\(\int (A+B x+C x^2) (4 c^3-27 c d^2 x^2-27 d^3 x^3)^4 \, dx\) [29]

Optimal result

Integrand size = 36, antiderivative size = 230 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=\frac {c^4 \left (4 c^2 C-6 B c d+9 A d^2\right ) (2 c+3 d x)^9}{3 d^3}-\frac {c^3 \left (28 c^2 C-33 B c d+36 A d^2\right ) (2 c+3 d x)^{10}}{10 d^3}+\frac {3 c^2 \left (9 c^2 C-8 B c d+6 A d^2\right ) (2 c+3 d x)^{11}}{11 d^3}-\frac {c \left (62 c^2 C-39 B c d+18 A d^2\right ) (2 c+3 d x)^{12}}{54 d^3}+\frac {\left (106 c^2 C-42 B c d+9 A d^2\right ) (2 c+3 d x)^{13}}{351 d^3}-\frac {(16 c C-3 B d) (2 c+3 d x)^{14}}{378 d^3}+\frac {C (2 c+3 d x)^{15}}{405 d^3} \] Output:

1/3*c^4*(9*A*d^2-6*B*c*d+4*C*c^2)*(3*d*x+2*c)^9/d^3-1/10*c^3*(36*A*d^2-33* 
B*c*d+28*C*c^2)*(3*d*x+2*c)^10/d^3+3/11*c^2*(6*A*d^2-8*B*c*d+9*C*c^2)*(3*d 
*x+2*c)^11/d^3-1/54*c*(18*A*d^2-39*B*c*d+62*C*c^2)*(3*d*x+2*c)^12/d^3+1/35 
1*(9*A*d^2-42*B*c*d+106*C*c^2)*(3*d*x+2*c)^13/d^3-1/378*(-3*B*d+16*C*c)*(3 
*d*x+2*c)^14/d^3+1/405*C*(3*d*x+2*c)^15/d^3
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.53 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=256 A c^{12} x+128 B c^{12} x^2+\frac {256}{3} c^{10} \left (c^2 C-27 A d^2\right ) x^3-1728 c^9 d^2 (B c+A d) x^4-\frac {864}{5} c^8 d^2 \left (8 c^2 C+8 B c d-81 A d^2\right ) x^5-144 c^7 d^3 \left (8 c^2 C-81 B c d-162 A d^2\right ) x^6+\frac {34992}{7} c^6 d^4 \left (2 c^2 C+4 B c d-7 A d^2\right ) x^7+4374 c^5 d^5 \left (4 c^2 C-7 B c d-27 A d^2\right ) x^8-243 c^4 d^6 \left (112 c^2 C+432 B c d+189 A d^2\right ) x^9-\frac {19683}{10} c^3 d^7 \left (48 c^2 C+21 B c d-92 A d^2\right ) x^{10}-\frac {19683}{11} c^2 d^8 \left (21 c^2 C-92 B c d-162 A d^2\right ) x^{11}+\frac {6561}{2} c d^9 \left (46 c^2 C+81 B c d+54 A d^2\right ) x^{12}+\frac {531441}{13} d^{10} \left (6 c^2 C+4 B c d+A d^2\right ) x^{13}+\frac {531441}{14} d^{11} (4 c C+B d) x^{14}+\frac {177147}{5} C d^{12} x^{15} \] Input:

Integrate[(A + B*x + C*x^2)*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^4,x]
 

Output:

256*A*c^12*x + 128*B*c^12*x^2 + (256*c^10*(c^2*C - 27*A*d^2)*x^3)/3 - 1728 
*c^9*d^2*(B*c + A*d)*x^4 - (864*c^8*d^2*(8*c^2*C + 8*B*c*d - 81*A*d^2)*x^5 
)/5 - 144*c^7*d^3*(8*c^2*C - 81*B*c*d - 162*A*d^2)*x^6 + (34992*c^6*d^4*(2 
*c^2*C + 4*B*c*d - 7*A*d^2)*x^7)/7 + 4374*c^5*d^5*(4*c^2*C - 7*B*c*d - 27* 
A*d^2)*x^8 - 243*c^4*d^6*(112*c^2*C + 432*B*c*d + 189*A*d^2)*x^9 - (19683* 
c^3*d^7*(48*c^2*C + 21*B*c*d - 92*A*d^2)*x^10)/10 - (19683*c^2*d^8*(21*c^2 
*C - 92*B*c*d - 162*A*d^2)*x^11)/11 + (6561*c*d^9*(46*c^2*C + 81*B*c*d + 5 
4*A*d^2)*x^12)/2 + (531441*d^10*(6*c^2*C + 4*B*c*d + A*d^2)*x^13)/13 + (53 
1441*d^11*(4*c*C + B*d)*x^14)/14 + (177147*C*d^12*x^15)/5
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2188, 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.

\(\displaystyle \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2188

\(\displaystyle \int \left (-6912 c^9 d^2 x^3 (A d+B c)+531441 d^{10} x^{12} \left (A d^2+4 B c d+6 c^2 C\right )+39366 c d^9 x^{11} \left (54 A d^2+81 B c d+46 c^2 C\right )-19683 c^2 d^8 x^{10} \left (-162 A d^2-92 B c d+21 c^2 C\right )-864 c^8 d^2 x^4 \left (-81 A d^2+8 B c d+8 c^2 C\right )-864 c^7 d^3 x^5 \left (-162 A d^2-81 B c d+8 c^2 C\right )+34992 c^6 d^4 x^6 \left (-7 A d^2+4 B c d+2 c^2 C\right )+34992 c^5 d^5 x^7 \left (-27 A d^2-7 B c d+4 c^2 C\right )-2187 c^4 d^6 x^8 \left (189 A d^2+432 B c d+112 c^2 C\right )-19683 c^3 d^7 x^9 \left (-92 A d^2+21 B c d+48 c^2 C\right )+256 A c^{12}+256 c^{10} x^2 \left (c^2 C-27 A d^2\right )+256 B c^{12} x+531441 d^{11} x^{13} (B d+4 c C)+531441 C d^{12} x^{14}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -1728 c^9 d^2 x^4 (A d+B c)+\frac {531441}{13} d^{10} x^{13} \left (A d^2+4 B c d+6 c^2 C\right )+\frac {6561}{2} c d^9 x^{12} \left (54 A d^2+81 B c d+46 c^2 C\right )-\frac {19683}{11} c^2 d^8 x^{11} \left (-162 A d^2-92 B c d+21 c^2 C\right )-\frac {864}{5} c^8 d^2 x^5 \left (-81 A d^2+8 B c d+8 c^2 C\right )-144 c^7 d^3 x^6 \left (-162 A d^2-81 B c d+8 c^2 C\right )+\frac {34992}{7} c^6 d^4 x^7 \left (-7 A d^2+4 B c d+2 c^2 C\right )+4374 c^5 d^5 x^8 \left (-27 A d^2-7 B c d+4 c^2 C\right )-243 c^4 d^6 x^9 \left (189 A d^2+432 B c d+112 c^2 C\right )-\frac {19683}{10} c^3 d^7 x^{10} \left (-92 A d^2+21 B c d+48 c^2 C\right )+256 A c^{12} x+\frac {256}{3} c^{10} x^3 \left (c^2 C-27 A d^2\right )+128 B c^{12} x^2+\frac {531441}{14} d^{11} x^{14} (B d+4 c C)+\frac {177147}{5} C d^{12} x^{15}\)

Input:

Int[(A + B*x + C*x^2)*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^4,x]
 

Output:

256*A*c^12*x + 128*B*c^12*x^2 + (256*c^10*(c^2*C - 27*A*d^2)*x^3)/3 - 1728 
*c^9*d^2*(B*c + A*d)*x^4 - (864*c^8*d^2*(8*c^2*C + 8*B*c*d - 81*A*d^2)*x^5 
)/5 - 144*c^7*d^3*(8*c^2*C - 81*B*c*d - 162*A*d^2)*x^6 + (34992*c^6*d^4*(2 
*c^2*C + 4*B*c*d - 7*A*d^2)*x^7)/7 + 4374*c^5*d^5*(4*c^2*C - 7*B*c*d - 27* 
A*d^2)*x^8 - 243*c^4*d^6*(112*c^2*C + 432*B*c*d + 189*A*d^2)*x^9 - (19683* 
c^3*d^7*(48*c^2*C + 21*B*c*d - 92*A*d^2)*x^10)/10 - (19683*c^2*d^8*(21*c^2 
*C - 92*B*c*d - 162*A*d^2)*x^11)/11 + (6561*c*d^9*(46*c^2*C + 81*B*c*d + 5 
4*A*d^2)*x^12)/2 + (531441*d^10*(6*c^2*C + 4*B*c*d + A*d^2)*x^13)/13 + (53 
1441*d^11*(4*c*C + B*d)*x^14)/14 + (177147*C*d^12*x^15)/5
 

Defintions of rubi rules used

rule 2009

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

rule 2188

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq 
, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.60

method	result	size
norman	\(256 c^{12} A x +128 B \,c^{12} x^{2}+\left (-2304 c^{10} A \,d^{2}+\frac {256}{3} C \,c^{12}\right ) x^{3}+\left (-1728 c^{9} A \,d^{3}-1728 B \,c^{10} d^{2}\right ) x^{4}+\left (\frac {69984}{5} c^{8} A \,d^{4}-\frac {6912}{5} B \,c^{9} d^{3}-\frac {6912}{5} C \,c^{10} d^{2}\right ) x^{5}+\left (23328 c^{7} A \,d^{5}+11664 B \,c^{8} d^{4}-1152 C \,c^{9} d^{3}\right ) x^{6}+\left (-34992 c^{6} A \,d^{6}+\frac {139968}{7} B \,c^{7} d^{5}+\frac {69984}{7} C \,d^{4} c^{8}\right ) x^{7}+\left (-118098 c^{5} A \,d^{7}-30618 B \,c^{6} d^{6}+17496 C \,d^{5} c^{7}\right ) x^{8}+\left (-45927 c^{4} A \,d^{8}-104976 B \,c^{5} d^{7}-27216 C \,d^{6} c^{6}\right ) x^{9}+\left (\frac {905418}{5} A \,c^{3} d^{9}-\frac {413343}{10} B \,c^{4} d^{8}-\frac {472392}{5} C \,d^{7} c^{5}\right ) x^{10}+\left (\frac {3188646}{11} A \,c^{2} d^{10}+\frac {1810836}{11} B \,c^{3} d^{9}-\frac {413343}{11} C \,d^{8} c^{4}\right ) x^{11}+\left (177147 A c \,d^{11}+\frac {531441}{2} B \,c^{2} d^{10}+150903 C \,d^{9} c^{3}\right ) x^{12}+\left (\frac {531441}{13} A \,d^{12}+\frac {2125764}{13} B c \,d^{11}+\frac {3188646}{13} C \,d^{10} c^{2}\right ) x^{13}+\left (\frac {531441}{14} B \,d^{12}+\frac {1062882}{7} C \,d^{11} c \right ) x^{14}+\frac {177147 C \,d^{12} x^{15}}{5}\)	\(369\)
default	\(\frac {177147 C \,d^{12} x^{15}}{5}+\frac {\left (531441 B \,d^{12}+2125764 C \,d^{11} c \right ) x^{14}}{14}+\frac {\left (531441 A \,d^{12}+2125764 B c \,d^{11}+3188646 C \,d^{10} c^{2}\right ) x^{13}}{13}+\frac {\left (2125764 A c \,d^{11}+3188646 B \,c^{2} d^{10}+1810836 C \,d^{9} c^{3}\right ) x^{12}}{12}+\frac {\left (3188646 A \,c^{2} d^{10}+1810836 B \,c^{3} d^{9}-413343 C \,d^{8} c^{4}\right ) x^{11}}{11}+\frac {\left (1810836 A \,c^{3} d^{9}-413343 B \,c^{4} d^{8}-944784 C \,d^{7} c^{5}\right ) x^{10}}{10}+\frac {\left (-413343 c^{4} A \,d^{8}-944784 B \,c^{5} d^{7}-244944 C \,d^{6} c^{6}\right ) x^{9}}{9}+\frac {\left (-944784 c^{5} A \,d^{7}-244944 B \,c^{6} d^{6}+139968 C \,d^{5} c^{7}\right ) x^{8}}{8}+\frac {\left (-244944 c^{6} A \,d^{6}+139968 B \,c^{7} d^{5}+69984 C \,d^{4} c^{8}\right ) x^{7}}{7}+\frac {\left (139968 c^{7} A \,d^{5}+69984 B \,c^{8} d^{4}-6912 C \,c^{9} d^{3}\right ) x^{6}}{6}+\frac {\left (69984 c^{8} A \,d^{4}-6912 B \,c^{9} d^{3}-6912 C \,c^{10} d^{2}\right ) x^{5}}{5}+\frac {\left (-6912 c^{9} A \,d^{3}-6912 B \,c^{10} d^{2}\right ) x^{4}}{4}+\frac {\left (-6912 c^{10} A \,d^{2}+256 C \,c^{12}\right ) x^{3}}{3}+128 B \,c^{12} x^{2}+256 c^{12} A x\)	\(381\)
gosper	\(\frac {x \left (1063944882 C \,d^{12} x^{14}[phone] x^{13} B \,d^{12}[phone] x^{13} C \,d^{11} c [phone] x^{12} A \,d^{12}[phone] x^{12} B c \,d^{11}[phone] x^{12} C \,d^{10} c^{2}[phone] x^{11} A c \,d^{11}[phone] x^{11} B \,c^{2} d^{10}[phone] x^{11} C \,d^{9} c^{3}[phone] x^{10} A \,c^{2} d^{10}[phone] x^{10} B \,c^{3} d^{9}-1128426390 x^{10} C \,d^{8} c^{4}[phone] x^{9} A \,c^{3} d^{9}-1241269029 x^{9} B \,c^{4} d^{8}-2837186352 x^{9} C \,d^{7} c^{5}-1379187810 A \,c^{4} d^{8} x^{8}-3152429280 B \,c^{5} d^{7} x^{8}-817296480 C \,c^{6} d^{6} x^{8}-3546482940 A \,c^{5} d^{7} x^{7}-919458540 B \,c^{6} d^{6} x^{7}[phone] C \,c^{7} d^{5} x^{7}-1050809760 x^{6} c^{6} A \,d^{6}[phone] x^{6} B \,c^{7} d^{5}[phone] x^{6} C \,d^{4} c^{8}[phone] A \,c^{7} d^{5} x^{5}[phone] B \,c^{8} d^{4} x^{5}-34594560 C \,c^{9} d^{3} x^{5}[phone] x^{4} c^{8} A \,d^{4}-41513472 x^{4} B \,c^{9} d^{3}-41513472 x^{4} C \,c^{10} d^{2}-51891840 A \,c^{9} d^{3} x^{3}-51891840 B \,c^{10} d^{2} x^{3}-69189120 x^{2} c^{10} A \,d^{2}+2562560 x^{2} C \,c^{12}+3843840 B \,c^{12} x +7687680 c^{12} A \right )}{30030}\)	\(408\)
risch	\(\frac {1062882}{7} x^{14} C \,d^{11} c +\frac {2125764}{13} x^{13} B c \,d^{11}+\frac {3188646}{13} x^{13} C \,d^{10} c^{2}+177147 x^{12} A c \,d^{11}+\frac {531441}{2} x^{12} B \,c^{2} d^{10}+150903 x^{12} C \,d^{9} c^{3}+\frac {3188646}{11} x^{11} A \,c^{2} d^{10}+\frac {1810836}{11} x^{11} B \,c^{3} d^{9}-\frac {413343}{11} x^{11} C \,d^{8} c^{4}+\frac {905418}{5} x^{10} A \,c^{3} d^{9}-\frac {413343}{10} x^{10} B \,c^{4} d^{8}-\frac {472392}{5} x^{10} C \,d^{7} c^{5}-34992 x^{7} c^{6} A \,d^{6}+\frac {139968}{7} x^{7} B \,c^{7} d^{5}+\frac {69984}{7} x^{7} C \,d^{4} c^{8}+\frac {69984}{5} x^{5} c^{8} A \,d^{4}-\frac {6912}{5} x^{5} B \,c^{9} d^{3}-\frac {6912}{5} x^{5} C \,c^{10} d^{2}-2304 x^{3} c^{10} A \,d^{2}-45927 A \,c^{4} d^{8} x^{9}-104976 B \,c^{5} d^{7} x^{9}-27216 C \,c^{6} d^{6} x^{9}-118098 A \,c^{5} d^{7} x^{8}-30618 B \,c^{6} d^{6} x^{8}+17496 C \,c^{7} d^{5} x^{8}+23328 A \,c^{7} d^{5} x^{6}+11664 B \,c^{8} d^{4} x^{6}-1152 C \,c^{9} d^{3} x^{6}-1728 A \,c^{9} d^{3} x^{4}-1728 B \,c^{10} d^{2} x^{4}+\frac {256}{3} x^{3} C \,c^{12}+256 c^{12} A x +128 B \,c^{12} x^{2}+\frac {177147}{5} C \,d^{12} x^{15}+\frac {531441}{13} x^{13} A \,d^{12}+\frac {531441}{14} x^{14} B \,d^{12}\)	\(408\)
parallelrisch	\(\frac {1062882}{7} x^{14} C \,d^{11} c +\frac {2125764}{13} x^{13} B c \,d^{11}+\frac {3188646}{13} x^{13} C \,d^{10} c^{2}+177147 x^{12} A c \,d^{11}+\frac {531441}{2} x^{12} B \,c^{2} d^{10}+150903 x^{12} C \,d^{9} c^{3}+\frac {3188646}{11} x^{11} A \,c^{2} d^{10}+\frac {1810836}{11} x^{11} B \,c^{3} d^{9}-\frac {413343}{11} x^{11} C \,d^{8} c^{4}+\frac {905418}{5} x^{10} A \,c^{3} d^{9}-\frac {413343}{10} x^{10} B \,c^{4} d^{8}-\frac {472392}{5} x^{10} C \,d^{7} c^{5}-34992 x^{7} c^{6} A \,d^{6}+\frac {139968}{7} x^{7} B \,c^{7} d^{5}+\frac {69984}{7} x^{7} C \,d^{4} c^{8}+\frac {69984}{5} x^{5} c^{8} A \,d^{4}-\frac {6912}{5} x^{5} B \,c^{9} d^{3}-\frac {6912}{5} x^{5} C \,c^{10} d^{2}-2304 x^{3} c^{10} A \,d^{2}-45927 A \,c^{4} d^{8} x^{9}-104976 B \,c^{5} d^{7} x^{9}-27216 C \,c^{6} d^{6} x^{9}-118098 A \,c^{5} d^{7} x^{8}-30618 B \,c^{6} d^{6} x^{8}+17496 C \,c^{7} d^{5} x^{8}+23328 A \,c^{7} d^{5} x^{6}+11664 B \,c^{8} d^{4} x^{6}-1152 C \,c^{9} d^{3} x^{6}-1728 A \,c^{9} d^{3} x^{4}-1728 B \,c^{10} d^{2} x^{4}+\frac {256}{3} x^{3} C \,c^{12}+256 c^{12} A x +128 B \,c^{12} x^{2}+\frac {177147}{5} C \,d^{12} x^{15}+\frac {531441}{13} x^{13} A \,d^{12}+\frac {531441}{14} x^{14} B \,d^{12}\)	\(408\)
orering	\(\frac {x \left (1063944882 C \,d^{12} x^{14}[phone] x^{13} B \,d^{12}[phone] x^{13} C \,d^{11} c [phone] x^{12} A \,d^{12}[phone] x^{12} B c \,d^{11}[phone] x^{12} C \,d^{10} c^{2}[phone] x^{11} A c \,d^{11}[phone] x^{11} B \,c^{2} d^{10}[phone] x^{11} C \,d^{9} c^{3}[phone] x^{10} A \,c^{2} d^{10}[phone] x^{10} B \,c^{3} d^{9}-1128426390 x^{10} C \,d^{8} c^{4}[phone] x^{9} A \,c^{3} d^{9}-1241269029 x^{9} B \,c^{4} d^{8}-2837186352 x^{9} C \,d^{7} c^{5}-1379187810 A \,c^{4} d^{8} x^{8}-3152429280 B \,c^{5} d^{7} x^{8}-817296480 C \,c^{6} d^{6} x^{8}-3546482940 A \,c^{5} d^{7} x^{7}-919458540 B \,c^{6} d^{6} x^{7}[phone] C \,c^{7} d^{5} x^{7}-1050809760 x^{6} c^{6} A \,d^{6}[phone] x^{6} B \,c^{7} d^{5}[phone] x^{6} C \,d^{4} c^{8}[phone] A \,c^{7} d^{5} x^{5}[phone] B \,c^{8} d^{4} x^{5}-34594560 C \,c^{9} d^{3} x^{5}[phone] x^{4} c^{8} A \,d^{4}-41513472 x^{4} B \,c^{9} d^{3}-41513472 x^{4} C \,c^{10} d^{2}-51891840 A \,c^{9} d^{3} x^{3}-51891840 B \,c^{10} d^{2} x^{3}-69189120 x^{2} c^{10} A \,d^{2}+2562560 x^{2} C \,c^{12}+3843840 B \,c^{12} x +7687680 c^{12} A \right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{4}}{30030 \left (-3 d x +c \right )^{4} \left (3 d x +2 c \right )^{8}}\)	\(451\)

Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^4,x,method=_RETURNVERBO 
SE)
 

Output:

256*c^12*A*x+128*B*c^12*x^2+(-2304*c^10*A*d^2+256/3*C*c^12)*x^3+(-1728*A*c 
^9*d^3-1728*B*c^10*d^2)*x^4+(69984/5*c^8*A*d^4-6912/5*B*c^9*d^3-6912/5*C*c 
^10*d^2)*x^5+(23328*A*c^7*d^5+11664*B*c^8*d^4-1152*C*c^9*d^3)*x^6+(-34992* 
c^6*A*d^6+139968/7*B*c^7*d^5+69984/7*C*d^4*c^8)*x^7+(-118098*A*c^5*d^7-306 
18*B*c^6*d^6+17496*C*c^7*d^5)*x^8+(-45927*A*c^4*d^8-104976*B*c^5*d^7-27216 
*C*c^6*d^6)*x^9+(905418/5*A*c^3*d^9-413343/10*B*c^4*d^8-472392/5*C*d^7*c^5 
)*x^10+(3188646/11*A*c^2*d^10+1810836/11*B*c^3*d^9-413343/11*C*d^8*c^4)*x^ 
11+(177147*A*c*d^11+531441/2*B*c^2*d^10+150903*C*d^9*c^3)*x^12+(531441/13* 
A*d^12+2125764/13*B*c*d^11+3188646/13*C*d^10*c^2)*x^13+(531441/14*B*d^12+1 
062882/7*C*d^11*c)*x^14+177147/5*C*d^12*x^15
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.63 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=\frac {177147}{5} \, C d^{12} x^{15} + 128 \, B c^{12} x^{2} + \frac {531441}{14} \, {\left (4 \, C c d^{11} + B d^{12}\right )} x^{14} + 256 \, A c^{12} x + \frac {531441}{13} \, {\left (6 \, C c^{2} d^{10} + 4 \, B c d^{11} + A d^{12}\right )} x^{13} + \frac {6561}{2} \, {\left (46 \, C c^{3} d^{9} + 81 \, B c^{2} d^{10} + 54 \, A c d^{11}\right )} x^{12} - \frac {19683}{11} \, {\left (21 \, C c^{4} d^{8} - 92 \, B c^{3} d^{9} - 162 \, A c^{2} d^{10}\right )} x^{11} - \frac {19683}{10} \, {\left (48 \, C c^{5} d^{7} + 21 \, B c^{4} d^{8} - 92 \, A c^{3} d^{9}\right )} x^{10} - 243 \, {\left (112 \, C c^{6} d^{6} + 432 \, B c^{5} d^{7} + 189 \, A c^{4} d^{8}\right )} x^{9} + 4374 \, {\left (4 \, C c^{7} d^{5} - 7 \, B c^{6} d^{6} - 27 \, A c^{5} d^{7}\right )} x^{8} + \frac {34992}{7} \, {\left (2 \, C c^{8} d^{4} + 4 \, B c^{7} d^{5} - 7 \, A c^{6} d^{6}\right )} x^{7} - 144 \, {\left (8 \, C c^{9} d^{3} - 81 \, B c^{8} d^{4} - 162 \, A c^{7} d^{5}\right )} x^{6} - \frac {864}{5} \, {\left (8 \, C c^{10} d^{2} + 8 \, B c^{9} d^{3} - 81 \, A c^{8} d^{4}\right )} x^{5} - 1728 \, {\left (B c^{10} d^{2} + A c^{9} d^{3}\right )} x^{4} + \frac {256}{3} \, {\left (C c^{12} - 27 \, A c^{10} d^{2}\right )} x^{3} \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^4,x, algorithm="f 
ricas")
 

Output:

177147/5*C*d^12*x^15 + 128*B*c^12*x^2 + 531441/14*(4*C*c*d^11 + B*d^12)*x^ 
14 + 256*A*c^12*x + 531441/13*(6*C*c^2*d^10 + 4*B*c*d^11 + A*d^12)*x^13 + 
6561/2*(46*C*c^3*d^9 + 81*B*c^2*d^10 + 54*A*c*d^11)*x^12 - 19683/11*(21*C* 
c^4*d^8 - 92*B*c^3*d^9 - 162*A*c^2*d^10)*x^11 - 19683/10*(48*C*c^5*d^7 + 2 
1*B*c^4*d^8 - 92*A*c^3*d^9)*x^10 - 243*(112*C*c^6*d^6 + 432*B*c^5*d^7 + 18 
9*A*c^4*d^8)*x^9 + 4374*(4*C*c^7*d^5 - 7*B*c^6*d^6 - 27*A*c^5*d^7)*x^8 + 3 
4992/7*(2*C*c^8*d^4 + 4*B*c^7*d^5 - 7*A*c^6*d^6)*x^7 - 144*(8*C*c^9*d^3 - 
81*B*c^8*d^4 - 162*A*c^7*d^5)*x^6 - 864/5*(8*C*c^10*d^2 + 8*B*c^9*d^3 - 81 
*A*c^8*d^4)*x^5 - 1728*(B*c^10*d^2 + A*c^9*d^3)*x^4 + 256/3*(C*c^12 - 27*A 
*c^10*d^2)*x^3
 

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.85 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=256 A c^{12} x + 128 B c^{12} x^{2} + \frac {177147 C d^{12} x^{15}}{5} + x^{14} \cdot \left (\frac {531441 B d^{12}}{14} + \frac {1062882 C c d^{11}}{7}\right ) + x^{13} \cdot \left (\frac {531441 A d^{12}}{13} + \frac {2125764 B c d^{11}}{13} + \frac {3188646 C c^{2} d^{10}}{13}\right ) + x^{12} \cdot \left (177147 A c d^{11} + \frac {531441 B c^{2} d^{10}}{2} + 150903 C c^{3} d^{9}\right ) + x^{11} \cdot \left (\frac {3188646 A c^{2} d^{10}}{11} + \frac {1810836 B c^{3} d^{9}}{11} - \frac {413343 C c^{4} d^{8}}{11}\right ) + x^{10} \cdot \left (\frac {905418 A c^{3} d^{9}}{5} - \frac {413343 B c^{4} d^{8}}{10} - \frac {472392 C c^{5} d^{7}}{5}\right ) + x^{9} \left (- 45927 A c^{4} d^{8} - 104976 B c^{5} d^{7} - 27216 C c^{6} d^{6}\right ) + x^{8} \left (- 118098 A c^{5} d^{7} - 30618 B c^{6} d^{6} + 17496 C c^{7} d^{5}\right ) + x^{7} \left (- 34992 A c^{6} d^{6} + \frac {139968 B c^{7} d^{5}}{7} + \frac {69984 C c^{8} d^{4}}{7}\right ) + x^{6} \cdot \left (23328 A c^{7} d^{5} + 11664 B c^{8} d^{4} - 1152 C c^{9} d^{3}\right ) + x^{5} \cdot \left (\frac {69984 A c^{8} d^{4}}{5} - \frac {6912 B c^{9} d^{3}}{5} - \frac {6912 C c^{10} d^{2}}{5}\right ) + x^{4} \left (- 1728 A c^{9} d^{3} - 1728 B c^{10} d^{2}\right ) + x^{3} \left (- 2304 A c^{10} d^{2} + \frac {256 C c^{12}}{3}\right ) \] Input:

integrate((C*x**2+B*x+A)*(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**4,x)
 

Output:

256*A*c**12*x + 128*B*c**12*x**2 + 177147*C*d**12*x**15/5 + x**14*(531441* 
B*d**12/14 + 1062882*C*c*d**11/7) + x**13*(531441*A*d**12/13 + 2125764*B*c 
*d**11/13 + 3188646*C*c**2*d**10/13) + x**12*(177147*A*c*d**11 + 531441*B* 
c**2*d**10/2 + 150903*C*c**3*d**9) + x**11*(3188646*A*c**2*d**10/11 + 1810 
836*B*c**3*d**9/11 - 413343*C*c**4*d**8/11) + x**10*(905418*A*c**3*d**9/5 
- 413343*B*c**4*d**8/10 - 472392*C*c**5*d**7/5) + x**9*(-45927*A*c**4*d**8 
 - 104976*B*c**5*d**7 - 27216*C*c**6*d**6) + x**8*(-118098*A*c**5*d**7 - 3 
0618*B*c**6*d**6 + 17496*C*c**7*d**5) + x**7*(-34992*A*c**6*d**6 + 139968* 
B*c**7*d**5/7 + 69984*C*c**8*d**4/7) + x**6*(23328*A*c**7*d**5 + 11664*B*c 
**8*d**4 - 1152*C*c**9*d**3) + x**5*(69984*A*c**8*d**4/5 - 6912*B*c**9*d** 
3/5 - 6912*C*c**10*d**2/5) + x**4*(-1728*A*c**9*d**3 - 1728*B*c**10*d**2) 
+ x**3*(-2304*A*c**10*d**2 + 256*C*c**12/3)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.63 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=\frac {177147}{5} \, C d^{12} x^{15} + 128 \, B c^{12} x^{2} + \frac {531441}{14} \, {\left (4 \, C c d^{11} + B d^{12}\right )} x^{14} + 256 \, A c^{12} x + \frac {531441}{13} \, {\left (6 \, C c^{2} d^{10} + 4 \, B c d^{11} + A d^{12}\right )} x^{13} + \frac {6561}{2} \, {\left (46 \, C c^{3} d^{9} + 81 \, B c^{2} d^{10} + 54 \, A c d^{11}\right )} x^{12} - \frac {19683}{11} \, {\left (21 \, C c^{4} d^{8} - 92 \, B c^{3} d^{9} - 162 \, A c^{2} d^{10}\right )} x^{11} - \frac {19683}{10} \, {\left (48 \, C c^{5} d^{7} + 21 \, B c^{4} d^{8} - 92 \, A c^{3} d^{9}\right )} x^{10} - 243 \, {\left (112 \, C c^{6} d^{6} + 432 \, B c^{5} d^{7} + 189 \, A c^{4} d^{8}\right )} x^{9} + 4374 \, {\left (4 \, C c^{7} d^{5} - 7 \, B c^{6} d^{6} - 27 \, A c^{5} d^{7}\right )} x^{8} + \frac {34992}{7} \, {\left (2 \, C c^{8} d^{4} + 4 \, B c^{7} d^{5} - 7 \, A c^{6} d^{6}\right )} x^{7} - 144 \, {\left (8 \, C c^{9} d^{3} - 81 \, B c^{8} d^{4} - 162 \, A c^{7} d^{5}\right )} x^{6} - \frac {864}{5} \, {\left (8 \, C c^{10} d^{2} + 8 \, B c^{9} d^{3} - 81 \, A c^{8} d^{4}\right )} x^{5} - 1728 \, {\left (B c^{10} d^{2} + A c^{9} d^{3}\right )} x^{4} + \frac {256}{3} \, {\left (C c^{12} - 27 \, A c^{10} d^{2}\right )} x^{3} \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^4,x, algorithm="m 
axima")
 

Output:

177147/5*C*d^12*x^15 + 128*B*c^12*x^2 + 531441/14*(4*C*c*d^11 + B*d^12)*x^ 
14 + 256*A*c^12*x + 531441/13*(6*C*c^2*d^10 + 4*B*c*d^11 + A*d^12)*x^13 + 
6561/2*(46*C*c^3*d^9 + 81*B*c^2*d^10 + 54*A*c*d^11)*x^12 - 19683/11*(21*C* 
c^4*d^8 - 92*B*c^3*d^9 - 162*A*c^2*d^10)*x^11 - 19683/10*(48*C*c^5*d^7 + 2 
1*B*c^4*d^8 - 92*A*c^3*d^9)*x^10 - 243*(112*C*c^6*d^6 + 432*B*c^5*d^7 + 18 
9*A*c^4*d^8)*x^9 + 4374*(4*C*c^7*d^5 - 7*B*c^6*d^6 - 27*A*c^5*d^7)*x^8 + 3 
4992/7*(2*C*c^8*d^4 + 4*B*c^7*d^5 - 7*A*c^6*d^6)*x^7 - 144*(8*C*c^9*d^3 - 
81*B*c^8*d^4 - 162*A*c^7*d^5)*x^6 - 864/5*(8*C*c^10*d^2 + 8*B*c^9*d^3 - 81 
*A*c^8*d^4)*x^5 - 1728*(B*c^10*d^2 + A*c^9*d^3)*x^4 + 256/3*(C*c^12 - 27*A 
*c^10*d^2)*x^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.77 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=\frac {177147}{5} \, C d^{12} x^{15} + \frac {1062882}{7} \, C c d^{11} x^{14} + \frac {531441}{14} \, B d^{12} x^{14} + \frac {3188646}{13} \, C c^{2} d^{10} x^{13} + \frac {2125764}{13} \, B c d^{11} x^{13} + \frac {531441}{13} \, A d^{12} x^{13} + 150903 \, C c^{3} d^{9} x^{12} + \frac {531441}{2} \, B c^{2} d^{10} x^{12} + 177147 \, A c d^{11} x^{12} - \frac {413343}{11} \, C c^{4} d^{8} x^{11} + \frac {1810836}{11} \, B c^{3} d^{9} x^{11} + \frac {3188646}{11} \, A c^{2} d^{10} x^{11} - \frac {472392}{5} \, C c^{5} d^{7} x^{10} - \frac {413343}{10} \, B c^{4} d^{8} x^{10} + \frac {905418}{5} \, A c^{3} d^{9} x^{10} - 27216 \, C c^{6} d^{6} x^{9} - 104976 \, B c^{5} d^{7} x^{9} - 45927 \, A c^{4} d^{8} x^{9} + 17496 \, C c^{7} d^{5} x^{8} - 30618 \, B c^{6} d^{6} x^{8} - 118098 \, A c^{5} d^{7} x^{8} + \frac {69984}{7} \, C c^{8} d^{4} x^{7} + \frac {139968}{7} \, B c^{7} d^{5} x^{7} - 34992 \, A c^{6} d^{6} x^{7} - 1152 \, C c^{9} d^{3} x^{6} + 11664 \, B c^{8} d^{4} x^{6} + 23328 \, A c^{7} d^{5} x^{6} - \frac {6912}{5} \, C c^{10} d^{2} x^{5} - \frac {6912}{5} \, B c^{9} d^{3} x^{5} + \frac {69984}{5} \, A c^{8} d^{4} x^{5} - 1728 \, B c^{10} d^{2} x^{4} - 1728 \, A c^{9} d^{3} x^{4} + \frac {256}{3} \, C c^{12} x^{3} - 2304 \, A c^{10} d^{2} x^{3} + 128 \, B c^{12} x^{2} + 256 \, A c^{12} x \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^4,x, algorithm="g 
iac")
                                                                                    
                                                                                    
 

Output:

177147/5*C*d^12*x^15 + 1062882/7*C*c*d^11*x^14 + 531441/14*B*d^12*x^14 + 3 
188646/13*C*c^2*d^10*x^13 + 2125764/13*B*c*d^11*x^13 + 531441/13*A*d^12*x^ 
13 + 150903*C*c^3*d^9*x^12 + 531441/2*B*c^2*d^10*x^12 + 177147*A*c*d^11*x^ 
12 - 413343/11*C*c^4*d^8*x^11 + 1810836/11*B*c^3*d^9*x^11 + 3188646/11*A*c 
^2*d^10*x^11 - 472392/5*C*c^5*d^7*x^10 - 413343/10*B*c^4*d^8*x^10 + 905418 
/5*A*c^3*d^9*x^10 - 27216*C*c^6*d^6*x^9 - 104976*B*c^5*d^7*x^9 - 45927*A*c 
^4*d^8*x^9 + 17496*C*c^7*d^5*x^8 - 30618*B*c^6*d^6*x^8 - 118098*A*c^5*d^7* 
x^8 + 69984/7*C*c^8*d^4*x^7 + 139968/7*B*c^7*d^5*x^7 - 34992*A*c^6*d^6*x^7 
 - 1152*C*c^9*d^3*x^6 + 11664*B*c^8*d^4*x^6 + 23328*A*c^7*d^5*x^6 - 6912/5 
*C*c^10*d^2*x^5 - 6912/5*B*c^9*d^3*x^5 + 69984/5*A*c^8*d^4*x^5 - 1728*B*c^ 
10*d^2*x^4 - 1728*A*c^9*d^3*x^4 + 256/3*C*c^12*x^3 - 2304*A*c^10*d^2*x^3 + 
 128*B*c^12*x^2 + 256*A*c^12*x
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.46 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=x^3\,\left (\frac {256\,C\,c^{12}}{3}-2304\,A\,c^{10}\,d^2\right )+128\,B\,c^{12}\,x^2+\frac {177147\,C\,d^{12}\,x^{15}}{5}+\frac {531441\,d^{11}\,x^{14}\,\left (B\,d+4\,C\,c\right )}{14}+\frac {531441\,d^{10}\,x^{13}\,\left (6\,C\,c^2+4\,B\,c\,d+A\,d^2\right )}{13}+256\,A\,c^{12}\,x-1728\,c^9\,d^2\,x^4\,\left (A\,d+B\,c\right )+\frac {34992\,c^6\,d^4\,x^7\,\left (2\,C\,c^2+4\,B\,c\,d-7\,A\,d^2\right )}{7}-4374\,c^5\,d^5\,x^8\,\left (-4\,C\,c^2+7\,B\,c\,d+27\,A\,d^2\right )-\frac {864\,c^8\,d^2\,x^5\,\left (8\,C\,c^2+8\,B\,c\,d-81\,A\,d^2\right )}{5}-\frac {19683\,c^3\,d^7\,x^{10}\,\left (48\,C\,c^2+21\,B\,c\,d-92\,A\,d^2\right )}{10}+144\,c^7\,d^3\,x^6\,\left (-8\,C\,c^2+81\,B\,c\,d+162\,A\,d^2\right )+\frac {19683\,c^2\,d^8\,x^{11}\,\left (-21\,C\,c^2+92\,B\,c\,d+162\,A\,d^2\right )}{11}-243\,c^4\,d^6\,x^9\,\left (112\,C\,c^2+432\,B\,c\,d+189\,A\,d^2\right )+\frac {6561\,c\,d^9\,x^{12}\,\left (46\,C\,c^2+81\,B\,c\,d+54\,A\,d^2\right )}{2} \] Input:

int((A + B*x + C*x^2)*(27*d^3*x^3 - 4*c^3 + 27*c*d^2*x^2)^4,x)
 

Output:

x^3*((256*C*c^12)/3 - 2304*A*c^10*d^2) + 128*B*c^12*x^2 + (177147*C*d^12*x 
^15)/5 + (531441*d^11*x^14*(B*d + 4*C*c))/14 + (531441*d^10*x^13*(A*d^2 + 
6*C*c^2 + 4*B*c*d))/13 + 256*A*c^12*x - 1728*c^9*d^2*x^4*(A*d + B*c) + (34 
992*c^6*d^4*x^7*(2*C*c^2 - 7*A*d^2 + 4*B*c*d))/7 - 4374*c^5*d^5*x^8*(27*A* 
d^2 - 4*C*c^2 + 7*B*c*d) - (864*c^8*d^2*x^5*(8*C*c^2 - 81*A*d^2 + 8*B*c*d) 
)/5 - (19683*c^3*d^7*x^10*(48*C*c^2 - 92*A*d^2 + 21*B*c*d))/10 + 144*c^7*d 
^3*x^6*(162*A*d^2 - 8*C*c^2 + 81*B*c*d) + (19683*c^2*d^8*x^11*(162*A*d^2 - 
 21*C*c^2 + 92*B*c*d))/11 - 243*c^4*d^6*x^9*(189*A*d^2 + 112*C*c^2 + 432*B 
*c*d) + (6561*c*d^9*x^12*(54*A*d^2 + 46*C*c^2 + 81*B*c*d))/2
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.73 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^4 \, dx=\frac {x \left (1063944882 c \,d^{12} x^{14}[phone] b \,d^{12} x^{13}[phone] c^{2} d^{11} x^{13}[phone] a \,d^{12} x^{12}[phone] b c \,d^{11} x^{12}[phone] c^{3} d^{10} x^{12}[phone] a c \,d^{11} x^{11}[phone] b \,c^{2} d^{10} x^{11}[phone] c^{4} d^{9} x^{11}[phone] a \,c^{2} d^{10} x^{10}[phone] b \,c^{3} d^{9} x^{10}-1128426390 c^{5} d^{8} x^{10}[phone] a \,c^{3} d^{9} x^{9}-1241269029 b \,c^{4} d^{8} x^{9}-2837186352 c^{6} d^{7} x^{9}-1379187810 a \,c^{4} d^{8} x^{8}-3152429280 b \,c^{5} d^{7} x^{8}-817296480 c^{7} d^{6} x^{8}-3546482940 a \,c^{5} d^{7} x^{7}-919458540 b \,c^{6} d^{6} x^{7}[phone] c^{8} d^{5} x^{7}-1050809760 a \,c^{6} d^{6} x^{6}[phone] b \,c^{7} d^{5} x^{6}[phone] c^{9} d^{4} x^{6}[phone] a \,c^{7} d^{5} x^{5}[phone] b \,c^{8} d^{4} x^{5}-34594560 c^{10} d^{3} x^{5}[phone] a \,c^{8} d^{4} x^{4}-41513472 b \,c^{9} d^{3} x^{4}-41513472 c^{11} d^{2} x^{4}-51891840 a \,c^{9} d^{3} x^{3}-51891840 b \,c^{10} d^{2} x^{3}-69189120 a \,c^{10} d^{2} x^{2}+2562560 c^{13} x^{2}+3843840 b \,c^{12} x +7687680 a \,c^{12}\right )}{30030} \] Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^4,x)
 

Output:

(x*(7687680*a*c**12 - 69189120*a*c**10*d**2*x**2 - 51891840*a*c**9*d**3*x* 
*3 + 420323904*a*c**8*d**4*x**4 + 700539840*a*c**7*d**5*x**5 - 1050809760* 
a*c**6*d**6*x**6 - 3546482940*a*c**5*d**7*x**7 - 1379187810*a*c**4*d**8*x* 
*8 + 5437940508*a*c**3*d**9*x**9 + 8705003580*a*c**2*d**10*x**10 + 5319724 
410*a*c*d**11*x**11 + 1227628710*a*d**12*x**12 + 3843840*b*c**12*x - 51891 
840*b*c**10*d**2*x**3 - 41513472*b*c**9*d**3*x**4 + 350269920*b*c**8*d**4* 
x**5 + 600462720*b*c**7*d**5*x**6 - 919458540*b*c**6*d**6*x**7 - 315242928 
0*b*c**5*d**7*x**8 - 1241269029*b*c**4*d**8*x**9 + 4943582280*b*c**3*d**9* 
x**10 + 7979586615*b*c**2*d**10*x**11 + 4910514840*b*c*d**11*x**12 + 11399 
40945*b*d**12*x**13 + 2562560*c**13*x**2 - 41513472*c**11*d**2*x**4 - 3459 
4560*c**10*d**3*x**5 + 300231360*c**9*d**4*x**6 + 525404880*c**8*d**5*x**7 
 - 817296480*c**7*d**6*x**8 - 2837186352*c**6*d**7*x**9 - 1128426390*c**5* 
d**8*x**10 + 4531617090*c**4*d**9*x**11 + 7365772260*c**3*d**10*x**12 + 45 
59763780*c**2*d**11*x**13 + 1063944882*c*d**12*x**14))/30030