\(\int \frac {A+B x+C x^2+D x^3}{(88-402 x+855 x^2-837 x^3+324 x^4)^2} \, dx\) [16]

Optimal result

Integrand size = 38, antiderivative size = 544 \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=-\frac {11961 A+2673 \sqrt {177} A-39735 B+2169 \sqrt {177} B-58236 C-300 \sqrt {177} C-56694 D-2614 \sqrt {177} D-8 (2673 A+2169 B-300 C-2614 D) (2-3 x)}{12415842 \left (16-\left (1+\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}+\frac {8 \left (4779 A+1053 \sqrt {177} A-9558 B+774 \sqrt {177} B-14868 C-204 \sqrt {177} C-18408 D-1112 \sqrt {177} D-2 \left (6372 C-\sqrt {177} \left (54 \left (1-\sqrt {177}\right ) A-36 \left (15+\sqrt {177}\right ) B-732 C-689 D-39 \sqrt {177} D\right )\right ) (2-3 x)\right )}{14337 \left (167+\sqrt {177}\right ) \left (16-\left (1-\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right ) \left (16-\left (1+\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}-\frac {\left (187371 A+59787 \sqrt {177} A-384747 B+39093 \sqrt {177} B-591872 C+29312 \sqrt {177} C-580980 D+23756 \sqrt {177} D\right ) \arctan \left (\frac {31-\sqrt {177}-48 x}{\sqrt {2 \left (167-\sqrt {177}\right )}}\right )}{8277228 \sqrt {354 \left (167-\sqrt {177}\right )}}+\frac {\left (187371 A-59787 \sqrt {177} A-384747 B-39093 \sqrt {177} B-591872 C-29312 \sqrt {177} C-580980 D-23756 \sqrt {177} D\right ) \arctan \left (\frac {31+\sqrt {177}-48 x}{\sqrt {2 \left (167+\sqrt {177}\right )}}\right )}{8277228 \sqrt {354 \left (167+\sqrt {177}\right )}}-\frac {(513 A+351 B+192 C+68 D) \log \left (16-\left (1-\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}{114696 \sqrt {177}}+\frac {(513 A+351 B+192 C+68 D) \log \left (16-\left (1+\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}{114696 \sqrt {177}} \] Output:

-1/12415842*(11961*A+2673*177^(1/2)*A-39735*B+2169*177^(1/2)*B-58236*C-300 
*177^(1/2)*C-56694*D-2614*177^(1/2)*D-8*(2673*A+2169*B-300*C-2614*D)*(2-3* 
x))/(16-(1+177^(1/2))*(2-3*x)+8*(2-3*x)^2)+8/14337*(4779*A+1053*177^(1/2)* 
A-9558*B+774*177^(1/2)*B-14868*C-204*177^(1/2)*C-18408*D-1112*177^(1/2)*D- 
2*(6372*C-177^(1/2)*(54*(1-177^(1/2))*A-36*(15+177^(1/2))*B-732*C-689*D-39 
*177^(1/2)*D))*(2-3*x))/(167+177^(1/2))/(16-(1-177^(1/2))*(2-3*x)+8*(2-3*x 
)^2)/(16-(1+177^(1/2))*(2-3*x)+8*(2-3*x)^2)-1/8277228*(187371*A+59787*177^ 
(1/2)*A-384747*B+39093*177^(1/2)*B-591872*C+29312*177^(1/2)*C-580980*D+237 
56*177^(1/2)*D)*arctan((31-177^(1/2)-48*x)/(334-2*177^(1/2))^(1/2))/(59118 
-354*177^(1/2))^(1/2)+1/8277228*(187371*A-59787*177^(1/2)*A-384747*B-39093 
*177^(1/2)*B-591872*C-29312*177^(1/2)*C-580980*D-23756*177^(1/2)*D)*arctan 
((31+177^(1/2)-48*x)/(334+2*177^(1/2))^(1/2))/(59118+354*177^(1/2))^(1/2)- 
1/20301192*(513*A+351*B+192*C+68*D)*ln(16-(1-177^(1/2))*(2-3*x)+8*(2-3*x)^ 
2)*177^(1/2)+1/20301192*(513*A+351*B+192*C+68*D)*ln(16-(1+177^(1/2))*(2-3* 
x)+8*(2-3*x)^2)*177^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.18 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.54 \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\frac {\frac {6 C \left (10604-35373 x+33102 x^2+5400 x^3\right )-27 B \left (-2904+15539 x-23613 x^2+8676 x^3\right )-27 A \left (2273+4602 x-18945 x^2+10692 x^3\right )+2 D \left (-4400+51912 x-148869 x^2+141156 x^3\right )}{88-402 x+855 x^2-837 x^3+324 x^4}-\text {RootSum}\left [88-402 \text {$\#$1}+855 \text {$\#$1}^2-837 \text {$\#$1}^3+324 \text {$\#$1}^4\&,\frac {-181899 A \log (x-\text {$\#$1})-20457 B \log (x-\text {$\#$1})+26136 C \log (x-\text {$\#$1})+21208 D \log (x-\text {$\#$1})-92421 A \log (x-\text {$\#$1}) \text {$\#$1}-223317 B \log (x-\text {$\#$1}) \text {$\#$1}-160308 C \log (x-\text {$\#$1}) \text {$\#$1}-44610 D \log (x-\text {$\#$1}) \text {$\#$1}+96228 A \log (x-\text {$\#$1}) \text {$\#$1}^2+78084 B \log (x-\text {$\#$1}) \text {$\#$1}^2-10800 C \log (x-\text {$\#$1}) \text {$\#$1}^2-94104 D \log (x-\text {$\#$1}) \text {$\#$1}^2}{-134+570 \text {$\#$1}-837 \text {$\#$1}^2+432 \text {$\#$1}^3}\&\right ]}{12415842} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/(88 - 402*x + 855*x^2 - 837*x^3 + 324* 
x^4)^2,x]
 

Output:

((6*C*(10604 - 35373*x + 33102*x^2 + 5400*x^3) - 27*B*(-2904 + 15539*x - 2 
3613*x^2 + 8676*x^3) - 27*A*(2273 + 4602*x - 18945*x^2 + 10692*x^3) + 2*D* 
(-4400 + 51912*x - 148869*x^2 + 141156*x^3))/(88 - 402*x + 855*x^2 - 837*x 
^3 + 324*x^4) - RootSum[88 - 402*#1 + 855*#1^2 - 837*#1^3 + 324*#1^4 & , ( 
-181899*A*Log[x - #1] - 20457*B*Log[x - #1] + 26136*C*Log[x - #1] + 21208* 
D*Log[x - #1] - 92421*A*Log[x - #1]*#1 - 223317*B*Log[x - #1]*#1 - 160308* 
C*Log[x - #1]*#1 - 44610*D*Log[x - #1]*#1 + 96228*A*Log[x - #1]*#1^2 + 780 
84*B*Log[x - #1]*#1^2 - 10800*C*Log[x - #1]*#1^2 - 94104*D*Log[x - #1]*#1^ 
2)/(-134 + 570*#1 - 837*#1^2 + 432*#1^3) & ])/12415842
 

Rubi [A] (verified)

Time = 9.01 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.47, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2494, 2492, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (324 x^4-837 x^3+855 x^2-402 x+88\right )^2} \, dx\)

\(\Big \downarrow \) 2494

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (324 \left (x-\frac {2}{3}\right )^4+27 \left (x-\frac {2}{3}\right )^3+45 \left (x-\frac {2}{3}\right )^2+6 \left (x-\frac {2}{3}\right )+16\right )^2}d\left (x-\frac {2}{3}\right )\)

\(\Big \downarrow \) 2492

\(\displaystyle \frac {\int \left (-\frac {108 \sqrt {\frac {3}{59}} \left (-513 \sqrt {177} A+729 A-351 \sqrt {177} B-657 B-192 \sqrt {177} C-1248 C-68 \sqrt {177} D-1148 D+24 (513 A+351 B+192 C+68 D) \left (x-\frac {2}{3}\right )\right )}{59 \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )}+\frac {108 \sqrt {\frac {3}{59}} \left (513 \sqrt {177} A+729 A+351 \sqrt {177} B-657 B+192 \sqrt {177} C-1248 C+68 \sqrt {177} D-1148 D+24 (513 A+351 B+192 C+68 D) \left (x-\frac {2}{3}\right )\right )}{59 \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )}+\frac {648 \left (27 \left (25-\sqrt {177}\right ) A+54 \left (7+\sqrt {177}\right ) B+84 \left (7+\sqrt {177}\right ) C+8 \left (109+11 \sqrt {177}\right ) D+12 \left (27 \left (1-\sqrt {177}\right ) A-18 \left (7+\sqrt {177}\right ) B-12 \left (15+\sqrt {177}\right ) C-8 \left (19+\sqrt {177}\right ) D\right ) \left (x-\frac {2}{3}\right )\right )}{59 \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )^2}+\frac {648 \left (27 \left (25+\sqrt {177}\right ) A+54 \left (7-\sqrt {177}\right ) B-84 \sqrt {177} C+588 C-88 \sqrt {177} D+872 D+12 \left (27 \left (1+\sqrt {177}\right ) A-18 \left (7-\sqrt {177}\right ) B-12 \left (15-\sqrt {177}\right ) C-8 \left (19-\sqrt {177}\right ) D\right ) \left (x-\frac {2}{3}\right )\right )}{59 \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )^2}\right )d\left (x-\frac {2}{3}\right )}{104976}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {648 \left (459 \sqrt {177} A+333 A+330 \sqrt {177} B+1158 B+172 \sqrt {177} C+1460 C+40 \sqrt {177} D+792 D-4 \left (27 \sqrt {177} A+1053 A-54 \sqrt {177} B+774 B-84 \sqrt {177} C-204 C-104 \sqrt {177} D-1112 D\right ) \left (x-\frac {2}{3}\right )\right )}{59 \left (167+\sqrt {177}\right ) \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )}-\frac {54}{59} \sqrt {\frac {2}{59 \left (167+\sqrt {177}\right )}} \left (-513 \sqrt {59} A+315 \sqrt {3} A-351 \sqrt {59} B-555 \sqrt {3} B-192 \sqrt {59} C-896 \sqrt {3} C-68 \sqrt {59} D-788 \sqrt {3} D\right ) \arctan \left (\frac {48 \left (x-\frac {2}{3}\right )-\sqrt {177}+1}{\sqrt {2 \left (167+\sqrt {177}\right )}}\right )-\frac {864 \sqrt {2} \left (27 \sqrt {177} A+1053 A-54 \sqrt {177} B+774 B-84 \sqrt {177} C-204 C-104 \sqrt {177} D-1112 D\right ) \arctan \left (\frac {48 \left (x-\frac {2}{3}\right )-\sqrt {177}+1}{\sqrt {2 \left (167+\sqrt {177}\right )}}\right )}{59 \left (167+\sqrt {177}\right )^{3/2}}-\frac {864 \sqrt {2} \left (-27 \sqrt {177} A+1053 A+54 \sqrt {177} B+774 B+84 \sqrt {177} C-204 C+104 \sqrt {177} D-1112 D\right ) \arctan \left (\frac {48 \left (x-\frac {2}{3}\right )+\sqrt {177}+1}{\sqrt {2 \left (167-\sqrt {177}\right )}}\right )}{59 \left (167-\sqrt {177}\right )^{3/2}}+\frac {18}{59} \sqrt {\frac {6}{59 \left (167-\sqrt {177}\right )}} \left (27 \left (35+19 \sqrt {177}\right ) A-9 \left (185-39 \sqrt {177}\right ) B-192 \left (14-\sqrt {177}\right ) C-4 \left (591-17 \sqrt {177}\right ) D\right ) \arctan \left (\frac {48 \left (x-\frac {2}{3}\right )+\sqrt {177}+1}{\sqrt {2 \left (167-\sqrt {177}\right )}}\right )-\frac {18}{59} \sqrt {\frac {3}{59}} (513 A+351 B+192 C+68 D) \log \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )+\frac {18}{59} \sqrt {\frac {3}{59}} (513 A+351 B+192 C+68 D) \log \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )+\frac {648 \left (-459 \sqrt {177} A+333 A-330 \sqrt {177} B+1158 B-172 \sqrt {177} C+1460 C-40 \sqrt {177} D+792 D-4 \left (-27 \sqrt {177} A+1053 A+54 \sqrt {177} B+774 B+84 \sqrt {177} C-204 C+104 \sqrt {177} D-1112 D\right ) \left (x-\frac {2}{3}\right )\right )}{59 \left (167-\sqrt {177}\right ) \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right )}}{104976}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/(88 - 402*x + 855*x^2 - 837*x^3 + 324*x^4)^2 
,x]
 

Output:

((648*(333*A + 459*Sqrt[177]*A + 1158*B + 330*Sqrt[177]*B + 1460*C + 172*S 
qrt[177]*C + 792*D + 40*Sqrt[177]*D - 4*(1053*A + 27*Sqrt[177]*A + 774*B - 
 54*Sqrt[177]*B - 204*C - 84*Sqrt[177]*C - 1112*D - 104*Sqrt[177]*D)*(-2/3 
 + x)))/(59*(167 + Sqrt[177])*(16 + 3*(1 - Sqrt[177])*(-2/3 + x) + 72*(-2/ 
3 + x)^2)) + (648*(333*A - 459*Sqrt[177]*A + 1158*B - 330*Sqrt[177]*B + 14 
60*C - 172*Sqrt[177]*C + 792*D - 40*Sqrt[177]*D - 4*(1053*A - 27*Sqrt[177] 
*A + 774*B + 54*Sqrt[177]*B - 204*C + 84*Sqrt[177]*C - 1112*D + 104*Sqrt[1 
77]*D)*(-2/3 + x)))/(59*(167 - Sqrt[177])*(16 + 3*(1 + Sqrt[177])*(-2/3 + 
x) + 72*(-2/3 + x)^2)) - (54*Sqrt[2/(59*(167 + Sqrt[177]))]*(315*Sqrt[3]*A 
 - 513*Sqrt[59]*A - 555*Sqrt[3]*B - 351*Sqrt[59]*B - 896*Sqrt[3]*C - 192*S 
qrt[59]*C - 788*Sqrt[3]*D - 68*Sqrt[59]*D)*ArcTan[(1 - Sqrt[177] + 48*(-2/ 
3 + x))/Sqrt[2*(167 + Sqrt[177])]])/59 - (864*Sqrt[2]*(1053*A + 27*Sqrt[17 
7]*A + 774*B - 54*Sqrt[177]*B - 204*C - 84*Sqrt[177]*C - 1112*D - 104*Sqrt 
[177]*D)*ArcTan[(1 - Sqrt[177] + 48*(-2/3 + x))/Sqrt[2*(167 + Sqrt[177])]] 
)/(59*(167 + Sqrt[177])^(3/2)) - (864*Sqrt[2]*(1053*A - 27*Sqrt[177]*A + 7 
74*B + 54*Sqrt[177]*B - 204*C + 84*Sqrt[177]*C - 1112*D + 104*Sqrt[177]*D) 
*ArcTan[(1 + Sqrt[177] + 48*(-2/3 + x))/Sqrt[2*(167 - Sqrt[177])]])/(59*(1 
67 - Sqrt[177])^(3/2)) + (18*Sqrt[6/(59*(167 - Sqrt[177]))]*(27*(35 + 19*S 
qrt[177])*A - 9*(185 - 39*Sqrt[177])*B - 192*(14 - Sqrt[177])*C - 4*(591 - 
 17*Sqrt[177])*D)*ArcTan[(1 + Sqrt[177] + 48*(-2/3 + x))/Sqrt[2*(167 - ...
 

Defintions of rubi rules used

rule 2009

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

rule 2492

Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) 
^(p_), x_Symbol] :> Simp[e^p   Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( 
(b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 
2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
 

rule 2494

Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) 
^(p_), x_Symbol] :> With[{S = Root[a*d^2 - b^2*e + (b*d^2 - 4*b*c*e + 8*a*d 
*e)*x + (c*d^2 - 4*c^2*e + 2*b*d*e + 16*a*e^2)*x^2 + (d^3 - 4*c*d*e + 8*b*e 
^2)*x^3, 3]}, Subst[Int[(Px /. x -> x + S)*ExpandToSum[a + b*(x + S) + c*(x 
 + S)^2 + d*(x + S)^3 + e*(x + S)^4, x]^p, x], x, x - S] /; RationalQ[S]] / 
; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && RationalQ[a, b 
, c, d, e] && NeQ[a*d^2 - b^2*e, 0]
 

Maple [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.33

method	result	size
default	\(\frac {\left (\frac {1307 D}{18623763}-\frac {11 A}{153282}-\frac {241 B}{4138614}+\frac {50 C}{6207921}\right ) x^{3}+\left (-\frac {16541 D}{223485156}+\frac {2105 A}{16554456}+\frac {7871 B}{49663368}+\frac {613 C}{12415842}\right ) x^{2}+\left (\frac {1442 D}{55871289}-\frac {13 A}{420876}-\frac {15539 B}{148990104}-\frac {11791 C}{223485156}\right ) x -\frac {1100 D}{502841601}-\frac {2273 A}{148990104}+\frac {121 B}{6207921}+\frac {2651 C}{167613867}}{x^{4}-\frac {31}{12} x^{3}+\frac {95}{36} x^{2}-\frac {67}{54} x +\frac {22}{81}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (324 \textit {\_Z}^{4}-837 \textit {\_Z}^{3}+855 \textit {\_Z}^{2}-402 \textit {\_Z} +88\right )}{\sum }\frac {\left (36 \left (-2673 A -2169 B +300 C +2614 D\right ) \textit {\_R}^{2}+3 \left (30807 A +74439 B +53436 C +14870 D\right ) \textit {\_R} +181899 A +20457 B -26136 C -21208 D\right ) \ln \left (x -\textit {\_R} \right )}{432 \textit {\_R}^{3}-837 \textit {\_R}^{2}+570 \textit {\_R} -134}\right )}{12415842}\)	\(182\)

Input:

int((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88)^2,x,method=_RET 
URNVERBOSE)
 

Output:

((1307/18623763*D-11/153282*A-241/4138614*B+50/6207921*C)*x^3+(-16541/2234 
85156*D+2105/16554456*A+7871/49663368*B+613/12415842*C)*x^2+(1442/55871289 
*D-13/420876*A-15539/148990104*B-11791/223485156*C)*x-1100/502841601*D-227 
3/148990104*A+121/6207921*B+2651/167613867*C)/(x^4-31/12*x^3+95/36*x^2-67/ 
54*x+22/81)+1/12415842*sum((36*(-2673*A-2169*B+300*C+2614*D)*_R^2+3*(30807 
*A+74439*B+53436*C+14870*D)*_R+181899*A+20457*B-26136*C-21208*D)/(432*_R^3 
-837*_R^2+570*_R-134)*ln(x-_R),_R=RootOf(324*_Z^4-837*_Z^3+855*_Z^2-402*_Z 
+88))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.89 (sec) , antiderivative size = 761015, normalized size of antiderivative = 1398.92 \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88)^2,x, algo 
rithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(324*x**4-837*x**3+855*x**2-402*x+88)**2,x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (324 \, x^{4} - 837 \, x^{3} + 855 \, x^{2} - 402 \, x + 88\right )}^{2}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88)^2,x, algo 
rithm="maxima")
 

Output:

-1/12415842*(108*(2673*A + 2169*B - 300*C - 2614*D)*x^3 - 9*(56835*A + 708 
39*B + 22068*C - 33082*D)*x^2 + 9*(13806*A + 46617*B + 23582*C - 11536*D)* 
x + 61371*A - 78408*B - 63624*C + 8800*D)/(324*x^4 - 837*x^3 + 855*x^2 - 4 
02*x + 88) + 1/4138614*integrate(-(36*(2673*A + 2169*B - 300*C - 2614*D)*x 
^2 - 3*(30807*A + 74439*B + 53436*C + 14870*D)*x - 181899*A - 20457*B + 26 
136*C + 21208*D)/(324*x^4 - 837*x^3 + 855*x^2 - 402*x + 88), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88)^2,x, algo 
rithm="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 [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (324\,x^4-837\,x^3+855\,x^2-402\,x+88\right )}^2} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/(855*x^2 - 402*x - 837*x^3 + 324*x^4 + 88)^2 
,x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/(855*x^2 - 402*x - 837*x^3 + 324*x^4 + 88)^2 
, x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (88-402 x+855 x^2-837 x^3+324 x^4\right )^2} \, dx=\text {too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88)^2,x)
                                                                                    
                                                                                    
 

Output:

(419904*int(x**3/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 
+ 1460997*x**4 - 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*a*x**4 - 1 
084752*int(x**3/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 
 1460997*x**4 - 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*a*x**3 + 11 
08080*int(x**3/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 
1460997*x**4 - 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*a*x**2 - 520 
992*int(x**3/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 14 
60997*x**4 - 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*a*x + 114048*i 
nt(x**3/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997 
*x**4 - 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*a + 130248*int(x**3 
/(104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997*x**4 - 
 834732*x**3 + 312084*x**2 - 70752*x + 7744),x)*d*x**4 - 336474*int(x**3/( 
104976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997*x**4 - 8 
34732*x**3 + 312084*x**2 - 70752*x + 7744),x)*d*x**3 + 343710*int(x**3/(10 
4976*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997*x**4 - 834 
732*x**3 + 312084*x**2 - 70752*x + 7744),x)*d*x**2 - 161604*int(x**3/(1049 
76*x**8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997*x**4 - 83473 
2*x**3 + 312084*x**2 - 70752*x + 7744),x)*d*x + 35376*int(x**3/(104976*x** 
8 - 542376*x**7 + 1254609*x**6 - 1691766*x**5 + 1460997*x**4 - 834732*x**3 
 + 312084*x**2 - 70752*x + 7744),x)*d - 813564*int(x**2/(104976*x**8 - ...