\left (x - 6\right )}^{5} + 979776 \, {\left (x - 6\right )}^{4} + 3919104 \, {\left (x - 6\right )}^{3} + 10077696 \, {\left (x - 6\right )}^{2} + 15116544 \, x - 80621568} \]

[In]

integrate(((24*x^2-153*x+54)*log(-x+6)-75*x^2+460*x-162)/(x^11-6*x^10),x, algorithm="giac")

[Out]

-(3*x - 1)*log(-x + 6)/((x - 6)^9 + 54*(x - 6)^8 + 1296*(x - 6)^7 + 18144*(x - 6)^6 + 163296*(x - 6)^5 + 97977
6*(x - 6)^4 + 3919104*(x - 6)^3 + 10077696*(x - 6)^2 + 15116544*x - 80621568) + 3*(3*x - 1)/((x - 6)^9 + 54*(x
- 6)^8 + 1296*(x - 6)^7 + 18144*(x - 6)^6 + 163296*(x - 6)^5 + 979776*(x - 6)^4 + 3919104*(x - 6)^3 + 1007769
6*(x - 6)^2 + 15116544*x - 80621568)

Mupad [B] (verification not implemented)

Time = 11.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-162+460 x-75 x^2+\left (54-153 x+24 x^2\right ) \log (6-x)}{-6 x^{10}+x^{11}} \, dx=-\frac {\left (\ln \left (6-x\right )-3\right )\,\left (3\,x-1\right )}{x^9} \]

[In]

int(-(460*x + log(6 - x)*(24*x^2 - 153*x + 54) - 75*x^2 - 162)/(6*x^10 - x^11),x)

[Out]

-((log(6 - x) - 3)*(3*x - 1))/x^9