3.9.68 \(\int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+(1440-2480 x+1200 x^2-160 x^3) \log (-2+3 x-x^2)}{7200-4320 x-5382 x^2+1809 x^3+873 x^4-189 x^5+9 x^6} \, dx\)

Optimal. Leaf size=34 \[ \frac {x (x+\log ((1-x) (-2+x)))}{9 \left (\frac {5}{9-x}+\frac {x}{4}\right )} \]

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Rubi [B]  time = 5.01, antiderivative size = 985, normalized size of antiderivative = 28.97, number of steps used = 106, number of rules used = 17, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.191, Rules used = {6741, 12, 6728, 6742, 638, 618, 206, 632, 31, 2528, 2525, 800, 2524, 2418, 2394, 2393, 2391}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 + (1440 - 2480*x + 1200*x^2 - 160*x^3)*Log[-2 + 3*
x - x^2])/(7200 - 4320*x - 5382*x^2 + 1809*x^3 + 873*x^4 - 189*x^5 + 9*x^6),x]

[Out]

(4*x)/9 + (41*(60 - 113*x))/(1173*(20 + 9*x - x^2)) + (20*(140 - 3*x))/(2737*(20 + 9*x - x^2)) + (19*(2260 + 9
57*x))/(1071*(20 + 9*x - x^2)) + (38*(19140 + 10873*x))/(24633*(20 + 9*x - x^2)) - (19*(217460 + 116997*x))/(2
4633*(20 + 9*x - x^2)) + (2339940 + 1270433*x)/(24633*(20 + 9*x - x^2)) + (160*ArcTanh[(9 - 2*x)/Sqrt[161]])/(
9*Sqrt[161]) + ((109154343 - 16160303*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/23450616 - (19*(1691213 - 357445*Sq
rt[161])*Log[9 - Sqrt[161] - 2*x])/7816872 - (5*(2369 - 9*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/651406 + (41*(1
1385 + 271*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/1116696 - (19*(29417 + 831*Sqrt[161])*Log[9 - Sqrt[161] - 2*x]
)/1019592 - (19*(21827 + 22229*Sqrt[161])*Log[9 - Sqrt[161] - 2*x])/3908436 - (19*(21827 - 22229*Sqrt[161])*Lo
g[9 + Sqrt[161] - 2*x])/3908436 - (19*(29417 - 831*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/1019592 + (41*(11385 -
271*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/1116696 - (5*(2369 + 9*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/651406 -
(19*(1691213 + 357445*Sqrt[161])*Log[9 + Sqrt[161] - 2*x])/7816872 + ((109154343 + 16160303*Sqrt[161])*Log[9
+ Sqrt[161] - 2*x])/23450616 + (8*Log[1 - x])/63 - (160*Log[1 - x])/(161*(7 - Sqrt[161])) + (160*(9 - Sqrt[161
])*Log[1 - x])/(1449*(7 - Sqrt[161])) - (160*Log[1 - x])/(161*(7 + Sqrt[161])) + (160*(9 + Sqrt[161])*Log[1 -
x])/(1449*(7 + Sqrt[161])) + (28*Log[2 - x])/153 - (160*Log[2 - x])/(161*(5 - Sqrt[161])) + (160*(9 - Sqrt[161
])*Log[2 - x])/(1449*(5 - Sqrt[161])) - (160*Log[2 - x])/(161*(5 + Sqrt[161])) + (160*(9 + Sqrt[161])*Log[2 -
x])/(1449*(5 + Sqrt[161])) - (400*(43 - 3*Sqrt[161])*Log[-2*(231 - 19*Sqrt[161]) + (49 - 3*Sqrt[161])*x])/(144
9*(49 - 3*Sqrt[161])) + (80*(6 - Sqrt[161])*Log[-2*(231 - 19*Sqrt[161]) + (49 - 3*Sqrt[161])*x])/(161*(49 - 3*
Sqrt[161])) + (80*(6 + Sqrt[161])*Log[2*(231 + 19*Sqrt[161]) - (49 + 3*Sqrt[161])*x])/(161*(49 + 3*Sqrt[161]))
- (400*(43 + 3*Sqrt[161])*Log[2*(231 + 19*Sqrt[161]) - (49 + 3*Sqrt[161])*x])/(1449*(49 + 3*Sqrt[161])) + (16
0*Log[-2 + 3*x - x^2])/(161*(9 - Sqrt[161] - 2*x)) - (160*(9 - Sqrt[161])*Log[-2 + 3*x - x^2])/(1449*(9 - Sqrt
[161] - 2*x)) + (160*Log[-2 + 3*x - x^2])/(161*(9 + Sqrt[161] - 2*x)) - (160*(9 + Sqrt[161])*Log[-2 + 3*x - x^
2])/(1449*(9 + Sqrt[161] - 2*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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

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

Rule 2394

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

Rule 2418

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 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

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

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 {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{9 \left (20+9 x-x^2\right )^2 \left (2-3 x+x^2\right )} \, dx\\ &=\frac {1}{9} \int \frac {720 x-3444 x^2+1748 x^3+152 x^4-76 x^5+4 x^6+\left (1440-2480 x+1200 x^2-160 x^3\right ) \log \left (-2+3 x-x^2\right )}{\left (20+9 x-x^2\right )^2 \left (2-3 x+x^2\right )} \, dx\\ &=\frac {1}{9} \int \left (\frac {720 x}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}-\frac {3444 x^2}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}+\frac {1748 x^3}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}+\frac {152 x^4}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}-\frac {76 x^5}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}+\frac {4 x^6}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2}-\frac {80 (-9+2 x) \log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2}\right ) \, dx\\ &=\frac {4}{9} \int \frac {x^6}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx-\frac {76}{9} \int \frac {x^5}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx-\frac {80}{9} \int \frac {(-9+2 x) \log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2} \, dx+\frac {152}{9} \int \frac {x^4}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx+80 \int \frac {x}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx+\frac {1748}{9} \int \frac {x^3}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx-\frac {1148}{3} \int \frac {x^2}{(-2+x) (-1+x) \left (-20-9 x+x^2\right )^2} \, dx\\ &=\frac {4}{9} \int \left (1+\frac {16}{289 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {1293820+700599 x}{476 \left (-20-9 x+x^2\right )^2}+\frac {35204776+4745841 x}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx-\frac {76}{9} \int \left (\frac {8}{289 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {118380+64691 x}{476 \left (-20-9 x+x^2\right )^2}+\frac {3 (920168+73531 x)}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx-\frac {80}{9} \int \left (-\frac {9 \log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2}+\frac {2 x \log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2}\right ) \, dx+\frac {152}{9} \int \left (\frac {4}{289 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {11420+5919 x}{476 \left (-20-9 x+x^2\right )^2}-\frac {3 (-82072+949 x)}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx+80 \int \left (\frac {1}{578 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {-60+11 x}{476 \left (-20-9 x+x^2\right )^2}+\frac {432-103 x}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx+\frac {1748}{9} \int \left (\frac {2}{289 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {780+571 x}{476 \left (-20-9 x+x^2\right )^2}+\frac {8664-1279 x}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx-\frac {1148}{3} \int \left (\frac {1}{289 (-2+x)}-\frac {1}{784 (-1+x)}+\frac {220+39 x}{476 \left (-20-9 x+x^2\right )^2}+\frac {3176-495 x}{226576 \left (-20-9 x+x^2\right )}\right ) \, dx\\ &=\frac {4 x}{9}+\frac {8}{63} \log (1-x)+\frac {28}{153} \log (2-x)+\frac {\int \frac {35204776+4745841 x}{-20-9 x+x^2} \, dx}{509796}-\frac {19 \int \frac {920168+73531 x}{-20-9 x+x^2} \, dx}{169932}-\frac {19 \int \frac {-82072+949 x}{-20-9 x+x^2} \, dx}{84966}+\frac {5 \int \frac {432-103 x}{-20-9 x+x^2} \, dx}{14161}+\frac {437 \int \frac {8664-1279 x}{-20-9 x+x^2} \, dx}{509796}+\frac {\int \frac {1293820+700599 x}{\left (-20-9 x+x^2\right )^2} \, dx}{1071}-\frac {41 \int \frac {3176-495 x}{-20-9 x+x^2} \, dx}{24276}-\frac {19 \int \frac {118380+64691 x}{\left (-20-9 x+x^2\right )^2} \, dx}{1071}+\frac {38 \int \frac {11420+5919 x}{\left (-20-9 x+x^2\right )^2} \, dx}{1071}+\frac {20}{119} \int \frac {-60+11 x}{\left (-20-9 x+x^2\right )^2} \, dx+\frac {437 \int \frac {780+571 x}{\left (-20-9 x+x^2\right )^2} \, dx}{1071}-\frac {41}{51} \int \frac {220+39 x}{\left (-20-9 x+x^2\right )^2} \, dx-\frac {160}{9} \int \frac {x \log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2} \, dx+80 \int \frac {\log \left (-2+3 x-x^2\right )}{\left (-20-9 x+x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 55, normalized size = 1.62 \begin {gather*} \frac {1}{9} \left (4 x+\frac {80 x}{-20-9 x+x^2}+\frac {80 \log \left (-2+3 x-x^2\right )}{-20-9 x+x^2}+4 \log \left (2-3 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 + (1440 - 2480*x + 1200*x^2 - 160*x^3)*Log[-
2 + 3*x - x^2])/(7200 - 4320*x - 5382*x^2 + 1809*x^3 + 873*x^4 - 189*x^5 + 9*x^6),x]

[Out]

(4*x + (80*x)/(-20 - 9*x + x^2) + (80*Log[-2 + 3*x - x^2])/(-20 - 9*x + x^2) + 4*Log[2 - 3*x + x^2])/9

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fricas [A]  time = 0.79, size = 40, normalized size = 1.18 \begin {gather*} \frac {4 \, {\left (x^{3} - 9 \, x^{2} + {\left (x^{2} - 9 \, x\right )} \log \left (-x^{2} + 3 \, x - 2\right )\right )}}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+152*x^4+1748*x^3-3444*x^2+720*x)/(9*x^
6-189*x^5+873*x^4+1809*x^3-5382*x^2-4320*x+7200),x, algorithm="fricas")

[Out]

4/9*(x^3 - 9*x^2 + (x^2 - 9*x)*log(-x^2 + 3*x - 2))/(x^2 - 9*x - 20)

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giac [B]  time = 0.37, size = 51, normalized size = 1.50 \begin {gather*} \frac {4}{9} \, x + \frac {80 \, x}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {80 \, \log \left (-x^{2} + 3 \, x - 2\right )}{9 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {4}{9} \, \log \left (x^{2} - 3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+152*x^4+1748*x^3-3444*x^2+720*x)/(9*x^
6-189*x^5+873*x^4+1809*x^3-5382*x^2-4320*x+7200),x, algorithm="giac")

[Out]

4/9*x + 80/9*x/(x^2 - 9*x - 20) + 80/9*log(-x^2 + 3*x - 2)/(x^2 - 9*x - 20) + 4/9*log(x^2 - 3*x + 2)

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maple [A]  time = 0.10, size = 52, normalized size = 1.53

method	result	size
norman	\(\frac {-36 x -4 \ln \left (-x^{2}+3 x -2\right ) x +\frac {4 \ln \left (-x^{2}+3 x -2\right ) x^{2}}{9}+\frac {4 x^{3}}{9}-80}{x^{2}-9 x -20}\)	\(52\)
risch	\(\frac {80 \ln \left (-x^{2}+3 x -2\right )}{9 \left (x^{2}-9 x -20\right )}+\frac {\frac {4 \ln \left (x^{2}-3 x +2\right ) x^{2}}{9}+\frac {4 x^{3}}{9}-4 \ln \left (x^{2}-3 x +2\right ) x -4 x^{2}-\frac {80 \ln \left (x^{2}-3 x +2\right )}{9}}{x^{2}-9 x -20}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-160*x^3+1200*x^2-2480*x+1440)*ln(-x^2+3*x-2)+4*x^6-76*x^5+152*x^4+1748*x^3-3444*x^2+720*x)/(9*x^6-189*x
^5+873*x^4+1809*x^3-5382*x^2-4320*x+7200),x,method=_RETURNVERBOSE)

[Out]

(-36*x-4*ln(-x^2+3*x-2)*x+4/9*ln(-x^2+3*x-2)*x^2+4/9*x^3-80)/(x^2-9*x-20)

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maxima [B]  time = 0.85, size = 161, normalized size = 4.74 \begin {gather*} \frac {4}{9} \, x + \frac {20 \, {\left (17 \, {\left (x^{2} - 9 \, x + 8\right )} \log \left (x - 1\right ) + 14 \, {\left (x^{2} - 9 \, x + 14\right )} \log \left (-x + 2\right )\right )}}{1071 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {1270433 \, x + 2339940}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {19 \, {\left (116997 \, x + 217460\right )}}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {38 \, {\left (10873 \, x + 19140\right )}}{24633 \, {\left (x^{2} - 9 \, x - 20\right )}} - \frac {19 \, {\left (957 \, x + 2260\right )}}{1071 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {41 \, {\left (113 \, x - 60\right )}}{1173 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {20 \, {\left (3 \, x - 140\right )}}{2737 \, {\left (x^{2} - 9 \, x - 20\right )}} + \frac {8}{63} \, \log \left (x - 1\right ) + \frac {28}{153} \, \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^3+1200*x^2-2480*x+1440)*log(-x^2+3*x-2)+4*x^6-76*x^5+152*x^4+1748*x^3-3444*x^2+720*x)/(9*x^
6-189*x^5+873*x^4+1809*x^3-5382*x^2-4320*x+7200),x, algorithm="maxima")

[Out]

4/9*x + 20/1071*(17*(x^2 - 9*x + 8)*log(x - 1) + 14*(x^2 - 9*x + 14)*log(-x + 2))/(x^2 - 9*x - 20) - 1/24633*(
1270433*x + 2339940)/(x^2 - 9*x - 20) + 19/24633*(116997*x + 217460)/(x^2 - 9*x - 20) - 38/24633*(10873*x + 19
140)/(x^2 - 9*x - 20) - 19/1071*(957*x + 2260)/(x^2 - 9*x - 20) + 41/1173*(113*x - 60)/(x^2 - 9*x - 20) + 20/2
737*(3*x - 140)/(x^2 - 9*x - 20) + 8/63*log(x - 1) + 28/153*log(x - 2)

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mupad [B]  time = 0.98, size = 55, normalized size = 1.62 \begin {gather*} \frac {4\,x}{9}+\frac {4\,\ln \left (x^2-3\,x+2\right )}{9}-\frac {80\,\ln \left (-x^2+3\,x-2\right )}{9\,\left (-x^2+9\,x+20\right )}-\frac {80\,x}{9\,\left (-x^2+9\,x+20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((720*x - 3444*x^2 + 1748*x^3 + 152*x^4 - 76*x^5 + 4*x^6 - log(3*x - x^2 - 2)*(2480*x - 1200*x^2 + 160*x^3
- 1440))/(1809*x^3 - 5382*x^2 - 4320*x + 873*x^4 - 189*x^5 + 9*x^6 + 7200),x)

[Out]

(4*x)/9 + (4*log(x^2 - 3*x + 2))/9 - (80*log(3*x - x^2 - 2))/(9*(9*x - x^2 + 20)) - (80*x)/(9*(9*x - x^2 + 20)
)

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sympy [B]  time = 0.27, size = 53, normalized size = 1.56 \begin {gather*} \frac {4 x}{9} + \frac {80 x}{9 x^{2} - 81 x - 180} + \frac {4 \log {\left (x^{2} - 3 x + 2 \right )}}{9} + \frac {80 \log {\left (- x^{2} + 3 x - 2 \right )}}{9 x^{2} - 81 x - 180} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x**3+1200*x**2-2480*x+1440)*ln(-x**2+3*x-2)+4*x**6-76*x**5+152*x**4+1748*x**3-3444*x**2+720*x
)/(9*x**6-189*x**5+873*x**4+1809*x**3-5382*x**2-4320*x+7200),x)

[Out]

4*x/9 + 80*x/(9*x**2 - 81*x - 180) + 4*log(x**2 - 3*x + 2)/9 + 80*log(-x**2 + 3*x - 2)/(9*x**2 - 81*x - 180)

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