\(\int \frac {A+B x+C x^2}{(2+3 x+4 x^2+x^3)^2} \, dx\) [175]

Optimal result

Integrand size = 26, antiderivative size = 1237 \[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx =\text {Too large to display} \] Output:

-1/3*C/(x^3+4*x^2+3*x+2)+3*(37-3*114^(1/2))^(1/3)*(7*A+3*B-15*C)/(532+76*( 
37-3*114^(1/2))^(2/3)+76*(37-3*114^(1/2))^(1/3)*(4+3*x))+3/76*(-1026+111*1 
14^(1/2))^(1/3)*(42*A+18*B-90*C-(3*(37-3*114^(1/2)+7*(37-3*114^(1/2))^(1/3 
))*A-3*(33-4*114^(1/2)-(37-3*114^(1/2))^(1/3)*(3-114^(1/2)))*B+(153-23*114 
^(1/2)-(45-8*114^(1/2))*(37-3*114^(1/2))^(1/3))*C)*(4+3*x)/(37-3*114^(1/2) 
)^(2/3))*38^(1/3)/(7-(37-3*114^(1/2))^(2/3))/(4+7/(37-3*114^(1/2))^(1/3)+( 
37-3*114^(1/2))^(1/3)+3*x)/(7-49/(37-3*114^(1/2))^(2/3)-(37-3*114^(1/2))^( 
2/3)+(7+(37-3*114^(1/2))^(2/3))*(4+3*x)/(37-3*114^(1/2))^(1/3)-(4+3*x)^2)- 
162*3^(1/2)/(2395-222*114^(1/2)+49*(37-3*114^(1/2))^(2/3)-14*(37-3*114^(1/ 
2))^(4/3))^(1/2)*((54904003-5140114*114^(1/2)+(4332457-405224*114^(1/2))*( 
37-3*114^(1/2))^(2/3))*A-3*(4661535-436022*114^(1/2)+(1167501-109300*114^( 
1/2))*(37-3*114^(1/2))^(2/3))*B-(17611723-1651938*114^(1/2)-(5007551-46917 
6*114^(1/2))*(37-3*114^(1/2))^(2/3))*C)*arctan((37-3*114^(1/2)+7*(37-3*114 
^(1/2))^(1/3)-2*(37-3*114^(1/2))^(2/3)*(4+3*x))/(7185-666*114^(1/2)+147*(3 
7-3*114^(1/2))^(2/3)-42*(37-3*114^(1/2))^(4/3))^(1/2))/(7-(37-3*114^(1/2)) 
^(2/3))^2/(49+7*(37-3*114^(1/2))^(2/3)+(37-3*114^(1/2))^(4/3))^3+27*((1151 
773-107793*114^(1/2)+49*(2395-222*114^(1/2))*(37-3*114^(1/2))^(1/3)-74*(23 
95-222*114^(1/2))*(37-3*114^(1/2))^(2/3))*A+3*(164539-15399*114^(1/2)+7*(2 
395-222*114^(1/2))*(37-3*114^(1/2))^(1/3)+22*(2395-222*114^(1/2))*(37-3*11 
4^(1/2))^(2/3))*B-3*(822695-76995*114^(1/2)+(83825-7770*114^(1/2))*(37-...
 

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.13 \[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx=\frac {1}{76} \left (\frac {B \left (-14-3 x+3 x^2\right )+A \left (18+31 x+7 x^2\right )-C \left (6+23 x+15 x^2\right )}{2+3 x+4 x^2+x^3}+\text {RootSum}\left [2+3 \text {$\#$1}+4 \text {$\#$1}^2+\text {$\#$1}^3\&,\frac {34 A \log (x-\text {$\#$1})-18 B \log (x-\text {$\#$1})+14 C \log (x-\text {$\#$1})+7 A \log (x-\text {$\#$1}) \text {$\#$1}+3 B \log (x-\text {$\#$1}) \text {$\#$1}-15 C \log (x-\text {$\#$1}) \text {$\#$1}}{3+8 \text {$\#$1}+3 \text {$\#$1}^2}\&\right ]\right ) \] Input:

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

Output:

((B*(-14 - 3*x + 3*x^2) + A*(18 + 31*x + 7*x^2) - C*(6 + 23*x + 15*x^2))/( 
2 + 3*x + 4*x^2 + x^3) + RootSum[2 + 3*#1 + 4*#1^2 + #1^3 & , (34*A*Log[x 
- #1] - 18*B*Log[x - #1] + 14*C*Log[x - #1] + 7*A*Log[x - #1]*#1 + 3*B*Log 
[x - #1]*#1 - 15*C*Log[x - #1]*#1)/(3 + 8*#1 + 3*#1^2) & ])/76
 

Rubi [A] (warning: unable to verify)

Time = 8.60 (sec) , antiderivative size = 1268, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2526, 2490, 2485, 27, 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.

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

\(\Big \downarrow \) 2526

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

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{3} \int \frac {\frac {1}{3} (9 (A-C)-4 (3 B-8 C))+(3 B-8 C) \left (x+\frac {4}{3}\right )}{\left (\left (x+\frac {4}{3}\right )^3-\frac {7}{3} \left (x+\frac {4}{3}\right )+\frac {74}{27}\right )^2}d\left (x+\frac {4}{3}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 2485

\(\displaystyle \frac {1}{3} \int \frac {243 \left (9 A-12 B+23 C+3 (3 B-8 C) \left (x+\frac {4}{3}\right )\right )}{\left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )^2 \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )^2}d\left (x+\frac {4}{3}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle 81 \int \frac {9 A-12 B+23 C+3 (3 B-8 C) \left (x+\frac {4}{3}\right )}{\left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )^2 \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )^2}d\left (x+\frac {4}{3}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle 81 \left (\frac {\sqrt [3]{\frac {37}{114 \sqrt {114}}-\frac {1}{38}} \left (2 (7 A+3 B-15 C)-\frac {\left (x+\frac {4}{3}\right ) \left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}\right )}{6 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right ) \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}-\frac {\sqrt [3]{\frac {37}{114 \sqrt {114}}-\frac {1}{38}} \int -\frac {243 \left (\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (49-21 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) (3 B-8 C)-2 \sqrt [3]{37-3 \sqrt {114}} \left (49-7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) (9 A-12 B+23 C)}{37-3 \sqrt {114}}-\frac {18 \left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right ) \left (x+\frac {4}{3}\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}\right )}{\left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )^2 \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}d\left (x+\frac {4}{3}\right )}{4374 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right )}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle 81 \left (\frac {\sqrt [3]{\frac {37}{114 \sqrt {114}}-\frac {1}{38}} \int \frac {\frac {\left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (49-21 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) (3 B-8 C)-2 \sqrt [3]{37-3 \sqrt {114}} \left (49-7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) (9 A-12 B+23 C)}{37-3 \sqrt {114}}-\frac {18 \left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right ) \left (x+\frac {4}{3}\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}}{\left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right )^2 \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}d\left (x+\frac {4}{3}\right )}{18 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right )}+\frac {\sqrt [3]{\frac {37}{114 \sqrt {114}}-\frac {1}{38}} \left (2 (7 A+3 B-15 C)-\frac {\left (x+\frac {4}{3}\right ) \left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}\right )}{6 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right ) \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle 81 \left (\frac {\sqrt [3]{-\frac {1}{38}+\frac {37}{114 \sqrt {114}}} \left (2 (7 A+3 B-15 C)-\frac {\left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right ) \left (x+\frac {4}{3}\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}\right )}{6 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right ) \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}+\frac {\sqrt [3]{-\frac {1}{38}+\frac {37}{114 \sqrt {114}}} \int \left (\frac {18 \left (2395-222 \sqrt {114}\right ) (-7 A-3 B+15 C)}{\left (37-3 \sqrt {114}\right )^{2/3} \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) \left (3 \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{2/3}+7\right )^2}+\frac {6 \left (-\left (\left (16765-1554 \sqrt {114}+49 \left (37-3 \sqrt {114}\right )^{4/3}-74 \left (37-3 \sqrt {114}\right )^{5/3}\right ) A\right )-3 \left (2395-222 \sqrt {114}+7 \left (37-3 \sqrt {114}\right )^{4/3}+22 \left (37-3 \sqrt {114}\right )^{5/3}\right ) B+3 \left (11975-1110 \sqrt {114}+35 \left (37-3 \sqrt {114}\right )^{4/3}+34 \left (37-3 \sqrt {114}\right )^{5/3}\right ) C\right )}{\left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right )^2 \left (3 \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{2/3}+7\right )}+\frac {6 \left (\left (25173257-2355198 \sqrt {114}+7 \left (518999-48255 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}-49 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) A-3 \left (\left (7122361-666678 \sqrt {114}+\left (573121-52977 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+7 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) B-\left (10601877-992742 \sqrt {114}+\left (317325-28677 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+\left (83825-7770 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) C\right )+3 \left (\left (1151773-107793 \sqrt {114}+49 \left (2395-222 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}-74 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) A+3 \left (\left (164539-15399 \sqrt {114}+7 \left (2395-222 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+22 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) B-\left (822695-76995 \sqrt {114}+\left (83825-7770 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+\left (81430-7548 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) C\right )\right ) \left (x+\frac {4}{3}\right )\right )}{\left (37-3 \sqrt {114}\right )^{2/3} \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right )^2 \left (9 \left (37-3 \sqrt {114}\right )^{2/3} \left (x+\frac {4}{3}\right )^2-3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{4/3}-7 \left (37-3 \sqrt {114}\right )^{2/3}+49\right )}\right )d\left (x+\frac {4}{3}\right )}{18 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right )}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle 81 \left (\frac {\sqrt [3]{-\frac {1}{38}+\frac {37}{114 \sqrt {114}}} \left (2 (7 A+3 B-15 C)-\frac {\left (3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) A-3 \left (33-4 \sqrt {114}-\sqrt [3]{37-3 \sqrt {114}} \left (3-\sqrt {114}\right )\right ) B+\left (153-23 \sqrt {114}-\left (45-8 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}\right ) C\right ) \left (x+\frac {4}{3}\right )}{\left (37-3 \sqrt {114}\right )^{2/3}}\right )}{6 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (3 \left (x+\frac {4}{3}\right )+\frac {7+\left (37-3 \sqrt {114}\right )^{2/3}}{\sqrt [3]{37-3 \sqrt {114}}}\right ) \left (-9 \left (x+\frac {4}{3}\right )^2+\frac {3 \left (7+\left (37-3 \sqrt {114}\right )^{2/3}\right ) \left (x+\frac {4}{3}\right )}{\sqrt [3]{37-3 \sqrt {114}}}-\left (37-3 \sqrt {114}\right )^{2/3}-\frac {49}{\left (37-3 \sqrt {114}\right )^{2/3}}+7\right )}+\frac {\sqrt [3]{-\frac {1}{38}+\frac {37}{114 \sqrt {114}}} \left (\frac {6 \left (2395-222 \sqrt {114}\right ) (7 A+3 B-15 C)}{\left (37-3 \sqrt {114}\right ) \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right ) \left (3 \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{2/3}+7\right )}-\frac {6 \sqrt {\frac {3}{2395-222 \sqrt {114}+49 \left (37-3 \sqrt {114}\right )^{2/3}-14 \left (37-3 \sqrt {114}\right )^{4/3}}} \left (\left (7 \left (7843429-734302 \sqrt {114}\right )+\left (4332457-405224 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) A-3 \left (4661535-436022 \sqrt {114}+\left (1167501-109300 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) B-\left (17611723-1651938 \sqrt {114}-\left (5007551-469176 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) C\right ) \arctan \left (\frac {-6 \left (37-3 \sqrt {114}\right )^{2/3} \left (x+\frac {4}{3}\right )+7 \sqrt [3]{37-3 \sqrt {114}}-3 \sqrt {114}+37}{\sqrt {3 \left (2395-222 \sqrt {114}+49 \left (37-3 \sqrt {114}\right )^{2/3}-14 \left (37-3 \sqrt {114}\right )^{4/3}\right )}}\right )}{\left (37-3 \sqrt {114}\right )^{4/3} \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right )^2}-\frac {2 \left (\left (16765-1554 \sqrt {114}+49 \left (37-3 \sqrt {114}\right )^{4/3}-74 \left (37-3 \sqrt {114}\right )^{5/3}\right ) A+3 \left (2395-222 \sqrt {114}+7 \left (37-3 \sqrt {114}\right )^{4/3}+22 \left (37-3 \sqrt {114}\right )^{5/3}\right ) B-3 \left (11975-1110 \sqrt {114}+35 \left (37-3 \sqrt {114}\right )^{4/3}+34 \left (37-3 \sqrt {114}\right )^{5/3}\right ) C\right ) \log \left (3 \sqrt [3]{37-3 \sqrt {114}} \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{2/3}+7\right )}{\sqrt [3]{37-3 \sqrt {114}} \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right )^2}+\frac {\left (\left (1151773-107793 \sqrt {114}+49 \left (2395-222 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}-74 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) A+3 \left (\left (164539-15399 \sqrt {114}+7 \left (2395-222 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+22 \left (2395-222 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) B-\left (822695-76995 \sqrt {114}+\left (83825-7770 \sqrt {114}\right ) \sqrt [3]{37-3 \sqrt {114}}+\left (81430-7548 \sqrt {114}\right ) \left (37-3 \sqrt {114}\right )^{2/3}\right ) C\right )\right ) \log \left (9 \left (37-3 \sqrt {114}\right )^{2/3} \left (x+\frac {4}{3}\right )^2-3 \left (37-3 \sqrt {114}+7 \sqrt [3]{37-3 \sqrt {114}}\right ) \left (x+\frac {4}{3}\right )+\left (37-3 \sqrt {114}\right )^{4/3}-7 \left (37-3 \sqrt {114}\right )^{2/3}+49\right )}{\left (37-3 \sqrt {114}\right )^{4/3} \left (49+7 \left (37-3 \sqrt {114}\right )^{2/3}+\left (37-3 \sqrt {114}\right )^{4/3}\right )^2}\right )}{18 \left (7-\left (37-3 \sqrt {114}\right )^{2/3}\right )}\right )-\frac {C}{3 \left (x^3+4 x^2+3 x+2\right )}\)

Input:

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

Output:

-1/3*C/(2 + 3*x + 4*x^2 + x^3) + 81*(((-1/38 + 37/(114*Sqrt[114]))^(1/3)*( 
2*(7*A + 3*B - 15*C) - ((3*(37 - 3*Sqrt[114] + 7*(37 - 3*Sqrt[114])^(1/3)) 
*A - 3*(33 - 4*Sqrt[114] - (37 - 3*Sqrt[114])^(1/3)*(3 - Sqrt[114]))*B + ( 
153 - 23*Sqrt[114] - (45 - 8*Sqrt[114])*(37 - 3*Sqrt[114])^(1/3))*C)*(4/3 
+ x))/(37 - 3*Sqrt[114])^(2/3)))/(6*(7 - (37 - 3*Sqrt[114])^(2/3))*((7 + ( 
37 - 3*Sqrt[114])^(2/3))/(37 - 3*Sqrt[114])^(1/3) + 3*(4/3 + x))*(7 - 49/( 
37 - 3*Sqrt[114])^(2/3) - (37 - 3*Sqrt[114])^(2/3) + (3*(7 + (37 - 3*Sqrt[ 
114])^(2/3))*(4/3 + x))/(37 - 3*Sqrt[114])^(1/3) - 9*(4/3 + x)^2)) + ((-1/ 
38 + 37/(114*Sqrt[114]))^(1/3)*((6*(2395 - 222*Sqrt[114])*(7*A + 3*B - 15* 
C))/((37 - 3*Sqrt[114])*(49 + 7*(37 - 3*Sqrt[114])^(2/3) + (37 - 3*Sqrt[11 
4])^(4/3))*(7 + (37 - 3*Sqrt[114])^(2/3) + 3*(37 - 3*Sqrt[114])^(1/3)*(4/3 
 + x))) - (6*Sqrt[3/(2395 - 222*Sqrt[114] + 49*(37 - 3*Sqrt[114])^(2/3) - 
14*(37 - 3*Sqrt[114])^(4/3))]*((7*(7843429 - 734302*Sqrt[114]) + (4332457 
- 405224*Sqrt[114])*(37 - 3*Sqrt[114])^(2/3))*A - 3*(4661535 - 436022*Sqrt 
[114] + (1167501 - 109300*Sqrt[114])*(37 - 3*Sqrt[114])^(2/3))*B - (176117 
23 - 1651938*Sqrt[114] - (5007551 - 469176*Sqrt[114])*(37 - 3*Sqrt[114])^( 
2/3))*C)*ArcTan[(37 - 3*Sqrt[114] + 7*(37 - 3*Sqrt[114])^(1/3) - 6*(37 - 3 
*Sqrt[114])^(2/3)*(4/3 + x))/Sqrt[3*(2395 - 222*Sqrt[114] + 49*(37 - 3*Sqr 
t[114])^(2/3) - 14*(37 - 3*Sqrt[114])^(4/3))]])/((37 - 3*Sqrt[114])^(4/3)* 
(49 + 7*(37 - 3*Sqrt[114])^(2/3) + (37 - 3*Sqrt[114])^(4/3))^2) - (2*((...
 

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 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 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 = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.09

method	result	size
default	\(\frac {\left (\frac {7 A}{76}+\frac {3 B}{76}-\frac {15 C}{76}\right ) x^{2}+\left (\frac {31 A}{76}-\frac {3 B}{76}-\frac {23 C}{76}\right ) x +\frac {9 A}{38}-\frac {7 B}{38}-\frac {3 C}{38}}{x^{3}+4 x^{2}+3 x +2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+3 \textit {\_Z} +2\right )}{\sum }\frac {\left (7 A \textit {\_R} +3 B \textit {\_R} -15 C \textit {\_R} +34 A -18 B +14 C \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +3}\right )}{76}\)	\(114\)
risch	\(\frac {\left (\frac {7 A}{76}+\frac {3 B}{76}-\frac {15 C}{76}\right ) x^{2}+\left (\frac {31 A}{76}-\frac {3 B}{76}-\frac {23 C}{76}\right ) x +\frac {9 A}{38}-\frac {7 B}{38}-\frac {3 C}{38}}{x^{3}+4 x^{2}+3 x +2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+4 \textit {\_Z}^{2}+3 \textit {\_Z} +2\right )}{\sum }\frac {\left (\left (7 A +3 B -15 C \right ) \textit {\_R} +34 A -18 B +14 C \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+8 \textit {\_R} +3}\right )}{76}\)	\(114\)

Input:

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

Output:

((7/76*A+3/76*B-15/76*C)*x^2+(31/76*A-3/76*B-23/76*C)*x+9/38*A-7/38*B-3/38 
*C)/(x^3+4*x^2+3*x+2)+1/76*sum((7*A*_R+3*B*_R-15*C*_R+34*A-18*B+14*C)/(3*_ 
R^2+8*_R+3)*ln(x-_R),_R=RootOf(_Z^3+4*_Z^2+3*_Z+2))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.54 (sec) , antiderivative size = 5342, normalized size of antiderivative = 4.32 \[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [A] (verification not implemented)

Time = 8.46 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.28 \[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx=\operatorname {RootSum} {\left (877952 t^{3} + t \left (8785 A^{2} - 9342 A B + 7342 A C + 1809 B^{2} - 306 B C - 3263 C^{2}\right ) - 148 A^{3} + 543 A^{2} B - 1004 A^{2} C - 486 A B^{2} + 1506 A B C - 1004 A C^{2} + 135 B^{3} - 594 B^{2} C + 831 B C^{2} - 404 C^{3}, \left ( t \mapsto t \log {\left (x + \frac {227389568 t^{2} A - 102720384 t^{2} B + 46531456 t^{2} C + 9882736 t A^{2} - 19372704 t A B + 31895072 t A C + 8265456 t B^{2} - 24709728 t B C + 16998768 t C^{2} + 2715246 A^{3} - 4609158 A^{2} B + 4145350 A^{2} C + 2758122 A B^{2} - 5491668 A B C + 3176874 A C^{2} - 591138 B^{3} + 1970982 B^{2} C - 2510118 B C^{2} + 1172258 C^{3}}{898777 A^{3} - 1733157 A^{2} B + 1925421 A^{2} C + 1287819 A B^{2} - 3402054 A B C + 2610651 A C^{2} - 337527 B^{3} + 1412397 B^{2} C - 2065365 B C^{2} + 965663 C^{3}} \right )} \right )\right )} + \frac {18 A - 14 B - 6 C + x^{2} \cdot \left (7 A + 3 B - 15 C\right ) + x \left (31 A - 3 B - 23 C\right )}{76 x^{3} + 304 x^{2} + 228 x + 152} \] Input:

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

Output:

RootSum(877952*_t**3 + _t*(8785*A**2 - 9342*A*B + 7342*A*C + 1809*B**2 - 3 
06*B*C - 3263*C**2) - 148*A**3 + 543*A**2*B - 1004*A**2*C - 486*A*B**2 + 1 
506*A*B*C - 1004*A*C**2 + 135*B**3 - 594*B**2*C + 831*B*C**2 - 404*C**3, L 
ambda(_t, _t*log(x + (227389568*_t**2*A - 102720384*_t**2*B + 46531456*_t* 
*2*C + 9882736*_t*A**2 - 19372704*_t*A*B + 31895072*_t*A*C + 8265456*_t*B* 
*2 - 24709728*_t*B*C + 16998768*_t*C**2 + 2715246*A**3 - 4609158*A**2*B + 
4145350*A**2*C + 2758122*A*B**2 - 5491668*A*B*C + 3176874*A*C**2 - 591138* 
B**3 + 1970982*B**2*C - 2510118*B*C**2 + 1172258*C**3)/(898777*A**3 - 1733 
157*A**2*B + 1925421*A**2*C + 1287819*A*B**2 - 3402054*A*B*C + 2610651*A*C 
**2 - 337527*B**3 + 1412397*B**2*C - 2065365*B*C**2 + 965663*C**3)))) + (1 
8*A - 14*B - 6*C + x**2*(7*A + 3*B - 15*C) + x*(31*A - 3*B - 23*C))/(76*x* 
*3 + 304*x**2 + 228*x + 152)
 

Maxima [F]

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

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

Output:

1/76*((7*A + 3*B - 15*C)*x^2 + (31*A - 3*B - 23*C)*x + 18*A - 14*B - 6*C)/ 
(x^3 + 4*x^2 + 3*x + 2) - 1/76*integrate(-((7*A + 3*B - 15*C)*x + 34*A - 1 
8*B + 14*C)/(x^3 + 4*x^2 + 3*x + 2), x)
 

Giac [F(-2)]

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

integrate((C*x^2+B*x+A)/(x^3+4*x^2+3*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 = 22.12 (sec) , antiderivative size = 989, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((49*A^2*x)/5776 + (9*B^2*x)/5776 + (225*C^2*x)/5776 - 14*root(z 
^3 + z*((8785*A^2)/877952 + (1809*B^2)/877952 - (3263*C^2)/877952 - (4671* 
A*B)/438976 + (3671*A*C)/438976 - (153*B*C)/438976) + (753*A*B*C)/438976 + 
 (831*B*C^2)/877952 - (297*B^2*C)/438976 - (251*A^2*C)/219488 - (251*A*C^2 
)/219488 + (543*A^2*B)/877952 - (243*A*B^2)/438976 - (37*A^3)/219488 - (10 
1*C^3)/219488 + (135*B^3)/877952, z, k)^2*x + (119*A^2)/2888 - (27*B^2)/28 
88 - (105*C^2)/2888 + 6*root(z^3 + z*((8785*A^2)/877952 + (1809*B^2)/87795 
2 - (3263*C^2)/877952 - (4671*A*B)/438976 + (3671*A*C)/438976 - (153*B*C)/ 
438976) + (753*A*B*C)/438976 + (831*B*C^2)/877952 - (297*B^2*C)/438976 - ( 
251*A^2*C)/219488 - (251*A*C^2)/219488 + (543*A^2*B)/877952 - (243*A*B^2)/ 
438976 - (37*A^3)/219488 - (101*C^3)/219488 + (135*B^3)/877952, z, k)^2 - 
(3*A*B)/722 - (103*A*C)/1444 + (39*B*C)/722 + (115*A*root(z^3 + z*((8785*A 
^2)/877952 + (1809*B^2)/877952 - (3263*C^2)/877952 - (4671*A*B)/438976 + ( 
3671*A*C)/438976 - (153*B*C)/438976) + (753*A*B*C)/438976 + (831*B*C^2)/87 
7952 - (297*B^2*C)/438976 - (251*A^2*C)/219488 - (251*A*C^2)/219488 + (543 
*A^2*B)/877952 - (243*A*B^2)/438976 - (37*A^3)/219488 - (101*C^3)/219488 + 
 (135*B^3)/877952, z, k))/76 - (81*B*root(z^3 + z*((8785*A^2)/877952 + (18 
09*B^2)/877952 - (3263*C^2)/877952 - (4671*A*B)/438976 + (3671*A*C)/438976 
 - (153*B*C)/438976) + (753*A*B*C)/438976 + (831*B*C^2)/877952 - (297*B^2* 
C)/438976 - (251*A^2*C)/219488 - (251*A*C^2)/219488 + (543*A^2*B)/87795...
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (2+3 x+4 x^2+x^3\right )^2} \, dx=\frac {6 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b \,x^{3}+24 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b \,x^{2}+18 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b x +12 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b -16 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c \,x^{3}-64 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c \,x^{2}-48 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c x -32 \left (\int \frac {x^{3}}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c +32 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) a \,x^{3}+128 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) a \,x^{2}+96 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) a x +64 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) a -18 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b \,x^{3}-72 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b \,x^{2}-54 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b x -36 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) b +16 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c \,x^{3}+64 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c \,x^{2}+48 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c x +32 \left (\int \frac {1}{x^{6}+8 x^{5}+22 x^{4}+28 x^{3}+25 x^{2}+12 x +4}d x \right ) c +3 b x -4 b -8 c x}{32 x^{3}+128 x^{2}+96 x +64} \] Input:

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

Output:

(6*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*b* 
x**3 + 24*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4 
),x)*b*x**2 + 18*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 1 
2*x + 4),x)*b*x + 12*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 
 + 12*x + 4),x)*b - 16*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x* 
*2 + 12*x + 4),x)*c*x**3 - 64*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 28*x**3 
+ 25*x**2 + 12*x + 4),x)*c*x**2 - 48*int(x**3/(x**6 + 8*x**5 + 22*x**4 + 2 
8*x**3 + 25*x**2 + 12*x + 4),x)*c*x - 32*int(x**3/(x**6 + 8*x**5 + 22*x**4 
 + 28*x**3 + 25*x**2 + 12*x + 4),x)*c + 32*int(1/(x**6 + 8*x**5 + 22*x**4 
+ 28*x**3 + 25*x**2 + 12*x + 4),x)*a*x**3 + 128*int(1/(x**6 + 8*x**5 + 22* 
x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*a*x**2 + 96*int(1/(x**6 + 8*x**5 + 
 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*a*x + 64*int(1/(x**6 + 8*x**5 
+ 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*a - 18*int(1/(x**6 + 8*x**5 + 
 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*b*x**3 - 72*int(1/(x**6 + 8*x* 
*5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*b*x**2 - 54*int(1/(x**6 + 
8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*b*x - 36*int(1/(x**6 + 
 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*b + 16*int(1/(x**6 + 
8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*c*x**3 + 64*int(1/(x** 
6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*c*x**2 + 48*int(1/ 
(x**6 + 8*x**5 + 22*x**4 + 28*x**3 + 25*x**2 + 12*x + 4),x)*c*x + 32*in...