\(\int \frac {A+B x+C x^2}{(2-4 x+3 x^3)^2} \, dx\) [19]

Optimal result

Integrand size = 23, antiderivative size = 1171 \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx =\text {Too large to display} \] Output:

(9-17^(1/2))^(1/3)*(36*A+27*B+16*C)/(136+34*(9-17^(1/2))^(2/3)+102*(9-17^( 
1/2))^(1/3)*x)+1/34*(-17+9*17^(1/2))^(1/3)*(72*A+54*B+32*C-3*(9*(9-17^(1/2 
)+4*(9-17^(1/2))^(1/3))*A+3*(16+(9-17^(1/2))^(4/3))*B+4*(9-17^(1/2)+4*(9-1 
7^(1/2))^(1/3))*C)*x/(9-17^(1/2))^(2/3))*17^(1/3)/(4-(9-17^(1/2))^(2/3))/( 
(4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(4-16/(9-17^(1/2))^(2/3)-(9 
-17^(1/2))^(2/3)+3*(4+(9-17^(1/2))^(2/3))*x/(9-17^(1/2))^(1/3)-9*x^2)-C/(2 
7*x^3-36*x+18)-1/17*(9*(117000-27592*17^(1/2)+(21753-5009*17^(1/2))*(9-17^ 
(1/2))^(2/3))*A+3*(231137-54585*17^(1/2)+(40916-9396*17^(1/2))*(9-17^(1/2) 
)^(2/3))*B+4*(117000-27592*17^(1/2)+(21753-5009*17^(1/2))*(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)/(29 
4-54*17^(1/2)+48*(9-17^(1/2))^(2/3)-24*(9-17^(1/2))^(4/3))^(1/2))/(2376-52 
0*17^(1/2)+(1889-441*17^(1/2))*(9-17^(1/2))^(1/3)+(594-130*17^(1/2))*(9-17 
^(1/2))^(2/3))/(294-54*17^(1/2)+48*(9-17^(1/2))^(2/3)-24*(9-17^(1/2))^(4/3 
))^(1/2)-4/3*(9-17^(1/2))*(9*(196-36*17^(1/2)+8*(9-17^(1/2))^(4/3)-9*(9-17 
^(1/2))^(5/3))*A+3*(441-81*17^(1/2)+18*(9-17^(1/2))^(4/3)-16*(9-17^(1/2))^ 
(5/3))*B+4*(196-36*17^(1/2)+8*(9-17^(1/2))^(4/3)-9*(9-17^(1/2))^(5/3))*C)* 
ln(4+(9-17^(1/2))^(2/3)+3*(9-17^(1/2))^(1/3)*x)/(4-(9-17^(1/2))^(2/3))^2/( 
16+4*(9-17^(1/2))^(2/3)+(9-17^(1/2))^(4/3))^3+8/3*(9*(7556-1764*17^(1/2)+8 
*(297-65*17^(1/2))*(9-17^(1/2))^(1/3)-9*(297-65*17^(1/2))*(9-17^(1/2))^(2/ 
3))*A+3*(17001-3969*17^(1/2)+18*(297-65*17^(1/2))*(9-17^(1/2))^(1/3)-16...
 

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.13 \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\frac {1}{34} \left (\frac {2 C \left (-9+6 x+8 x^2\right )+B \left (-24+16 x+27 x^2\right )+A \left (-32+27 x+36 x^2\right )}{2-4 x+3 x^3}+\text {RootSum}\left [2-4 \text {$\#$1}+3 \text {$\#$1}^3\&,\frac {54 A \log (x-\text {$\#$1})+32 B \log (x-\text {$\#$1})+24 C \log (x-\text {$\#$1})+36 A \log (x-\text {$\#$1}) \text {$\#$1}+27 B \log (x-\text {$\#$1}) \text {$\#$1}+16 C \log (x-\text {$\#$1}) \text {$\#$1}}{-4+9 \text {$\#$1}^2}\&\right ]\right ) \] Input:

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

Output:

((2*C*(-9 + 6*x + 8*x^2) + B*(-24 + 16*x + 27*x^2) + A*(-32 + 27*x + 36*x^ 
2))/(2 - 4*x + 3*x^3) + RootSum[2 - 4*#1 + 3*#1^3 & , (54*A*Log[x - #1] + 
32*B*Log[x - #1] + 24*C*Log[x - #1] + 36*A*Log[x - #1]*#1 + 27*B*Log[x - # 
1]*#1 + 16*C*Log[x - #1]*#1)/(-4 + 9*#1^2) & ])/34
 

Rubi [A] (warning: unable to verify)

Time = 5.45 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2526, 2485, 1235, 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.

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

\(\Big \downarrow \) 2526

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

\(\Big \downarrow \) 2485

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

\(\Big \downarrow \) 1235

\(\displaystyle 9 \left (\frac {\sqrt [3]{9 \sqrt {17}-17} \left (2 (36 A+27 B+16 C)-\frac {3 x \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right )}{\left (9-\sqrt {17}\right )^{2/3}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}-\frac {\sqrt [3]{9 \sqrt {17}-17} \int -\frac {486 \left (-\frac {\left (4+\left (9-\sqrt {17}\right )^{2/3}\right )^3 B}{9-\sqrt {17}}+45 B-\frac {\left (16-4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) (9 A+4 C)}{\left (9-\sqrt {17}\right )^{2/3}}-\frac {3 \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}\right )}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^2 \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx}{1458\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle 9 \left (\frac {\sqrt [3]{9 \sqrt {17}-17} \int \frac {-\frac {\left (4+\left (9-\sqrt {17}\right )^{2/3}\right )^3 B}{9-\sqrt {17}}+45 B-\frac {\left (16-4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right ) (9 A+4 C)}{\left (9-\sqrt {17}\right )^{2/3}}-\frac {3 \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}}{\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^2 \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}dx}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}+\frac {\sqrt [3]{9 \sqrt {17}-17} \left (2 (36 A+27 B+16 C)-\frac {3 x \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right )}{\left (9-\sqrt {17}\right )^{2/3}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (-9 x^2+\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}-\left (9-\sqrt {17}\right )^{2/3}-\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}+4\right )}\right )-\frac {C}{9 \left (3 x^3-4 x+2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle 9 \left (\frac {\sqrt [3]{-17+9 \sqrt {17}} \left (2 (36 A+27 B+16 C)-\frac {3 \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \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 )}+\frac {\sqrt [3]{-17+9 \sqrt {17}} \int \left (\frac {2 \left (49-9 \sqrt {17}\right ) (-36 A-27 B-16 C)}{\left (9-\sqrt {17}\right )^{2/3} \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 )^2}+\frac {2 \left (-9 \left (196-36 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{4/3}-9 \left (9-\sqrt {17}\right )^{5/3}\right ) A-3 \left (441-81 \sqrt {17}+18 \left (9-\sqrt {17}\right )^{4/3}-16 \left (9-\sqrt {17}\right )^{5/3}\right ) B-4 \left (196-36 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{4/3}-9 \left (9-\sqrt {17}\right )^{5/3}\right ) C\right )}{3 \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2 \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}+\frac {4 \left (36 \left (3065-657 \sqrt {17}+\left (1179-227 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-2 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) A+3 \left (22536-4808 \sqrt {17}+\left (8945-1737 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-18 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) B+16 \left (3065-657 \sqrt {17}+\left (1179-227 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-2 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) C+3 \left (9 \left (1188-260 \sqrt {17}+8 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-9 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) A+3 \left (2673-585 \sqrt {17}+18 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-16 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \left (1188-260 \sqrt {17}+8 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-9 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) C\right ) x\right )}{3 \left (9-\sqrt {17}\right )^{2/3} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2 \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}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x+2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle 9 \left (\frac {\sqrt [3]{-17+9 \sqrt {17}} \left (2 (36 A+27 B+16 C)-\frac {3 \left (9 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) A+3 \left (16+\left (9-\sqrt {17}\right )^{4/3}\right ) B+4 \left (9-\sqrt {17}+4 \sqrt [3]{9-\sqrt {17}}\right ) C\right ) x}{\left (9-\sqrt {17}\right )^{2/3}}\right )}{18\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right ) \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 )}+\frac {\sqrt [3]{-17+9 \sqrt {17}} \left (\frac {2 \left (49-9 \sqrt {17}\right ) (36 A+27 B+16 C)}{3 \left (9-\sqrt {17}\right ) \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 {4 \sqrt {\frac {2}{3 \left (49-9 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{2/3}-4 \left (9-\sqrt {17}\right )^{4/3}\right )}} \left (9 \left (18248-4104 \sqrt {17}+\left (3457-729 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) A+3 \left (36009-8129 \sqrt {17}+\left (6516-1364 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \left (18248-4104 \sqrt {17}+\left (3457-729 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) C\right ) \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 )}{3 \left (9-\sqrt {17}\right )^{4/3} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}-\frac {2 \left (9 \left (196-36 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{4/3}-9 \left (9-\sqrt {17}\right )^{5/3}\right ) A+3 \left (441-81 \sqrt {17}+18 \left (9-\sqrt {17}\right )^{4/3}-16 \left (9-\sqrt {17}\right )^{5/3}\right ) B+4 \left (196-36 \sqrt {17}+8 \left (9-\sqrt {17}\right )^{4/3}-9 \left (9-\sqrt {17}\right )^{5/3}\right ) C\right ) \log \left (3 \sqrt [3]{9-\sqrt {17}} x+\left (9-\sqrt {17}\right )^{2/3}+4\right )}{9 \sqrt [3]{9-\sqrt {17}} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}+\frac {2 \left (9 \left (1188-260 \sqrt {17}+8 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-9 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) A+3 \left (2673-585 \sqrt {17}+18 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-16 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) B+4 \left (1188-260 \sqrt {17}+8 \left (49-9 \sqrt {17}\right ) \sqrt [3]{9-\sqrt {17}}-9 \left (49-9 \sqrt {17}\right ) \left (9-\sqrt {17}\right )^{2/3}\right ) C\right ) \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 )}{9 \left (9-\sqrt {17}\right )^{4/3} \left (16+4 \left (9-\sqrt {17}\right )^{2/3}+\left (9-\sqrt {17}\right )^{4/3}\right )^2}\right )}{3\ 17^{2/3} \left (4-\left (9-\sqrt {17}\right )^{2/3}\right )}\right )-\frac {C}{9 \left (3 x^3-4 x+2\right )}\)

Input:

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

Output:

-1/9*C/(2 - 4*x + 3*x^3) + 9*(((-17 + 9*Sqrt[17])^(1/3)*(2*(36*A + 27*B + 
16*C) - (3*(9*(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3))*A + 3*(16 + (9 - Sqr 
t[17])^(4/3))*B + 4*(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3))*C)*x)/(9 - Sqr 
t[17])^(2/3)))/(18*17^(2/3)*(4 - (9 - Sqrt[17])^(2/3))*((4 + (9 - Sqrt[17] 
)^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x)*(4 - 16/(9 - Sqrt[17])^(2/3) - (9 - S 
qrt[17])^(2/3) + (3*(4 + (9 - Sqrt[17])^(2/3))*x)/(9 - Sqrt[17])^(1/3) - 9 
*x^2)) + ((-17 + 9*Sqrt[17])^(1/3)*((2*(49 - 9*Sqrt[17])*(36*A + 27*B + 16 
*C))/(3*(9 - Sqrt[17])*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4/3) 
)*(4 + (9 - Sqrt[17])^(2/3) + 3*(9 - Sqrt[17])^(1/3)*x)) - (4*Sqrt[2/(3*(4 
9 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 4*(9 - Sqrt[17])^(4/3)))]*(9*(18 
248 - 4104*Sqrt[17] + (3457 - 729*Sqrt[17])*(9 - Sqrt[17])^(2/3))*A + 3*(3 
6009 - 8129*Sqrt[17] + (6516 - 1364*Sqrt[17])*(9 - Sqrt[17])^(2/3))*B + 4* 
(18248 - 4104*Sqrt[17] + (3457 - 729*Sqrt[17])*(9 - Sqrt[17])^(2/3))*C)*Ar 
cTan[(9 - Sqrt[17] + 4*(9 - Sqrt[17])^(1/3) - 6*(9 - Sqrt[17])^(2/3)*x)/Sq 
rt[6*(49 - 9*Sqrt[17] + 8*(9 - Sqrt[17])^(2/3) - 4*(9 - Sqrt[17])^(4/3))]] 
)/(3*(9 - Sqrt[17])^(4/3)*(16 + 4*(9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(4 
/3))^2) - (2*(9*(196 - 36*Sqrt[17] + 8*(9 - Sqrt[17])^(4/3) - 9*(9 - Sqrt[ 
17])^(5/3))*A + 3*(441 - 81*Sqrt[17] + 18*(9 - Sqrt[17])^(4/3) - 16*(9 - S 
qrt[17])^(5/3))*B + 4*(196 - 36*Sqrt[17] + 8*(9 - Sqrt[17])^(4/3) - 9*(9 - 
 Sqrt[17])^(5/3))*C)*Log[4 + (9 - Sqrt[17])^(2/3) + 3*(9 - Sqrt[17])^(1...
 

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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

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 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*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^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.09

method	result	size
default	\(\frac {\left (\frac {8 C}{51}+\frac {6 A}{17}+\frac {9 B}{34}\right ) x^{2}+\left (\frac {2 C}{17}+\frac {9 A}{34}+\frac {8 B}{51}\right ) x -\frac {3 C}{17}-\frac {16 A}{51}-\frac {4 B}{17}}{x^{3}-\frac {4}{3} x +\frac {2}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (36 A \textit {\_R} +27 B \textit {\_R} +16 C \textit {\_R} +54 A +32 B +24 C \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\right )}{34}\)	\(103\)
risch	\(\frac {\left (\frac {8 C}{51}+\frac {6 A}{17}+\frac {9 B}{34}\right ) x^{2}+\left (\frac {2 C}{17}+\frac {9 A}{34}+\frac {8 B}{51}\right ) x -\frac {3 C}{17}-\frac {16 A}{51}-\frac {4 B}{17}}{x^{3}-\frac {4}{3} x +\frac {2}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-4 \textit {\_Z} +2\right )}{\sum }\frac {\left (\left (36 A +27 B +16 C \right ) \textit {\_R} +24 C +54 A +32 B \right ) \ln \left (x -\textit {\_R} \right )}{9 \textit {\_R}^{2}-4}\right )}{34}\)	\(103\)

Input:

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

Output:

((8/51*C+6/17*A+9/34*B)*x^2+(2/17*C+9/34*A+8/51*B)*x-3/17*C-16/51*A-4/17*B 
)/(x^3-4/3*x+2/3)+1/34*sum((36*A*_R+27*B*_R+16*C*_R+54*A+32*B+24*C)/(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 = 0.91 (sec) , antiderivative size = 5268, normalized size of antiderivative = 4.50 \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Too large to display} \] 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 = 8.83 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.29 \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\operatorname {RootSum} {\left (235824 t^{3} + t \left (230688 A^{2} + 296460 A B + 205056 A C + 95184 B^{2} + 131760 B C + 45568 C^{2}\right ) - 2916 A^{3} - 1296 A^{2} B - 3888 A^{2} C + 1944 A B^{2} - 1152 A B C - 1728 A C^{2} + 1011 B^{3} + 864 B^{2} C - 256 B C^{2} - 256 C^{3}, \left ( t \mapsto t \log {\left (x + \frac {114610464 t^{2} A + 73930824 t^{2} B + 50937984 t^{2} C + 12172680 t A^{2} + 13483584 t A B + 10820160 t A C + 3641400 t B^{2} + 5992704 t B C + 2404480 t C^{2} + 74742912 A^{3} + 144266832 A^{2} B + 99657216 A^{2} C + 92799756 A B^{2} + 128237184 A B C + 44292096 A C^{2} + 19893456 B^{3} + 41244336 B^{2} C + 28497152 B C^{2} + 6561792 C^{3}}{55430244 A^{3} + 106760592 A^{2} B + 73906992 A^{2} C + 68586264 A B^{2} + 94898304 A B C + 32847552 A C^{2} + 14696883 B^{3} + 30482784 B^{2} C + 21088512 B C^{2} + 4866304 C^{3}} \right )} \right )\right )} + \frac {- 32 A - 24 B - 18 C + x^{2} \cdot \left (36 A + 27 B + 16 C\right ) + x \left (27 A + 16 B + 12 C\right )}{102 x^{3} - 136 x + 68} \] Input:

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

Output:

RootSum(235824*_t**3 + _t*(230688*A**2 + 296460*A*B + 205056*A*C + 95184*B 
**2 + 131760*B*C + 45568*C**2) - 2916*A**3 - 1296*A**2*B - 3888*A**2*C + 1 
944*A*B**2 - 1152*A*B*C - 1728*A*C**2 + 1011*B**3 + 864*B**2*C - 256*B*C** 
2 - 256*C**3, Lambda(_t, _t*log(x + (114610464*_t**2*A + 73930824*_t**2*B 
+ 50937984*_t**2*C + 12172680*_t*A**2 + 13483584*_t*A*B + 10820160*_t*A*C 
+ 3641400*_t*B**2 + 5992704*_t*B*C + 2404480*_t*C**2 + 74742912*A**3 + 144 
266832*A**2*B + 99657216*A**2*C + 92799756*A*B**2 + 128237184*A*B*C + 4429 
2096*A*C**2 + 19893456*B**3 + 41244336*B**2*C + 28497152*B*C**2 + 6561792* 
C**3)/(55430244*A**3 + 106760592*A**2*B + 73906992*A**2*C + 68586264*A*B** 
2 + 94898304*A*B*C + 32847552*A*C**2 + 14696883*B**3 + 30482784*B**2*C + 2 
1088512*B*C**2 + 4866304*C**3)))) + (-32*A - 24*B - 18*C + x**2*(36*A + 27 
*B + 16*C) + x*(27*A + 16*B + 12*C))/(102*x**3 - 136*x + 68)
 

Maxima [F]

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

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

Output:

1/34*((36*A + 27*B + 16*C)*x^2 + (27*A + 16*B + 12*C)*x - 32*A - 24*B - 18 
*C)/(3*x^3 - 4*x + 2) + 1/34*integrate(((36*A + 27*B + 16*C)*x + 54*A + 32 
*B + 24*C)/(3*x^3 - 4*x + 2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Exception raised: TypeError} \] 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 = 12.88 (sec) , antiderivative size = 987, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((972*A^2*x)/289 + (2187*B^2*x)/1156 + (192*C^2*x)/289 - 216*roo 
t(z^3 + z*((4806*A^2)/4913 + (1983*B^2)/4913 + (2848*C^2)/14739 + (24705*A 
*B)/19652 + (4272*A*C)/4913 + (2745*B*C)/4913) - (24*A*B*C)/4913 - (16*B*C 
^2)/14739 + (18*B^2*C)/4913 - (81*A^2*C)/4913 - (36*A*C^2)/4913 + (81*A*B^ 
2)/9826 - (27*A^2*B)/4913 - (243*A^3)/19652 - (16*C^3)/14739 + (337*B^3)/7 
8608, z, k)^2*x + (1458*A^2)/289 + (648*B^2)/289 + (288*C^2)/289 + 162*roo 
t(z^3 + z*((4806*A^2)/4913 + (1983*B^2)/4913 + (2848*C^2)/14739 + (24705*A 
*B)/19652 + (4272*A*C)/4913 + (2745*B*C)/4913) - (24*A*B*C)/4913 - (16*B*C 
^2)/14739 + (18*B^2*C)/4913 - (81*A^2*C)/4913 - (36*A*C^2)/4913 + (81*A*B^ 
2)/9826 - (27*A^2*B)/4913 - (243*A^3)/19652 - (16*C^3)/14739 + (337*B^3)/7 
8608, z, k)^2 + (3915*A*B)/578 + (1296*A*C)/289 + (870*B*C)/289 + (216*A*r 
oot(z^3 + z*((4806*A^2)/4913 + (1983*B^2)/4913 + (2848*C^2)/14739 + (24705 
*A*B)/19652 + (4272*A*C)/4913 + (2745*B*C)/4913) - (24*A*B*C)/4913 - (16*B 
*C^2)/14739 + (18*B^2*C)/4913 - (81*A^2*C)/4913 - (36*A*C^2)/4913 + (81*A* 
B^2)/9826 - (27*A^2*B)/4913 - (243*A^3)/19652 - (16*C^3)/14739 + (337*B^3) 
/78608, z, k))/17 + (162*B*root(z^3 + z*((4806*A^2)/4913 + (1983*B^2)/4913 
 + (2848*C^2)/14739 + (24705*A*B)/19652 + (4272*A*C)/4913 + (2745*B*C)/491 
3) - (24*A*B*C)/4913 - (16*B*C^2)/14739 + (18*B^2*C)/4913 - (81*A^2*C)/491 
3 - (36*A*C^2)/4913 + (81*A*B^2)/9826 - (27*A^2*B)/4913 - (243*A^3)/19652 
- (16*C^3)/14739 + (337*B^3)/78608, z, k))/17 + (96*C*root(z^3 + z*((48...
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (2-4 x+3 x^3\right )^2} \, dx=\frac {81 \left (\int \frac {x^{4}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b \,x^{3}-108 \left (\int \frac {x^{4}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b x +54 \left (\int \frac {x^{4}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b +324 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) a \,x^{3}-432 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) a x +216 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) a +144 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b \,x^{3}-192 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b x +96 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) b +144 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) c \,x^{3}-192 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) c x +96 \left (\int \frac {x^{3}}{9 x^{6}-24 x^{4}+12 x^{3}+16 x^{2}-16 x +4}d x \right ) c +18 a x +9 b \,x^{2}+8 b x -4 b +6 c \,x^{3}}{108 x^{3}-144 x +72} \] Input:

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

Output:

(81*int(x**4/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b*x**3 - 
 108*int(x**4/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b*x + 5 
4*int(x**4/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b + 324*in 
t(x**3/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*a*x**3 - 432*i 
nt(x**3/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*a*x + 216*int 
(x**3/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*a + 144*int(x** 
3/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b*x**3 - 192*int(x* 
*3/(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b*x + 96*int(x**3/ 
(9*x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*b + 144*int(x**3/(9*x 
**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*c*x**3 - 192*int(x**3/(9* 
x**6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*c*x + 96*int(x**3/(9*x** 
6 - 24*x**4 + 12*x**3 + 16*x**2 - 16*x + 4),x)*c + 18*a*x + 9*b*x**2 + 8*b 
*x - 4*b + 6*c*x**3)/(36*(3*x**3 - 4*x + 2))