\(\int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx\) [18]

Optimal result

Integrand size = 23, antiderivative size = 574 \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=-\frac {\left (9 \left (9-\sqrt {17}\right ) \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) A+2 \left (3 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}\right ) B+2 \left (9-\sqrt {17}\right ) \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) C\right )\right ) \arctan \left (\frac {9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}-6 \left (9-\sqrt {17}\right )^{2/3} x}{\sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}}\right )}{3 \sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}+\frac {\sqrt [3]{9-\sqrt {17}} \left (9 \sqrt [3]{9-\sqrt {17}} A-3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \sqrt [3]{9-\sqrt {17}} C\right ) \log \left (4+\left (9-\sqrt {17}\right )^{2/3}+3 \sqrt [3]{9-\sqrt {17}} x\right )}{9 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}-\frac {\left (9 \left (9-\sqrt {17}\right )^{2/3} A-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) B+4 \left (9-\sqrt {17}\right )^{2/3} C\right ) \log \left (16-4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+9 \left (9-\sqrt {17}\right )^{2/3} x^2\right )}{18 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}+\frac {1}{9} C \log \left (2-4 x+3 x^3\right ) \] Output:

-1/3*(9*(9-17^(1/2))*(4+(9-17^(1/2))^(2/3))*A+6*(49-9*17^(1/2)+8*(9-17^(1/ 
2))^(2/3))*B+4*(9-17^(1/2))*(4+(9-17^(1/2))^(2/3))*C)*arctan((9-17^(1/2)+4 
*(9-17^(1/2))^(1/3)-6*(9-17^(1/2))^(2/3)*x)/(294-54*17^(1/2)+48*(9-17^(1/2 
))^(2/3)-24*(9-17^(1/2))^(4/3))^(1/2))/(294-54*17^(1/2)+48*(9-17^(1/2))^(2 
/3)-24*(9-17^(1/2))^(4/3))^(1/2)/(16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/ 
3))+(9-17^(1/2))^(1/3)*(9*(9-17^(1/2))^(1/3)*A-3*(4+(9-17^(1/2))^(2/3))*B+ 
4*(9-17^(1/2))^(1/3)*C)*ln(4+(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(1/3)*x)/(1 
44+36*(9-17^(1/2))^(2/3)+9*(9-17^(1/2))^(4/3))-(9*(9-17^(1/2))^(2/3)*A-3*( 
9-17^(1/2)+4*(9-17^(1/2))^(1/3))*B+4*(9-17^(1/2))^(2/3)*C)*ln(16-4*(9-17^( 
1/2))^(2/3)+(9-17^(1/2))^(4/3)-3*(9-17^(1/2)+4*(9-17^(1/2))^(1/3))*x+9*(9- 
17^(1/2))^(2/3)*x^2)/(288+72*(9-17^(1/2))^(2/3)+18*(9-17^(1/2))^(4/3))+1/9 
*C*ln(3*x^3-4*x+2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.10 \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\text {RootSum}\left [2-4 \text {$\#$1}+3 \text {$\#$1}^3\&,\frac {A \log (x-\text {$\#$1})+B \log (x-\text {$\#$1}) \text {$\#$1}+C \log (x-\text {$\#$1}) \text {$\#$1}^2}{-4+9 \text {$\#$1}^2}\&\right ] \] Input:

Integrate[(A + B*x + C*x^2)/(2 - 4*x + 3*x^3),x]
 

Output:

RootSum[2 - 4*#1 + 3*#1^3 & , (A*Log[x - #1] + B*Log[x - #1]*#1 + C*Log[x 
- #1]*#1^2)/(-4 + 9*#1^2) & ]
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 571, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2525, 2485, 25, 1200, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2}{3 x^3-4 x+2} \, dx\)

\(\Big \downarrow \) 2525

\(\displaystyle \frac {1}{9} \int \frac {9 A+4 C+9 B x}{3 x^3-4 x+2}dx+\frac {1}{9} C \log \left (3 x^3-4 x+2\right )\)

\(\Big \downarrow \) 2485

\(\displaystyle \int -\frac {9 A+4 C+9 B x}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx+\frac {1}{9} C \log \left (3 x^3-4 x+2\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{9} C \log \left (3 x^3-4 x+2\right )-\int \frac {9 A+4 C+9 B x}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{9} C \log \left (3 x^3-4 x+2\right )-\int \left (\frac {\left (9-\sqrt {17}\right )^{2/3} \left (-9 \sqrt [3]{9-\sqrt {17}} A+3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B-4 \sqrt [3]{9-\sqrt {17}} C\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}+\frac {\left (9-\sqrt {17}\right )^{2/3} \left (-18 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A-3 \left (16-4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) B-8 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C+3 \left (9 \left (9-\sqrt {17}\right )^{2/3} A-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) B+4 \left (9-\sqrt {17}\right )^{2/3} C\right ) x\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) \left (9 \left (9-\sqrt {17}\right )^{2/3} x^2-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+\left (9-\sqrt {17}\right )^{4/3}-4 \left (9-\sqrt {17}\right )^{2/3}+16\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {-6 \left (9-\sqrt {17}\right )^{2/3} x+4 \sqrt [3]{9-\sqrt {17}}-\sqrt {17}+9}{\sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}}\right ) \left (9 \left (9-\sqrt {17}\right ) \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) A+6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \left (9-\sqrt {17}\right ) \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) C\right )}{3 \sqrt {6 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}-\frac {\log \left (9 \left (9-\sqrt {17}\right )^{2/3} x^2-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) x+\left (9-\sqrt {17}\right )^{4/3}-4 \left (9-\sqrt {17}\right )^{2/3}+16\right ) \left (9 \left (9-\sqrt {17}\right )^{2/3} A-3 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) B+4 \left (9-\sqrt {17}\right )^{2/3} C\right )}{18 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}+\frac {\sqrt [3]{9-\sqrt {17}} \log \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right ) \left (9 \sqrt [3]{9-\sqrt {17}} A-3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \sqrt [3]{9-\sqrt {17}} C\right )}{9 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )}+\frac {1}{9} C \log \left (3 x^3-4 x+2\right )\)

Input:

Int[(A + B*x + C*x^2)/(2 - 4*x + 3*x^3),x]
 

Output:

-1/3*((9*(9 - Sqrt[17])*(4 + (9 - Sqrt[17])^(2/3))*A + 6*(49 - 9*Sqrt[17] 
+ 8*(9 - Sqrt[17])^(2/3))*B + 4*(9 - Sqrt[17])*(4 + (9 - Sqrt[17])^(2/3))* 
C)*ArcTan[(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3) - 6*(9 - Sqrt[17])^(2/3)* 
x)/Sqrt[6*(49 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 4*(9 - Sqrt[17])^(4/ 
3))]])/(Sqrt[6*(49 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 4*(9 - Sqrt[17] 
)^(4/3))]*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4/3))) + ((9 - Sq 
rt[17])^(1/3)*(9*(9 - Sqrt[17])^(1/3)*A - 3*(4 + (9 - Sqrt[17])^(2/3))*B + 
 4*(9 - Sqrt[17])^(1/3)*C)*Log[4 + (9 - Sqrt[17])^(2/3) + 3*(9 - Sqrt[17]) 
^(1/3)*x])/(9*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4/3))) - ((9* 
(9 - Sqrt[17])^(2/3)*A - 3*(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3))*B + 4*( 
9 - Sqrt[17])^(2/3)*C)*Log[16 - 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4 
/3) - 3*(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3))*x + 9*(9 - Sqrt[17])^(2/3) 
*x^2])/(18*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4/3))) + (C*Log[ 
2 - 4*x + 3*x^3])/9
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2485

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[1/d^(2*p)   Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + 
 d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d 
*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Fre 
eQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]
 

rule 2525

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Si 
mp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coeff[Qn, x, n])), x] + Simp[1/(n*Coeff[Qn, 
x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x 
]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.07

method	result	size
default	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\)	\(41\)
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\)	\(41\)

Input:

int((C*x^2+B*x+A)/(3*x^3-4*x+2),x,method=_RETURNVERBOSE)
 

Output:

sum((C*_R^2+B*_R+A)/(9*_R^2-4)*ln(x-_R),_R=RootOf(3*_Z^3-4*_Z+2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 7302, normalized size of antiderivative = 12.72 \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x+2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [A] (verification not implemented)

Time = 8.09 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.48 \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\operatorname {RootSum} {\left (612 t^{3} - 204 t^{2} C + t \left (108 A^{2} + 162 A B + 96 A C + 48 B^{2} + 72 B C + 44 C^{2}\right ) - 9 A^{3} - 24 A^{2} C + 12 A B^{2} - 18 A B C - 16 A C^{2} + 6 B^{3} - 8 B C^{2} - 4 C^{3}, \left ( t \mapsto t \log {\left (x + \frac {7344 t^{2} A + 5508 t^{2} B + 3264 t^{2} C + 918 t A^{2} - 816 t A C - 408 t B^{2} - 1224 t B C - 544 t C^{2} + 864 A^{3} + 1944 A^{2} B + 1050 A^{2} C + 1356 A B^{2} + 1728 A B C + 512 A C^{2} + 288 B^{3} + 648 B^{2} C + 452 B C^{2} + 96 C^{3}}{729 A^{3} + 1296 A^{2} B + 972 A^{2} C + 972 A B^{2} + 1152 A B C + 432 A C^{2} + 294 B^{3} + 432 B^{2} C + 256 B C^{2} + 64 C^{3}} \right )} \right )\right )} \] Input:

integrate((C*x**2+B*x+A)/(3*x**3-4*x+2),x)
 

Output:

RootSum(612*_t**3 - 204*_t**2*C + _t*(108*A**2 + 162*A*B + 96*A*C + 48*B** 
2 + 72*B*C + 44*C**2) - 9*A**3 - 24*A**2*C + 12*A*B**2 - 18*A*B*C - 16*A*C 
**2 + 6*B**3 - 8*B*C**2 - 4*C**3, Lambda(_t, _t*log(x + (7344*_t**2*A + 55 
08*_t**2*B + 3264*_t**2*C + 918*_t*A**2 - 816*_t*A*C - 408*_t*B**2 - 1224* 
_t*B*C - 544*_t*C**2 + 864*A**3 + 1944*A**2*B + 1050*A**2*C + 1356*A*B**2 
+ 1728*A*B*C + 512*A*C**2 + 288*B**3 + 648*B**2*C + 452*B*C**2 + 96*C**3)/ 
(729*A**3 + 1296*A**2*B + 972*A**2*C + 972*A*B**2 + 1152*A*B*C + 432*A*C** 
2 + 294*B**3 + 432*B**2*C + 256*B*C**2 + 64*C**3))))
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\int { \frac {C x^{2} + B x + A}{3 \, x^{3} - 4 \, x + 2} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x+2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(3*x^3 - 4*x + 2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((C*x^2+B*x+A)/(3*x^3-4*x+2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 12.54 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\sum _{k=1}^3\ln \left (-x\,\left (-3\,B^2+4\,C^2+3\,A\,C\right )+2\,C^2+\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (108\,A^2+48\,B^2+44\,C^2+162\,A\,B+96\,A\,C+72\,B\,C\right )}{612}-\frac {A\,B\,C}{34}-\frac {2\,B\,C^2}{153}-\frac {4\,A\,C^2}{153}-\frac {2\,A^2\,C}{51}+\frac {A\,B^2}{51}-\frac {A^3}{68}-\frac {C^3}{153}+\frac {B^3}{102},z,k\right )\,\left (12\,B-36\,C+x\,\left (27\,A+60\,C\right )-\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (108\,A^2+48\,B^2+44\,C^2+162\,A\,B+96\,A\,C+72\,B\,C\right )}{612}-\frac {A\,B\,C}{34}-\frac {2\,B\,C^2}{153}-\frac {4\,A\,C^2}{153}-\frac {2\,A^2\,C}{51}+\frac {A\,B^2}{51}-\frac {A^3}{68}-\frac {C^3}{153}+\frac {B^3}{102},z,k\right )\,\left (216\,x-162\right )\right )+3\,A\,B\right )\,\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (108\,A^2+48\,B^2+44\,C^2+162\,A\,B+96\,A\,C+72\,B\,C\right )}{612}-\frac {A\,B\,C}{34}-\frac {2\,B\,C^2}{153}-\frac {4\,A\,C^2}{153}-\frac {2\,A^2\,C}{51}+\frac {A\,B^2}{51}-\frac {A^3}{68}-\frac {C^3}{153}+\frac {B^3}{102},z,k\right ) \] Input:

int((A + B*x + C*x^2)/(3*x^3 - 4*x + 2),x)
 

Output:

symsum(log(2*C^2 - x*(4*C^2 - 3*B^2 + 3*A*C) + root(z^3 - (C*z^2)/3 + (z*( 
108*A^2 + 48*B^2 + 44*C^2 + 162*A*B + 96*A*C + 72*B*C))/612 - (A*B*C)/34 - 
 (2*B*C^2)/153 - (4*A*C^2)/153 - (2*A^2*C)/51 + (A*B^2)/51 - A^3/68 - C^3/ 
153 + B^3/102, z, k)*(12*B - 36*C + x*(27*A + 60*C) - root(z^3 - (C*z^2)/3 
 + (z*(108*A^2 + 48*B^2 + 44*C^2 + 162*A*B + 96*A*C + 72*B*C))/612 - (A*B* 
C)/34 - (2*B*C^2)/153 - (4*A*C^2)/153 - (2*A^2*C)/51 + (A*B^2)/51 - A^3/68 
 - C^3/153 + B^3/102, z, k)*(216*x - 162)) + 3*A*B)*root(z^3 - (C*z^2)/3 + 
 (z*(108*A^2 + 48*B^2 + 44*C^2 + 162*A*B + 96*A*C + 72*B*C))/612 - (A*B*C) 
/34 - (2*B*C^2)/153 - (4*A*C^2)/153 - (2*A^2*C)/51 + (A*B^2)/51 - A^3/68 - 
 C^3/153 + B^3/102, z, k), k, 1, 3)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{2-4 x+3 x^3} \, dx=\left (\int \frac {x^{2}}{3 x^{3}-4 x +2}d x \right ) c +\left (\int \frac {x}{3 x^{3}-4 x +2}d x \right ) b +\left (\int \frac {1}{3 x^{3}-4 x +2}d x \right ) a \] Input:

int((C*x^2+B*x+A)/(3*x^3-4*x+2),x)
 

Output:

int(x**2/(3*x**3 - 4*x + 2),x)*c + int(x/(3*x**3 - 4*x + 2),x)*b + int(1/( 
3*x**3 - 4*x + 2),x)*a