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

Optimal. Leaf size=21 $${(-3{(3-\frac{2}{3-x})}^2+x+\log (x))}^2$$

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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} $$\begin{array}{c}{x}^2-54x-\frac{1704}{3-x}+\frac{1872}{(3-x{)}^2}-\frac{864}{(3-x{)}^3}+\frac{144}{(3-x{)}^4}+{\log }^2(x)+\frac{24x\log (x)}{3-x}+2x\log (x)-\frac{24\log (x)}{(3-x{)}^2}-30\log (x)\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{2(27+84x-54x^2+8x^3-x^4)(-147+135x-33x^2+x^3+(-3+x{)}^2\log (x))}{(3-x{)}^{5}x}dx\\ & =2\int \frac{(27+84x-54x^2+8x^3-x^4)(-147+135x-33x^2+x^3+(-3+x{)}^2\log (x))}{(3-x{)}^{5}x}dx\\ & =2\int (\frac{135(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^5}-\frac{147(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^{5}x}-\frac{33x(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^5}+\frac{x^2(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^5}+\frac{(-27-84x+54x^2-8x^3+x^4)\log (x)}{(-3+x{)}^{3}x})dx\\ & =2\int \frac{x^2(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^5}dx+2\int \frac{(-27-84x+54x^2-8x^3+x^4)\log (x)}{(-3+x{)}^{3}x}dx-66\int \frac{x(-27-84x+54x^2-8x^3+x^4)}{(-3+x{)}^5}dx+270\int \frac{-27-84x+54x^2-8x^3+x^4}{(-3+x{)}^5}dx-294\int \frac{-27-84x+54x^2-8x^3+x^4}{(-3+x{)}^{5}x}dx\\ & =2\int (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)dx+2\int (\log (x)+\frac{24\log (x)}{(-3+x{)}^3}+\frac{36\log (x)}{(-3+x{)}^2}+\frac{\log (x)}{x})dx-66\int (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})dx+270\int (\frac{72}{(-3+x{)}^5}+\frac{132}{(-3+x{)}^4}+\frac{36}{(-3+x{)}^3}+\frac{4}{(-3+x{)}^2}+\frac{1}{-3+x})dx-294\int (\frac{24}{(-3+x{)}^5}+\frac{36}{(-3+x{)}^4}+\frac{4}{3(-3+x{)}^2}-\frac{1}{9(-3+x)}+\frac{1}{9x})dx\\ & =\frac{144}{(3-x{)}^4}-\frac{864}{(3-x{)}^3}+\frac{1872}{(3-x{)}^2}-\frac{1712}{3-x}-52x+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}-54x+x^2-\frac{64}{3}\log (3-x)-\frac{98\log (x)}{3}-\frac{24\log (x)}{(3-x{)}^2}+2x\log (x)+\frac{24x\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}-54x+x^2+\frac{8}{3}\log (3-x)-\frac{98\log (x)}{3}-\frac{24\log (x)}{(3-x{)}^2}+2x\log (x)+\frac{24x\log (x)}{3-x}+{\log }^2(x)+24\int (\frac{1}{3(-3+x{)}^2}-\frac{1}{9(-3+x)}+\frac{1}{9x})dx\\ & =\frac{144}{(3-x{)}^4}-\frac{864}{(3-x{)}^3}+\frac{1872}{(3-x{)}^2}-\frac{1704}{3-x}-54x+x^2-30\log (x)-\frac{24\log (x)}{(3-x{)}^2}+2x\log (x)+\frac{24x\log (x)}{3-x}+{\log }^2(x)\end{array}\end{array}$$

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

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

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

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)

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

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)

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

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

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)

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

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)

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

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

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

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)

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