\(\int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx\) [45]

Optimal result

Integrand size = 25, antiderivative size = 593 \[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=-\frac {\sqrt {\frac {3}{2 \left (29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}\right )}} \left (9 \left (179-9 \sqrt {345}\right ) \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) A+2 \left (3969-211 \sqrt {345}+2 \left (179-9 \sqrt {345}\right )^{2/3} \left (27-\sqrt {345}\right )\right ) (9 B+8 C)\right ) \arctan \left (\frac {179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}+2 \left (179-9 \sqrt {345}\right )^{2/3} (4-9 x)}{\sqrt {6 \left (29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}\right )}}\right )}{256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}}+\frac {\sqrt [3]{179-9 \sqrt {345}} \left (81 \sqrt [3]{179-9 \sqrt {345}} A-\left (16-4 \sqrt [3]{179-9 \sqrt {345}}+\left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \log \left (16+\left (179-9 \sqrt {345}\right )^{2/3}-\sqrt [3]{179-9 \sqrt {345}} (4-9 x)\right )}{9 \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )}-\frac {\left (81 \left (179-9 \sqrt {345}\right )^{2/3} A-\left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}-4 \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \log \left (\left (27-\sqrt {345}\right ) \left (4+\sqrt [3]{179-9 \sqrt {345}}\right )-179 x+9 \sqrt {345} x-16 \sqrt [3]{179-9 \sqrt {345}} x-8 \left (179-9 \sqrt {345}\right )^{2/3} x+9 \left (179-9 \sqrt {345}\right )^{2/3} x^2\right )}{18 \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )}+\frac {1}{9} C \log \left (2-4 x^2+3 x^3\right ) \] Output:

-3^(1/2)/(59986-3222*345^(1/2)+256*(179-9*345^(1/2))^(2/3)-32*(179-9*345^( 
1/2))^(4/3))^(1/2)*(9*(179-9*345^(1/2))*(16+(179-9*345^(1/2))^(2/3))*A+2*( 
3969-211*345^(1/2)+2*(179-9*345^(1/2))^(2/3)*(27-345^(1/2)))*(9*B+8*C))*ar 
ctan((179-9*345^(1/2)+16*(179-9*345^(1/2))^(1/3)+2*(179-9*345^(1/2))^(2/3) 
*(4-9*x))/(179958-9666*345^(1/2)+768*(179-9*345^(1/2))^(2/3)-96*(179-9*345 
^(1/2))^(4/3))^(1/2))/(256+16*(179-9*345^(1/2))^(2/3)+(179-9*345^(1/2))^(4 
/3))+(179-9*345^(1/2))^(1/3)*(81*(179-9*345^(1/2))^(1/3)*A-((179-9*345^(1/ 
2))^(2/3)-4*(179-9*345^(1/2))^(1/3)+16)*(9*B+8*C))*ln(16+(179-9*345^(1/2)) 
^(2/3)-(179-9*345^(1/2))^(1/3)*(4-9*x))/(2304+144*(179-9*345^(1/2))^(2/3)+ 
9*(179-9*345^(1/2))^(4/3))-(81*(179-9*345^(1/2))^(2/3)*A-(179-9*345^(1/2)+ 
16*(179-9*345^(1/2))^(1/3)-4*(179-9*345^(1/2))^(2/3))*(9*B+8*C))*ln((27-34 
5^(1/2))*(4+(179-9*345^(1/2))^(1/3))-179*x+9*x*345^(1/2)-16*(179-9*345^(1/ 
2))^(1/3)*x-8*(179-9*345^(1/2))^(2/3)*x+9*(179-9*345^(1/2))^(2/3)*x^2)/(46 
08+288*(179-9*345^(1/2))^(2/3)+18*(179-9*345^(1/2))^(4/3))+1/9*C*ln(3*x^3- 
4*x^2+2)
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.11 \[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=\text {RootSum}\left [2-4 \text {$\#$1}^2+3 \text {$\#$1}^3\&,\frac {A \log (x-\text {$\#$1})+B \log (x-\text {$\#$1}) \text {$\#$1}+C \log (x-\text {$\#$1}) \text {$\#$1}^2}{-8 \text {$\#$1}+9 \text {$\#$1}^2}\&\right ] \] Input:

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

Output:

RootSum[2 - 4*#1^2 + 3*#1^3 & , (A*Log[x - #1] + B*Log[x - #1]*#1 + C*Log[ 
x - #1]*#1^2)/(-8*#1 + 9*#1^2) & ]
 

Rubi [A] (verified)

Time = 4.86 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2525, 2490, 2485, 27, 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.

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

\(\Big \downarrow \) 2525

\(\displaystyle \frac {1}{9} \int \frac {9 A+(9 B+8 C) x}{3 x^3-4 x^2+2}dx+\frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{9} \int \frac {\frac {1}{9} (81 A+4 (9 B+8 C))+(9 B+8 C) \left (x-\frac {4}{9}\right )}{3 \left (x-\frac {4}{9}\right )^3-\frac {16}{9} \left (x-\frac {4}{9}\right )+\frac {358}{243}}d\left (x-\frac {4}{9}\right )+\frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )\)

\(\Big \downarrow \) 2485

\(\displaystyle \int -\frac {3 \left (81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )\right )}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}d\left (x-\frac {4}{9}\right )+\frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )-3 \int \frac {81 A+36 B+32 C+9 (9 B+8 C) \left (x-\frac {4}{9}\right )}{\left (9 \left (x-\frac {4}{9}\right )+\frac {16+\left (179-9 \sqrt {345}\right )^{2/3}}{\sqrt [3]{179-9 \sqrt {345}}}\right ) \left (-81 \left (x-\frac {4}{9}\right )^2+\frac {9 \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) \left (x-\frac {4}{9}\right )}{\sqrt [3]{179-9 \sqrt {345}}}-\left (179-9 \sqrt {345}\right )^{2/3}-\frac {256}{\left (179-9 \sqrt {345}\right )^{2/3}}+16\right )}d\left (x-\frac {4}{9}\right )\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )-3 \int \left (\frac {-81 \left (179-9 \sqrt {345}\right ) A-\left (716-36 \sqrt {345}-16 \left (179-9 \sqrt {345}\right )^{2/3}-\left (179-9 \sqrt {345}\right )^{4/3}\right ) (9 B+8 C)}{3 \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right ) \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right )}+\frac {\left (179-9 \sqrt {345}\right )^{2/3} \left (-162 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) A+\left (16 \left (179-9 \sqrt {345}\right )^{2/3}-8 \left (211-9 \sqrt {345}\right )-\sqrt [3]{179-9 \sqrt {345}} \left (307-9 \sqrt {345}\right )\right ) (9 B+8 C)+9 \left (81 \left (179-9 \sqrt {345}\right )^{2/3} A-\left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}-4 \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right ) \left (x-\frac {4}{9}\right )\right )}{3 \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right ) \left (81 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )^2-9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{4/3}-16 \left (179-9 \sqrt {345}\right )^{2/3}+256\right )}\right )d\left (x-\frac {4}{9}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{9} C \log \left (3 x^3-4 x^2+2\right )-3 \left (\frac {\arctan \left (\frac {-18 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )+16 \sqrt [3]{179-9 \sqrt {345}}-9 \sqrt {345}+179}{\sqrt {6 \left (29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}\right )}}\right ) \left (9 \left (179-9 \sqrt {345}\right ) \left (16+\left (179-9 \sqrt {345}\right )^{2/3}\right ) A+2 \left (3969-211 \sqrt {345}+2 \left (179-9 \sqrt {345}\right )^{2/3} \left (27-\sqrt {345}\right )\right ) (9 B+8 C)\right )}{\sqrt {6 \left (29993-1611 \sqrt {345}+128 \left (179-9 \sqrt {345}\right )^{2/3}-16 \left (179-9 \sqrt {345}\right )^{4/3}\right )} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )}-\frac {\log \left (9 \sqrt [3]{179-9 \sqrt {345}} \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{2/3}+16\right ) \left (81 \left (179-9 \sqrt {345}\right ) A+\left (716-36 \sqrt {345}-16 \left (179-9 \sqrt {345}\right )^{2/3}-\left (179-9 \sqrt {345}\right )^{4/3}\right ) (9 B+8 C)\right )}{27 \sqrt [3]{179-9 \sqrt {345}} \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )}+\frac {\log \left (81 \left (179-9 \sqrt {345}\right )^{2/3} \left (x-\frac {4}{9}\right )^2-9 \left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}\right ) \left (x-\frac {4}{9}\right )+\left (179-9 \sqrt {345}\right )^{4/3}-16 \left (179-9 \sqrt {345}\right )^{2/3}+256\right ) \left (81 \left (179-9 \sqrt {345}\right )^{2/3} A-\left (179-9 \sqrt {345}+16 \sqrt [3]{179-9 \sqrt {345}}-4 \left (179-9 \sqrt {345}\right )^{2/3}\right ) (9 B+8 C)\right )}{54 \left (256+16 \left (179-9 \sqrt {345}\right )^{2/3}+\left (179-9 \sqrt {345}\right )^{4/3}\right )}\right )\)

Input:

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

Output:

-3*(((9*(179 - 9*Sqrt[345])*(16 + (179 - 9*Sqrt[345])^(2/3))*A + 2*(3969 - 
 211*Sqrt[345] + 2*(179 - 9*Sqrt[345])^(2/3)*(27 - Sqrt[345]))*(9*B + 8*C) 
)*ArcTan[(179 - 9*Sqrt[345] + 16*(179 - 9*Sqrt[345])^(1/3) - 18*(179 - 9*S 
qrt[345])^(2/3)*(-4/9 + x))/Sqrt[6*(29993 - 1611*Sqrt[345] + 128*(179 - 9* 
Sqrt[345])^(2/3) - 16*(179 - 9*Sqrt[345])^(4/3))]])/(Sqrt[6*(29993 - 1611* 
Sqrt[345] + 128*(179 - 9*Sqrt[345])^(2/3) - 16*(179 - 9*Sqrt[345])^(4/3))] 
*(256 + 16*(179 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(4/3))) - ((81* 
(179 - 9*Sqrt[345])*A + (716 - 36*Sqrt[345] - 16*(179 - 9*Sqrt[345])^(2/3) 
 - (179 - 9*Sqrt[345])^(4/3))*(9*B + 8*C))*Log[16 + (179 - 9*Sqrt[345])^(2 
/3) + 9*(179 - 9*Sqrt[345])^(1/3)*(-4/9 + x)])/(27*(179 - 9*Sqrt[345])^(1/ 
3)*(256 + 16*(179 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(4/3))) + ((8 
1*(179 - 9*Sqrt[345])^(2/3)*A - (179 - 9*Sqrt[345] + 16*(179 - 9*Sqrt[345] 
)^(1/3) - 4*(179 - 9*Sqrt[345])^(2/3))*(9*B + 8*C))*Log[256 - 16*(179 - 9* 
Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(4/3) - 9*(179 - 9*Sqrt[345] + 16*( 
179 - 9*Sqrt[345])^(1/3))*(-4/9 + x) + 81*(179 - 9*Sqrt[345])^(2/3)*(-4/9 
+ x)^2])/(54*(256 + 16*(179 - 9*Sqrt[345])^(2/3) + (179 - 9*Sqrt[345])^(4/ 
3)))) + (C*Log[2 - 4*x^2 + 3*x^3])/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

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 = 45, normalized size of antiderivative = 0.08

method	result	size
default	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-8 \textit {\_R}}\)	\(45\)
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-8 \textit {\_R}}\)	\(45\)

Input:

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

Output:

sum((C*_R^2+B*_R+A)/(9*_R^2-8*_R)*ln(x-_R),_R=RootOf(3*_Z^3-4*_Z^2+2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 7319, normalized size of antiderivative = 12.34 \[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [A] (verification not implemented)

Time = 8.77 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.46 \[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=\operatorname {RootSum} {\left (1380 t^{3} - 460 t^{2} C + t \left (48 A^{2} + 162 A B + 144 A C + 72 B^{2} + 128 B C + 108 C^{2}\right ) - 9 A^{3} - 12 A^{2} B - 16 A^{2} C - 18 A B C - 16 A C^{2} + 6 B^{3} + 8 B^{2} C - 4 C^{3}, \left ( t \mapsto t \log {\left (x + \frac {22080 t^{2} A + 37260 t^{2} B + 33120 t^{2} C + 6210 t A^{2} + 5520 t A B - 8280 t B C - 7360 t C^{2} - 204 A^{3} + 1296 A^{2} B + 462 A^{2} C + 2532 A B^{2} + 3888 A B C + 1728 A C^{2} + 648 B^{3} + 1728 B^{2} C + 1996 B C^{2} + 864 C^{3}}{1611 A^{3} + 2916 A^{2} B + 2592 A^{2} C + 2592 A B^{2} + 4608 A B C + 2048 A C^{2} + 1458 B^{3} + 3888 B^{2} C + 3456 B C^{2} + 1024 C^{3}} \right )} \right )\right )} \] Input:

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

Output:

RootSum(1380*_t**3 - 460*_t**2*C + _t*(48*A**2 + 162*A*B + 144*A*C + 72*B* 
*2 + 128*B*C + 108*C**2) - 9*A**3 - 12*A**2*B - 16*A**2*C - 18*A*B*C - 16* 
A*C**2 + 6*B**3 + 8*B**2*C - 4*C**3, Lambda(_t, _t*log(x + (22080*_t**2*A 
+ 37260*_t**2*B + 33120*_t**2*C + 6210*_t*A**2 + 5520*_t*A*B - 8280*_t*B*C 
 - 7360*_t*C**2 - 204*A**3 + 1296*A**2*B + 462*A**2*C + 2532*A*B**2 + 3888 
*A*B*C + 1728*A*C**2 + 648*B**3 + 1728*B**2*C + 1996*B*C**2 + 864*C**3)/(1 
611*A**3 + 2916*A**2*B + 2592*A**2*C + 2592*A*B**2 + 4608*A*B*C + 2048*A*C 
**2 + 1458*B**3 + 3888*B**2*C + 3456*B*C**2 + 1024*C**3))))
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((C*x^2+B*x+A)/(3*x^3-4*x^2+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 = 12.42 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=\sum _{k=1}^3\ln \left (-\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (48\,A^2+72\,B^2+108\,C^2+162\,A\,B+144\,A\,C+128\,B\,C\right )}{1380}-\frac {3\,A\,B\,C}{230}+\frac {2\,B^2\,C}{345}-\frac {4\,A^2\,C}{345}-\frac {4\,A\,C^2}{345}-\frac {A^2\,B}{115}-\frac {3\,A^3}{460}-\frac {C^3}{345}+\frac {B^3}{230},z,k\right )\,\left (12\,A+36\,C-x\,\left (27\,A+12\,B+32\,C\right )+\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (48\,A^2+72\,B^2+108\,C^2+162\,A\,B+144\,A\,C+128\,B\,C\right )}{1380}-\frac {3\,A\,B\,C}{230}+\frac {2\,B^2\,C}{345}-\frac {4\,A^2\,C}{345}-\frac {4\,A\,C^2}{345}-\frac {A^2\,B}{115}-\frac {3\,A^3}{460}-\frac {C^3}{345}+\frac {B^3}{230},z,k\right )\,\left (96\,x-162\right )\right )+2\,C^2+3\,A\,B+4\,A\,C+x\,\left (3\,B^2+4\,C\,B-3\,A\,C\right )\right )\,\mathrm {root}\left (z^3-\frac {C\,z^2}{3}+\frac {z\,\left (48\,A^2+72\,B^2+108\,C^2+162\,A\,B+144\,A\,C+128\,B\,C\right )}{1380}-\frac {3\,A\,B\,C}{230}+\frac {2\,B^2\,C}{345}-\frac {4\,A^2\,C}{345}-\frac {4\,A\,C^2}{345}-\frac {A^2\,B}{115}-\frac {3\,A^3}{460}-\frac {C^3}{345}+\frac {B^3}{230},z,k\right ) \] Input:

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

Output:

symsum(log(2*C^2 - root(z^3 - (C*z^2)/3 + (z*(48*A^2 + 72*B^2 + 108*C^2 + 
162*A*B + 144*A*C + 128*B*C))/1380 - (3*A*B*C)/230 + (2*B^2*C)/345 - (4*A^ 
2*C)/345 - (4*A*C^2)/345 - (A^2*B)/115 - (3*A^3)/460 - C^3/345 + B^3/230, 
z, k)*(12*A + 36*C - x*(27*A + 12*B + 32*C) + root(z^3 - (C*z^2)/3 + (z*(4 
8*A^2 + 72*B^2 + 108*C^2 + 162*A*B + 144*A*C + 128*B*C))/1380 - (3*A*B*C)/ 
230 + (2*B^2*C)/345 - (4*A^2*C)/345 - (4*A*C^2)/345 - (A^2*B)/115 - (3*A^3 
)/460 - C^3/345 + B^3/230, z, k)*(96*x - 162)) + 3*A*B + 4*A*C + x*(3*B^2 
- 3*A*C + 4*B*C))*root(z^3 - (C*z^2)/3 + (z*(48*A^2 + 72*B^2 + 108*C^2 + 1 
62*A*B + 144*A*C + 128*B*C))/1380 - (3*A*B*C)/230 + (2*B^2*C)/345 - (4*A^2 
*C)/345 - (4*A*C^2)/345 - (A^2*B)/115 - (3*A^3)/460 - C^3/345 + B^3/230, z 
, k), k, 1, 3)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{2-4 x^2+3 x^3} \, dx=\left (\int \frac {x}{3 x^{3}-4 x^{2}+2}d x \right ) b +\frac {8 \left (\int \frac {x}{3 x^{3}-4 x^{2}+2}d x \right ) c}{9}+\left (\int \frac {1}{3 x^{3}-4 x^{2}+2}d x \right ) a +\frac {\mathrm {log}\left (3 x^{3}-4 x^{2}+2\right ) c}{9} \] Input:

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

Output:

(9*int(x/(3*x**3 - 4*x**2 + 2),x)*b + 8*int(x/(3*x**3 - 4*x**2 + 2),x)*c + 
 9*int(1/(3*x**3 - 4*x**2 + 2),x)*a + log(3*x**3 - 4*x**2 + 2)*c)/9