3.95.45 \(\int \frac {7938+17406 x-36774 x^2+22422 x^3-6186 x^4+906 x^5-82 x^6+2 x^7+(-486-1188 x+1926 x^2-960 x^3+222 x^4-28 x^5+2 x^6) \log (x)}{-243 x+405 x^2-270 x^3+90 x^4-15 x^5+x^6} \, dx\)

Optimal. Leaf size=21 \[ \left (-3 \left (3-\frac {2}{3-x}\right )^2+x+\log (x)\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.75, antiderivative size = 79, normalized size of antiderivative = 3.76, number of steps used = 21, number of rules used = 12, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6688, 12, 6742, 1850, 1620, 2357, 2295, 2319, 44, 2314, 31, 2301} \begin {gather*} x^2-54 x-\frac {1704}{3-x}+\frac {1872}{(3-x)^2}-\frac {864}{(3-x)^3}+\frac {144}{(3-x)^4}+\log ^2(x)+\frac {24 x \log (x)}{3-x}+2 x \log (x)-\frac {24 \log (x)}{(3-x)^2}-30 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7938 + 17406*x - 36774*x^2 + 22422*x^3 - 6186*x^4 + 906*x^5 - 82*x^6 + 2*x^7 + (-486 - 1188*x + 1926*x^2
- 960*x^3 + 222*x^4 - 28*x^5 + 2*x^6)*Log[x])/(-243*x + 405*x^2 - 270*x^3 + 90*x^4 - 15*x^5 + x^6),x]

[Out]

144/(3 - x)^4 - 864/(3 - x)^3 + 1872/(3 - x)^2 - 1704/(3 - x) - 54*x + x^2 - 30*Log[x] - (24*Log[x])/(3 - x)^2
+ 2*x*Log[x] + (24*x*Log[x])/(3 - x) + Log[x]^2

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

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 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 2314

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

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (27+84 x-54 x^2+8 x^3-x^4\right ) \left (-147+135 x-33 x^2+x^3+(-3+x)^2 \log (x)\right )}{(3-x)^5 x} \, dx\\ &=2 \int \frac {\left (27+84 x-54 x^2+8 x^3-x^4\right ) \left (-147+135 x-33 x^2+x^3+(-3+x)^2 \log (x)\right )}{(3-x)^5 x} \, dx\\ &=2 \int \left (\frac {135 \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5}-\frac {147 \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5 x}-\frac {33 x \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5}+\frac {x^2 \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5}+\frac {\left (-27-84 x+54 x^2-8 x^3+x^4\right ) \log (x)}{(-3+x)^3 x}\right ) \, dx\\ &=2 \int \frac {x^2 \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5} \, dx+2 \int \frac {\left (-27-84 x+54 x^2-8 x^3+x^4\right ) \log (x)}{(-3+x)^3 x} \, dx-66 \int \frac {x \left (-27-84 x+54 x^2-8 x^3+x^4\right )}{(-3+x)^5} \, dx+270 \int \frac {-27-84 x+54 x^2-8 x^3+x^4}{(-3+x)^5} \, dx-294 \int \frac {-27-84 x+54 x^2-8 x^3+x^4}{(-3+x)^5 x} \, dx\\ &=2 \int \left (7+\frac {648}{(-3+x)^5}+\frac {1620}{(-3+x)^4}+\frac {1188}{(-3+x)^3}+\frac {384}{(-3+x)^2}+\frac {69}{-3+x}+x\right ) \, dx+2 \int \left (\log (x)+\frac {24 \log (x)}{(-3+x)^3}+\frac {36 \log (x)}{(-3+x)^2}+\frac {\log (x)}{x}\right ) \, dx-66 \int \left (1+\frac {216}{(-3+x)^5}+\frac {468}{(-3+x)^4}+\frac {240}{(-3+x)^3}+\frac {48}{(-3+x)^2}+\frac {7}{-3+x}\right ) \, dx+270 \int \left (\frac {72}{(-3+x)^5}+\frac {132}{(-3+x)^4}+\frac {36}{(-3+x)^3}+\frac {4}{(-3+x)^2}+\frac {1}{-3+x}\right ) \, dx-294 \int \left (\frac {24}{(-3+x)^5}+\frac {36}{(-3+x)^4}+\frac {4}{3 (-3+x)^2}-\frac {1}{9 (-3+x)}+\frac {1}{9 x}\right ) \, dx\\ &=\frac {144}{(3-x)^4}-\frac {864}{(3-x)^3}+\frac {1872}{(3-x)^2}-\frac {1712}{3-x}-52 x+x^2-\frac {64}{3} \log (3-x)-\frac {98 \log (x)}{3}+2 \int \log (x) \, dx+2 \int \frac {\log (x)}{x} \, dx+48 \int \frac {\log (x)}{(-3+x)^3} \, dx+72 \int \frac {\log (x)}{(-3+x)^2} \, dx\\ &=\frac {144}{(3-x)^4}-\frac {864}{(3-x)^3}+\frac {1872}{(3-x)^2}-\frac {1712}{3-x}-54 x+x^2-\frac {64}{3} \log (3-x)-\frac {98 \log (x)}{3}-\frac {24 \log (x)}{(3-x)^2}+2 x \log (x)+\frac {24 x \log (x)}{3-x}+\log ^2(x)+24 \int \frac {1}{-3+x} \, dx+24 \int \frac {1}{(-3+x)^2 x} \, dx\\ &=\frac {144}{(3-x)^4}-\frac {864}{(3-x)^3}+\frac {1872}{(3-x)^2}-\frac {1712}{3-x}-54 x+x^2+\frac {8}{3} \log (3-x)-\frac {98 \log (x)}{3}-\frac {24 \log (x)}{(3-x)^2}+2 x \log (x)+\frac {24 x \log (x)}{3-x}+\log ^2(x)+24 \int \left (\frac {1}{3 (-3+x)^2}-\frac {1}{9 (-3+x)}+\frac {1}{9 x}\right ) \, dx\\ &=\frac {144}{(3-x)^4}-\frac {864}{(3-x)^3}+\frac {1872}{(3-x)^2}-\frac {1704}{3-x}-54 x+x^2-30 \log (x)-\frac {24 \log (x)}{(3-x)^2}+2 x \log (x)+\frac {24 x \log (x)}{3-x}+\log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.09, size = 66, normalized size = 3.14 \begin {gather*} \frac {-31608+31266 x-7551 x^2-1320 x^3+702 x^4-66 x^5+x^6+2 (-3+x)^2 \left (-147+135 x-33 x^2+x^3\right ) \log (x)+(-3+x)^4 \log ^2(x)}{(-3+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7938 + 17406*x - 36774*x^2 + 22422*x^3 - 6186*x^4 + 906*x^5 - 82*x^6 + 2*x^7 + (-486 - 1188*x + 192
6*x^2 - 960*x^3 + 222*x^4 - 28*x^5 + 2*x^6)*Log[x])/(-243*x + 405*x^2 - 270*x^3 + 90*x^4 - 15*x^5 + x^6),x]

[Out]

(-31608 + 31266*x - 7551*x^2 - 1320*x^3 + 702*x^4 - 66*x^5 + x^6 + 2*(-3 + x)^2*(-147 + 135*x - 33*x^2 + x^3)*
Log[x] + (-3 + x)^4*Log[x]^2)/(-3 + x)^4

________________________________________________________________________________________

fricas [B]  time = 0.56, size = 99, normalized size = 4.71 \begin {gather*} \frac {x^{6} - 66 \, x^{5} + 702 \, x^{4} - 1320 \, x^{3} + {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} \log \relax (x)^{2} - 7551 \, x^{2} + 2 \, {\left (x^{5} - 39 \, x^{4} + 342 \, x^{3} - 1254 \, x^{2} + 2097 \, x - 1323\right )} \log \relax (x) + 31266 \, x - 31608}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-28*x^5+222*x^4-960*x^3+1926*x^2-1188*x-486)*log(x)+2*x^7-82*x^6+906*x^5-6186*x^4+22422*x^3-3
6774*x^2+17406*x+7938)/(x^6-15*x^5+90*x^4-270*x^3+405*x^2-243*x),x, algorithm="fricas")

[Out]

(x^6 - 66*x^5 + 702*x^4 - 1320*x^3 + (x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*log(x)^2 - 7551*x^2 + 2*(x^5 - 39*x^
4 + 342*x^3 - 1254*x^2 + 2097*x - 1323)*log(x) + 31266*x - 31608)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (x^{7} - 41 \, x^{6} + 453 \, x^{5} - 3093 \, x^{4} + 11211 \, x^{3} - 18387 \, x^{2} + {\left (x^{6} - 14 \, x^{5} + 111 \, x^{4} - 480 \, x^{3} + 963 \, x^{2} - 594 \, x - 243\right )} \log \relax (x) + 8703 \, x + 3969\right )}}{x^{6} - 15 \, x^{5} + 90 \, x^{4} - 270 \, x^{3} + 405 \, x^{2} - 243 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-28*x^5+222*x^4-960*x^3+1926*x^2-1188*x-486)*log(x)+2*x^7-82*x^6+906*x^5-6186*x^4+22422*x^3-3
6774*x^2+17406*x+7938)/(x^6-15*x^5+90*x^4-270*x^3+405*x^2-243*x),x, algorithm="giac")

[Out]

integrate(2*(x^7 - 41*x^6 + 453*x^5 - 3093*x^4 + 11211*x^3 - 18387*x^2 + (x^6 - 14*x^5 + 111*x^4 - 480*x^3 + 9
63*x^2 - 594*x - 243)*log(x) + 8703*x + 3969)/(x^6 - 15*x^5 + 90*x^4 - 270*x^3 + 405*x^2 - 243*x), x)

________________________________________________________________________________________

maple [B]  time = 0.10, size = 72, normalized size = 3.43

method	result	size
default	\(x^{2}-54 x -\frac {98 \ln \relax (x )}{3}+\frac {144}{\left (x -3\right )^{4}}+\frac {864}{\left (x -3\right )^{3}}+\frac {1872}{\left (x -3\right )^{2}}+\frac {1704}{x -3}+2 x \ln \relax (x )+\ln \relax (x )^{2}+\frac {8 \ln \relax (x ) x \left (x -6\right )}{3 \left (x -3\right )^{2}}-\frac {24 \ln \relax (x ) x}{x -3}\)	\(72\)
norman	\(\frac {x^{6}+x^{4} \ln \relax (x )^{2}-2998 x^{3}+1116 \ln \relax (x )+16164 x +\frac {5051 x^{4}}{6}-822 x \ln \relax (x )-\frac {284 x^{4} \ln \relax (x )}{9}+\frac {380 x^{3} \ln \relax (x )}{3}-66 x^{5}+81 \ln \relax (x )^{2}-108 x \ln \relax (x )^{2}+54 x^{2} \ln \relax (x )^{2}-12 x^{3} \ln \relax (x )^{2}+2 x^{5} \ln \relax (x )-\frac {40563}{2}}{\left (x -3\right )^{4}}-\frac {418 \ln \relax (x )}{9}\)	\(104\)
risch	\(\ln \relax (x )^{2}+\frac {2 \left (x^{3}-6 x^{2}-27 x +96\right ) \ln \relax (x )}{x^{2}-6 x +9}-\frac {-x^{6}+54 x^{4} \ln \relax (x )+66 x^{5}-648 x^{3} \ln \relax (x )-702 x^{4}+2916 x^{2} \ln \relax (x )+1320 x^{3}-5832 x \ln \relax (x )+7551 x^{2}+4374 \ln \relax (x )-31266 x +31608}{\left (x^{2}-6 x +9\right )^{2}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6-28*x^5+222*x^4-960*x^3+1926*x^2-1188*x-486)*ln(x)+2*x^7-82*x^6+906*x^5-6186*x^4+22422*x^3-36774*x^
2+17406*x+7938)/(x^6-15*x^5+90*x^4-270*x^3+405*x^2-243*x),x,method=_RETURNVERBOSE)

[Out]

x^2-54*x-98/3*ln(x)+144/(x-3)^4+864/(x-3)^3+1872/(x-3)^2+1704/(x-3)+2*x*ln(x)+ln(x)^2+8/3*ln(x)*x*(x-6)/(x-3)^
2-24*ln(x)*x/(x-3)

________________________________________________________________________________________

maxima [B]  time = 0.41, size = 336, normalized size = 16.00 \begin {gather*} x^{2} - 52 \, x - \frac {27 \, {\left (80 \, x^{3} - 630 \, x^{2} + 1692 \, x - 1539\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} + \frac {369 \, {\left (40 \, x^{3} - 300 \, x^{2} + 780 \, x - 693\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} - \frac {1359 \, {\left (16 \, x^{3} - 108 \, x^{2} + 264 \, x - 225\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} + \frac {3093 \, {\left (4 \, x^{3} - 18 \, x^{2} + 36 \, x - 27\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} + \frac {49 \, {\left (4 \, x^{3} - 42 \, x^{2} + 156 \, x - 225\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} - \frac {11211 \, {\left (2 \, x^{2} - 4 \, x + 3\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} - \frac {6 \, x^{3} - 3 \, {\left (x^{2} - 6 \, x + 9\right )} \log \relax (x)^{2} - 36 \, x^{2} - 2 \, {\left (3 \, x^{3} - 50 \, x^{2} + 111 \, x\right )} \log \relax (x) + 78 \, x - 72}{3 \, {\left (x^{2} - 6 \, x + 9\right )}} + \frac {6129 \, {\left (4 \, x - 3\right )}}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} - \frac {8703}{2 \, {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}} - \frac {98}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6-28*x^5+222*x^4-960*x^3+1926*x^2-1188*x-486)*log(x)+2*x^7-82*x^6+906*x^5-6186*x^4+22422*x^3-3
6774*x^2+17406*x+7938)/(x^6-15*x^5+90*x^4-270*x^3+405*x^2-243*x),x, algorithm="maxima")

[Out]

x^2 - 52*x - 27/2*(80*x^3 - 630*x^2 + 1692*x - 1539)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) + 369/2*(40*x^3 - 30
0*x^2 + 780*x - 693)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) - 1359/2*(16*x^3 - 108*x^2 + 264*x - 225)/(x^4 - 12*
x^3 + 54*x^2 - 108*x + 81) + 3093/2*(4*x^3 - 18*x^2 + 36*x - 27)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) + 49/2*(
4*x^3 - 42*x^2 + 156*x - 225)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) - 11211/2*(2*x^2 - 4*x + 3)/(x^4 - 12*x^3 +
54*x^2 - 108*x + 81) - 1/3*(6*x^3 - 3*(x^2 - 6*x + 9)*log(x)^2 - 36*x^2 - 2*(3*x^3 - 50*x^2 + 111*x)*log(x) +
78*x - 72)/(x^2 - 6*x + 9) + 6129/2*(4*x - 3)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81) - 8703/2/(x^4 - 12*x^3 + 5
4*x^2 - 108*x + 81) - 98/3*log(x)

________________________________________________________________________________________

mupad [B]  time = 7.12, size = 62, normalized size = 2.95 \begin {gather*} {\ln \relax (x)}^2-54\,\ln \relax (x)-\frac {x\,\left (1800\,\ln \relax (x)-35640\right )-1728\,\ln \relax (x)+x^3\,\left (72\,\ln \relax (x)-1704\right )-x^2\,\left (624\,\ln \relax (x)-13464\right )+31608}{{\left (x-3\right )}^4}+x\,\left (2\,\ln \relax (x)-54\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(17406*x - log(x)*(1188*x - 1926*x^2 + 960*x^3 - 222*x^4 + 28*x^5 - 2*x^6 + 486) - 36774*x^2 + 22422*x^3
- 6186*x^4 + 906*x^5 - 82*x^6 + 2*x^7 + 7938)/(243*x - 405*x^2 + 270*x^3 - 90*x^4 + 15*x^5 - x^6),x)

[Out]

log(x)^2 - 54*log(x) - (x*(1800*log(x) - 35640) - 1728*log(x) + x^3*(72*log(x) - 1704) - x^2*(624*log(x) - 134
64) + 31608)/(x - 3)^4 + x*(2*log(x) - 54) + x^2

________________________________________________________________________________________

sympy [B]  time = 0.30, size = 76, normalized size = 3.62 \begin {gather*} x^{2} - 54 x + \frac {1704 x^{3} - 13464 x^{2} + 35640 x - 31608}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} + \log {\relax (x )}^{2} - 54 \log {\relax (x )} + \frac {\left (2 x^{3} - 12 x^{2} - 54 x + 192\right ) \log {\relax (x )}}{x^{2} - 6 x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6-28*x**5+222*x**4-960*x**3+1926*x**2-1188*x-486)*ln(x)+2*x**7-82*x**6+906*x**5-6186*x**4+224
22*x**3-36774*x**2+17406*x+7938)/(x**6-15*x**5+90*x**4-270*x**3+405*x**2-243*x),x)

[Out]

x**2 - 54*x + (1704*x**3 - 13464*x**2 + 35640*x - 31608)/(x**4 - 12*x**3 + 54*x**2 - 108*x + 81) + log(x)**2 -
54*log(x) + (2*x**3 - 12*x**2 - 54*x + 192)*log(x)/(x**2 - 6*x + 9)

________________________________________________________________________________________