\(\int (A+B x+C x^2) (27 a^3+27 a^2 b x-4 b^3 x^3)^3 \, dx\) [1]

Optimal result

Integrand size = 34, antiderivative size = 192 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=\frac {729 a^3 \left (4 A b^2-6 a b B+9 a^2 C\right ) (3 a+2 b x)^7}{448 b^3}-\frac {243 a^2 \left (4 A b^2-12 a b B+27 a^2 C\right ) (3 a+2 b x)^8}{512 b^3}+\frac {3 a \left (2 A b^2-12 a b B+45 a^2 C\right ) (3 a+2 b x)^9}{32 b^3}-\frac {\left (2 A b^2-30 a b B+207 a^2 C\right ) (3 a+2 b x)^{10}}{320 b^3}-\frac {(2 b B-33 a C) (3 a+2 b x)^{11}}{704 b^3}-\frac {C (3 a+2 b x)^{12}}{768 b^3} \] Output:

729/448*a^3*(4*A*b^2-6*B*a*b+9*C*a^2)*(2*b*x+3*a)^7/b^3-243/512*a^2*(4*A*b 
^2-12*B*a*b+27*C*a^2)*(2*b*x+3*a)^8/b^3+3/32*a*(2*A*b^2-12*B*a*b+45*C*a^2) 
*(2*b*x+3*a)^9/b^3-1/320*(2*A*b^2-30*B*a*b+207*C*a^2)*(2*b*x+3*a)^10/b^3-1 
/704*(2*B*b-33*C*a)*(2*b*x+3*a)^11/b^3-1/768*C*(2*b*x+3*a)^12/b^3
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.25 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=-\frac {16}{165} b^9 x^{10} \left (66 A+60 B x+55 C x^2\right )+\frac {6561}{2} a^9 x (6 A+x (3 B+2 C x))+\frac {19683}{4} a^8 b x^2 (6 A+x (4 B+3 C x))+\frac {19683}{20} a^7 b^2 x^3 (20 A+3 x (5 B+4 C x))+\frac {729}{4} a^6 b^3 x^4 (15 A+2 x (6 B+5 C x))-\frac {2916}{35} a^5 b^4 x^5 (42 A+5 x (7 B+6 C x))-\frac {729}{14} a^4 b^5 x^6 (28 A+3 x (8 B+7 C x))+\frac {18}{7} a^3 b^6 x^7 (72 A+7 x (9 B+8 C x))+\frac {18}{5} a^2 b^7 x^8 (45 A+4 x (10 B+9 C x)) \] Input:

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

Output:

(-16*b^9*x^10*(66*A + 60*B*x + 55*C*x^2))/165 + (6561*a^9*x*(6*A + x*(3*B 
+ 2*C*x)))/2 + (19683*a^8*b*x^2*(6*A + x*(4*B + 3*C*x)))/4 + (19683*a^7*b^ 
2*x^3*(20*A + 3*x*(5*B + 4*C*x)))/20 + (729*a^6*b^3*x^4*(15*A + 2*x*(6*B + 
 5*C*x)))/4 - (2916*a^5*b^4*x^5*(42*A + 5*x*(7*B + 6*C*x)))/35 - (729*a^4* 
b^5*x^6*(28*A + 3*x*(8*B + 7*C*x)))/14 + (18*a^3*b^6*x^7*(72*A + 7*x*(9*B 
+ 8*C*x)))/7 + (18*a^2*b^7*x^8*(45*A + 4*x*(10*B + 9*C*x)))/5
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.

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

\(\Big \downarrow \) 2188

\(\displaystyle \int \left (19683 a^9 A+19683 a^8 x (a B+3 A b)+19683 a^7 x^2 \left (a (a C+3 b B)+3 A b^2\right )+2187 a^6 b x^3 \left (27 a (a C+b B)+5 A b^2\right )+2187 a^5 b^2 x^4 \left (a (27 a C+5 b B)-8 A b^2\right )+2187 a^4 b^3 x^5 \left (-a (8 b B-5 a C)-4 A b^2\right )+324 a^3 b^4 x^6 \left (4 A b^2-27 a (2 a C+b B)\right )-16 b^7 x^9 \left (4 A b^2-81 a^2 C\right )+324 a^2 b^5 x^7 \left (a (4 b B-27 a C)+4 A b^2\right )+1296 a^2 b^6 x^8 (a C+b B)-64 b^9 B x^{10}-64 b^9 C x^{11}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 19683 a^9 A x+\frac {19683}{2} a^8 x^2 (a B+3 A b)+6561 a^7 x^3 \left (a (a C+3 b B)+3 A b^2\right )+\frac {2187}{4} a^6 b x^4 \left (27 a (a C+b B)+5 A b^2\right )-\frac {2187}{5} a^5 b^2 x^5 \left (8 A b^2-a (27 a C+5 b B)\right )-\frac {729}{2} a^4 b^3 x^6 \left (a (8 b B-5 a C)+4 A b^2\right )+\frac {324}{7} a^3 b^4 x^7 \left (4 A b^2-27 a (2 a C+b B)\right )-\frac {8}{5} b^7 x^{10} \left (4 A b^2-81 a^2 C\right )+\frac {81}{2} a^2 b^5 x^8 \left (a (4 b B-27 a C)+4 A b^2\right )+144 a^2 b^6 x^9 (a C+b B)-\frac {64}{11} b^9 B x^{11}-\frac {16}{3} b^9 C x^{12}\)

Input:

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

Output:

19683*a^9*A*x + (19683*a^8*(3*A*b + a*B)*x^2)/2 + 6561*a^7*(3*A*b^2 + a*(3 
*b*B + a*C))*x^3 + (2187*a^6*b*(5*A*b^2 + 27*a*(b*B + a*C))*x^4)/4 - (2187 
*a^5*b^2*(8*A*b^2 - a*(5*b*B + 27*a*C))*x^5)/5 - (729*a^4*b^3*(4*A*b^2 + a 
*(8*b*B - 5*a*C))*x^6)/2 + (324*a^3*b^4*(4*A*b^2 - 27*a*(b*B + 2*a*C))*x^7 
)/7 + (81*a^2*b^5*(4*A*b^2 + a*(4*b*B - 27*a*C))*x^8)/2 + 144*a^2*b^6*(b*B 
 + a*C)*x^9 - (8*b^7*(4*A*b^2 - 81*a^2*C)*x^10)/5 - (64*b^9*B*x^11)/11 - ( 
16*b^9*C*x^12)/3
 

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 = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.42

method	result	size
norman	\(-\frac {16 C \,b^{9} x^{12}}{3}-\frac {64 B \,b^{9} x^{11}}{11}+\left (-\frac {32}{5} A \,b^{9}+\frac {648}{5} C \,b^{7} a^{2}\right ) x^{10}+\left (144 B \,a^{2} b^{7}+144 C \,b^{6} a^{3}\right ) x^{9}+\left (162 A \,a^{2} b^{7}+162 B \,a^{3} b^{6}-\frac {2187}{2} C \,b^{5} a^{4}\right ) x^{8}+\left (\frac {1296}{7} A \,a^{3} b^{6}-\frac {8748}{7} B \,a^{4} b^{5}-\frac {17496}{7} C \,b^{4} a^{5}\right ) x^{7}+\left (-1458 A \,a^{4} b^{5}-2916 B \,a^{5} b^{4}+\frac {3645}{2} C \,b^{3} a^{6}\right ) x^{6}+\left (-\frac {17496}{5} A \,a^{5} b^{4}+2187 B \,a^{6} b^{3}+\frac {59049}{5} C \,a^{7} b^{2}\right ) x^{5}+\left (\frac {10935}{4} A \,a^{6} b^{3}+\frac {59049}{4} B \,a^{7} b^{2}+\frac {59049}{4} C \,a^{8} b \right ) x^{4}+\left (19683 A \,a^{7} b^{2}+19683 B \,a^{8} b +6561 C \,a^{9}\right ) x^{3}+\left (\frac {59049}{2} A \,a^{8} b +\frac {19683}{2} B \,a^{9}\right ) x^{2}+19683 A \,a^{9} x\)	\(273\)
default	\(-\frac {16 C \,b^{9} x^{12}}{3}-\frac {64 B \,b^{9} x^{11}}{11}+\frac {\left (-64 A \,b^{9}+1296 C \,b^{7} a^{2}\right ) x^{10}}{10}+\frac {\left (1296 B \,a^{2} b^{7}+1296 C \,b^{6} a^{3}\right ) x^{9}}{9}+\frac {\left (1296 A \,a^{2} b^{7}+1296 B \,a^{3} b^{6}-8748 C \,b^{5} a^{4}\right ) x^{8}}{8}+\frac {\left (1296 A \,a^{3} b^{6}-8748 B \,a^{4} b^{5}-17496 C \,b^{4} a^{5}\right ) x^{7}}{7}+\frac {\left (-8748 A \,a^{4} b^{5}-17496 B \,a^{5} b^{4}+10935 C \,b^{3} a^{6}\right ) x^{6}}{6}+\frac {\left (-17496 A \,a^{5} b^{4}+10935 B \,a^{6} b^{3}+59049 C \,a^{7} b^{2}\right ) x^{5}}{5}+\frac {\left (10935 A \,a^{6} b^{3}+59049 B \,a^{7} b^{2}+59049 C \,a^{8} b \right ) x^{4}}{4}+\frac {\left (59049 A \,a^{7} b^{2}+59049 B \,a^{8} b +19683 C \,a^{9}\right ) x^{3}}{3}+\frac {\left (59049 A \,a^{8} b +19683 B \,a^{9}\right ) x^{2}}{2}+19683 A \,a^{9} x\)	\(282\)
gosper	\(\frac {x \left (-24640 C \,b^{9} x^{11}-26880 B \,b^{9} x^{10}-29568 x^{9} A \,b^{9}+598752 x^{9} C \,b^{7} a^{2}+665280 B \,a^{2} b^{7} x^{8}+665280 C \,a^{3} b^{6} x^{8}+748440 x^{7} A \,a^{2} b^{7}+748440 x^{7} B \,a^{3} b^{6}-5051970 x^{7} C \,b^{5} a^{4}+855360 x^{6} A \,a^{3} b^{6}-5773680 x^{6} B \,a^{4} b^{5}-11547360 x^{6} C \,b^{4} a^{5}-6735960 x^{5} A \,a^{4} b^{5}-13471920 x^{5} B \,a^{5} b^{4}+8419950 x^{5} C \,b^{3} a^{6}-16166304 x^{4} A \,a^{5} b^{4}[phone] x^{4} B \,a^{6} b^{3}[phone] x^{4} C \,a^{7} b^{2}[phone] x^{3} A \,a^{6} b^{3}[phone] x^{3} B \,a^{7} b^{2}[phone] x^{3} C \,a^{8} b [phone] A \,a^{7} b^{2} x^{2}[phone] B \,a^{8} b \,x^{2}[phone] C \,a^{9} x^{2}[phone] x A \,a^{8} b [phone] x B \,a^{9}[phone] A \,a^{9}\right )}{4620}\)	\(298\)
risch	\(\frac {648}{5} x^{10} C \,b^{7} a^{2}+162 x^{8} A \,a^{2} b^{7}+162 x^{8} B \,a^{3} b^{6}-\frac {2187}{2} x^{8} C \,b^{5} a^{4}+\frac {1296}{7} x^{7} A \,a^{3} b^{6}-\frac {8748}{7} x^{7} B \,a^{4} b^{5}-\frac {17496}{7} x^{7} C \,b^{4} a^{5}-1458 x^{6} A \,a^{4} b^{5}-2916 x^{6} B \,a^{5} b^{4}+\frac {3645}{2} x^{6} C \,b^{3} a^{6}-\frac {17496}{5} x^{5} A \,a^{5} b^{4}+2187 x^{5} B \,a^{6} b^{3}+\frac {59049}{5} x^{5} C \,a^{7} b^{2}+\frac {10935}{4} x^{4} A \,a^{6} b^{3}+\frac {59049}{4} x^{4} B \,a^{7} b^{2}+\frac {59049}{4} x^{4} C \,a^{8} b +\frac {59049}{2} x^{2} A \,a^{8} b +144 B \,a^{2} b^{7} x^{9}+144 C \,a^{3} b^{6} x^{9}+19683 A \,a^{7} b^{2} x^{3}+19683 B \,a^{8} b \,x^{3}-\frac {64}{11} B \,b^{9} x^{11}+19683 A \,a^{9} x -\frac {16}{3} C \,b^{9} x^{12}-\frac {32}{5} x^{10} A \,b^{9}+\frac {19683}{2} x^{2} B \,a^{9}+6561 C \,a^{9} x^{3}\)	\(300\)
parallelrisch	\(\frac {648}{5} x^{10} C \,b^{7} a^{2}+162 x^{8} A \,a^{2} b^{7}+162 x^{8} B \,a^{3} b^{6}-\frac {2187}{2} x^{8} C \,b^{5} a^{4}+\frac {1296}{7} x^{7} A \,a^{3} b^{6}-\frac {8748}{7} x^{7} B \,a^{4} b^{5}-\frac {17496}{7} x^{7} C \,b^{4} a^{5}-1458 x^{6} A \,a^{4} b^{5}-2916 x^{6} B \,a^{5} b^{4}+\frac {3645}{2} x^{6} C \,b^{3} a^{6}-\frac {17496}{5} x^{5} A \,a^{5} b^{4}+2187 x^{5} B \,a^{6} b^{3}+\frac {59049}{5} x^{5} C \,a^{7} b^{2}+\frac {10935}{4} x^{4} A \,a^{6} b^{3}+\frac {59049}{4} x^{4} B \,a^{7} b^{2}+\frac {59049}{4} x^{4} C \,a^{8} b +\frac {59049}{2} x^{2} A \,a^{8} b +144 B \,a^{2} b^{7} x^{9}+144 C \,a^{3} b^{6} x^{9}+19683 A \,a^{7} b^{2} x^{3}+19683 B \,a^{8} b \,x^{3}-\frac {64}{11} B \,b^{9} x^{11}+19683 A \,a^{9} x -\frac {16}{3} C \,b^{9} x^{12}-\frac {32}{5} x^{10} A \,b^{9}+\frac {19683}{2} x^{2} B \,a^{9}+6561 C \,a^{9} x^{3}\)	\(300\)
orering	\(\frac {x \left (-24640 C \,b^{9} x^{11}-26880 B \,b^{9} x^{10}-29568 x^{9} A \,b^{9}+598752 x^{9} C \,b^{7} a^{2}+665280 B \,a^{2} b^{7} x^{8}+665280 C \,a^{3} b^{6} x^{8}+748440 x^{7} A \,a^{2} b^{7}+748440 x^{7} B \,a^{3} b^{6}-5051970 x^{7} C \,b^{5} a^{4}+855360 x^{6} A \,a^{3} b^{6}-5773680 x^{6} B \,a^{4} b^{5}-11547360 x^{6} C \,b^{4} a^{5}-6735960 x^{5} A \,a^{4} b^{5}-13471920 x^{5} B \,a^{5} b^{4}+8419950 x^{5} C \,b^{3} a^{6}-16166304 x^{4} A \,a^{5} b^{4}[phone] x^{4} B \,a^{6} b^{3}[phone] x^{4} C \,a^{7} b^{2}[phone] x^{3} A \,a^{6} b^{3}[phone] x^{3} B \,a^{7} b^{2}[phone] x^{3} C \,a^{8} b [phone] A \,a^{7} b^{2} x^{2}[phone] B \,a^{8} b \,x^{2}[phone] C \,a^{9} x^{2}[phone] x A \,a^{8} b [phone] x B \,a^{9}[phone] A \,a^{9}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{3}}{4620 \left (-b x +3 a \right )^{3} \left (2 b x +3 a \right )^{6}}\)	\(341\)

Input:

int((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x,method=_RETURNVERBOSE 
)
 

Output:

-16/3*C*b^9*x^12-64/11*B*b^9*x^11+(-32/5*A*b^9+648/5*C*b^7*a^2)*x^10+(144* 
B*a^2*b^7+144*C*a^3*b^6)*x^9+(162*A*a^2*b^7+162*B*a^3*b^6-2187/2*C*b^5*a^4 
)*x^8+(1296/7*A*a^3*b^6-8748/7*B*a^4*b^5-17496/7*C*b^4*a^5)*x^7+(-1458*A*a 
^4*b^5-2916*B*a^5*b^4+3645/2*C*b^3*a^6)*x^6+(-17496/5*A*a^5*b^4+2187*B*a^6 
*b^3+59049/5*C*a^7*b^2)*x^5+(10935/4*A*a^6*b^3+59049/4*B*a^7*b^2+59049/4*C 
*a^8*b)*x^4+(19683*A*a^7*b^2+19683*B*a^8*b+6561*C*a^9)*x^3+(59049/2*A*a^8* 
b+19683/2*B*a^9)*x^2+19683*A*a^9*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.44 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=-\frac {16}{3} \, C b^{9} x^{12} - \frac {64}{11} \, B b^{9} x^{11} + 19683 \, A a^{9} x + \frac {8}{5} \, {\left (81 \, C a^{2} b^{7} - 4 \, A b^{9}\right )} x^{10} + 144 \, {\left (C a^{3} b^{6} + B a^{2} b^{7}\right )} x^{9} - \frac {81}{2} \, {\left (27 \, C a^{4} b^{5} - 4 \, B a^{3} b^{6} - 4 \, A a^{2} b^{7}\right )} x^{8} - \frac {324}{7} \, {\left (54 \, C a^{5} b^{4} + 27 \, B a^{4} b^{5} - 4 \, A a^{3} b^{6}\right )} x^{7} + \frac {729}{2} \, {\left (5 \, C a^{6} b^{3} - 8 \, B a^{5} b^{4} - 4 \, A a^{4} b^{5}\right )} x^{6} + \frac {2187}{5} \, {\left (27 \, C a^{7} b^{2} + 5 \, B a^{6} b^{3} - 8 \, A a^{5} b^{4}\right )} x^{5} + \frac {2187}{4} \, {\left (27 \, C a^{8} b + 27 \, B a^{7} b^{2} + 5 \, A a^{6} b^{3}\right )} x^{4} + 6561 \, {\left (C a^{9} + 3 \, B a^{8} b + 3 \, A a^{7} b^{2}\right )} x^{3} + \frac {19683}{2} \, {\left (B a^{9} + 3 \, A a^{8} b\right )} x^{2} \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="fri 
cas")
 

Output:

-16/3*C*b^9*x^12 - 64/11*B*b^9*x^11 + 19683*A*a^9*x + 8/5*(81*C*a^2*b^7 - 
4*A*b^9)*x^10 + 144*(C*a^3*b^6 + B*a^2*b^7)*x^9 - 81/2*(27*C*a^4*b^5 - 4*B 
*a^3*b^6 - 4*A*a^2*b^7)*x^8 - 324/7*(54*C*a^5*b^4 + 27*B*a^4*b^5 - 4*A*a^3 
*b^6)*x^7 + 729/2*(5*C*a^6*b^3 - 8*B*a^5*b^4 - 4*A*a^4*b^5)*x^6 + 2187/5*( 
27*C*a^7*b^2 + 5*B*a^6*b^3 - 8*A*a^5*b^4)*x^5 + 2187/4*(27*C*a^8*b + 27*B* 
a^7*b^2 + 5*A*a^6*b^3)*x^4 + 6561*(C*a^9 + 3*B*a^8*b + 3*A*a^7*b^2)*x^3 + 
19683/2*(B*a^9 + 3*A*a^8*b)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.64 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=19683 A a^{9} x - \frac {64 B b^{9} x^{11}}{11} - \frac {16 C b^{9} x^{12}}{3} + x^{10} \left (- \frac {32 A b^{9}}{5} + \frac {648 C a^{2} b^{7}}{5}\right ) + x^{9} \cdot \left (144 B a^{2} b^{7} + 144 C a^{3} b^{6}\right ) + x^{8} \cdot \left (162 A a^{2} b^{7} + 162 B a^{3} b^{6} - \frac {2187 C a^{4} b^{5}}{2}\right ) + x^{7} \cdot \left (\frac {1296 A a^{3} b^{6}}{7} - \frac {8748 B a^{4} b^{5}}{7} - \frac {17496 C a^{5} b^{4}}{7}\right ) + x^{6} \left (- 1458 A a^{4} b^{5} - 2916 B a^{5} b^{4} + \frac {3645 C a^{6} b^{3}}{2}\right ) + x^{5} \left (- \frac {17496 A a^{5} b^{4}}{5} + 2187 B a^{6} b^{3} + \frac {59049 C a^{7} b^{2}}{5}\right ) + x^{4} \cdot \left (\frac {10935 A a^{6} b^{3}}{4} + \frac {59049 B a^{7} b^{2}}{4} + \frac {59049 C a^{8} b}{4}\right ) + x^{3} \cdot \left (19683 A a^{7} b^{2} + 19683 B a^{8} b + 6561 C a^{9}\right ) + x^{2} \cdot \left (\frac {59049 A a^{8} b}{2} + \frac {19683 B a^{9}}{2}\right ) \] Input:

integrate((C*x**2+B*x+A)*(-4*b**3*x**3+27*a**2*b*x+27*a**3)**3,x)
 

Output:

19683*A*a**9*x - 64*B*b**9*x**11/11 - 16*C*b**9*x**12/3 + x**10*(-32*A*b** 
9/5 + 648*C*a**2*b**7/5) + x**9*(144*B*a**2*b**7 + 144*C*a**3*b**6) + x**8 
*(162*A*a**2*b**7 + 162*B*a**3*b**6 - 2187*C*a**4*b**5/2) + x**7*(1296*A*a 
**3*b**6/7 - 8748*B*a**4*b**5/7 - 17496*C*a**5*b**4/7) + x**6*(-1458*A*a** 
4*b**5 - 2916*B*a**5*b**4 + 3645*C*a**6*b**3/2) + x**5*(-17496*A*a**5*b**4 
/5 + 2187*B*a**6*b**3 + 59049*C*a**7*b**2/5) + x**4*(10935*A*a**6*b**3/4 + 
 59049*B*a**7*b**2/4 + 59049*C*a**8*b/4) + x**3*(19683*A*a**7*b**2 + 19683 
*B*a**8*b + 6561*C*a**9) + x**2*(59049*A*a**8*b/2 + 19683*B*a**9/2)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.44 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=-\frac {16}{3} \, C b^{9} x^{12} - \frac {64}{11} \, B b^{9} x^{11} + 19683 \, A a^{9} x + \frac {8}{5} \, {\left (81 \, C a^{2} b^{7} - 4 \, A b^{9}\right )} x^{10} + 144 \, {\left (C a^{3} b^{6} + B a^{2} b^{7}\right )} x^{9} - \frac {81}{2} \, {\left (27 \, C a^{4} b^{5} - 4 \, B a^{3} b^{6} - 4 \, A a^{2} b^{7}\right )} x^{8} - \frac {324}{7} \, {\left (54 \, C a^{5} b^{4} + 27 \, B a^{4} b^{5} - 4 \, A a^{3} b^{6}\right )} x^{7} + \frac {729}{2} \, {\left (5 \, C a^{6} b^{3} - 8 \, B a^{5} b^{4} - 4 \, A a^{4} b^{5}\right )} x^{6} + \frac {2187}{5} \, {\left (27 \, C a^{7} b^{2} + 5 \, B a^{6} b^{3} - 8 \, A a^{5} b^{4}\right )} x^{5} + \frac {2187}{4} \, {\left (27 \, C a^{8} b + 27 \, B a^{7} b^{2} + 5 \, A a^{6} b^{3}\right )} x^{4} + 6561 \, {\left (C a^{9} + 3 \, B a^{8} b + 3 \, A a^{7} b^{2}\right )} x^{3} + \frac {19683}{2} \, {\left (B a^{9} + 3 \, A a^{8} b\right )} x^{2} \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="max 
ima")
 

Output:

-16/3*C*b^9*x^12 - 64/11*B*b^9*x^11 + 19683*A*a^9*x + 8/5*(81*C*a^2*b^7 - 
4*A*b^9)*x^10 + 144*(C*a^3*b^6 + B*a^2*b^7)*x^9 - 81/2*(27*C*a^4*b^5 - 4*B 
*a^3*b^6 - 4*A*a^2*b^7)*x^8 - 324/7*(54*C*a^5*b^4 + 27*B*a^4*b^5 - 4*A*a^3 
*b^6)*x^7 + 729/2*(5*C*a^6*b^3 - 8*B*a^5*b^4 - 4*A*a^4*b^5)*x^6 + 2187/5*( 
27*C*a^7*b^2 + 5*B*a^6*b^3 - 8*A*a^5*b^4)*x^5 + 2187/4*(27*C*a^8*b + 27*B* 
a^7*b^2 + 5*A*a^6*b^3)*x^4 + 6561*(C*a^9 + 3*B*a^8*b + 3*A*a^7*b^2)*x^3 + 
19683/2*(B*a^9 + 3*A*a^8*b)*x^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.56 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=-\frac {16}{3} \, C b^{9} x^{12} - \frac {64}{11} \, B b^{9} x^{11} + \frac {648}{5} \, C a^{2} b^{7} x^{10} - \frac {32}{5} \, A b^{9} x^{10} + 144 \, C a^{3} b^{6} x^{9} + 144 \, B a^{2} b^{7} x^{9} - \frac {2187}{2} \, C a^{4} b^{5} x^{8} + 162 \, B a^{3} b^{6} x^{8} + 162 \, A a^{2} b^{7} x^{8} - \frac {17496}{7} \, C a^{5} b^{4} x^{7} - \frac {8748}{7} \, B a^{4} b^{5} x^{7} + \frac {1296}{7} \, A a^{3} b^{6} x^{7} + \frac {3645}{2} \, C a^{6} b^{3} x^{6} - 2916 \, B a^{5} b^{4} x^{6} - 1458 \, A a^{4} b^{5} x^{6} + \frac {59049}{5} \, C a^{7} b^{2} x^{5} + 2187 \, B a^{6} b^{3} x^{5} - \frac {17496}{5} \, A a^{5} b^{4} x^{5} + \frac {59049}{4} \, C a^{8} b x^{4} + \frac {59049}{4} \, B a^{7} b^{2} x^{4} + \frac {10935}{4} \, A a^{6} b^{3} x^{4} + 6561 \, C a^{9} x^{3} + 19683 \, B a^{8} b x^{3} + 19683 \, A a^{7} b^{2} x^{3} + \frac {19683}{2} \, B a^{9} x^{2} + \frac {59049}{2} \, A a^{8} b x^{2} + 19683 \, A a^{9} x \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="gia 
c")
                                                                                    
                                                                                    
 

Output:

-16/3*C*b^9*x^12 - 64/11*B*b^9*x^11 + 648/5*C*a^2*b^7*x^10 - 32/5*A*b^9*x^ 
10 + 144*C*a^3*b^6*x^9 + 144*B*a^2*b^7*x^9 - 2187/2*C*a^4*b^5*x^8 + 162*B* 
a^3*b^6*x^8 + 162*A*a^2*b^7*x^8 - 17496/7*C*a^5*b^4*x^7 - 8748/7*B*a^4*b^5 
*x^7 + 1296/7*A*a^3*b^6*x^7 + 3645/2*C*a^6*b^3*x^6 - 2916*B*a^5*b^4*x^6 - 
1458*A*a^4*b^5*x^6 + 59049/5*C*a^7*b^2*x^5 + 2187*B*a^6*b^3*x^5 - 17496/5* 
A*a^5*b^4*x^5 + 59049/4*C*a^8*b*x^4 + 59049/4*B*a^7*b^2*x^4 + 10935/4*A*a^ 
6*b^3*x^4 + 6561*C*a^9*x^3 + 19683*B*a^8*b*x^3 + 19683*A*a^7*b^2*x^3 + 196 
83/2*B*a^9*x^2 + 59049/2*A*a^8*b*x^2 + 19683*A*a^9*x
 

Mupad [B] (verification not implemented)

Time = 12.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.30 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=\frac {19683\,a^8\,x^2\,\left (3\,A\,b+B\,a\right )}{2}-\frac {64\,B\,b^9\,x^{11}}{11}-\frac {16\,C\,b^9\,x^{12}}{3}-x^{10}\,\left (\frac {32\,A\,b^9}{5}-\frac {648\,C\,a^2\,b^7}{5}\right )+6561\,a^7\,x^3\,\left (C\,a^2+3\,B\,a\,b+3\,A\,b^2\right )+19683\,A\,a^9\,x+144\,a^2\,b^6\,x^9\,\left (B\,b+C\,a\right )-\frac {729\,a^4\,b^3\,x^6\,\left (-5\,C\,a^2+8\,B\,a\,b+4\,A\,b^2\right )}{2}+\frac {81\,a^2\,b^5\,x^8\,\left (-27\,C\,a^2+4\,B\,a\,b+4\,A\,b^2\right )}{2}+\frac {2187\,a^5\,b^2\,x^5\,\left (27\,C\,a^2+5\,B\,a\,b-8\,A\,b^2\right )}{5}-\frac {324\,a^3\,b^4\,x^7\,\left (54\,C\,a^2+27\,B\,a\,b-4\,A\,b^2\right )}{7}+\frac {2187\,a^6\,b\,x^4\,\left (27\,C\,a^2+27\,B\,a\,b+5\,A\,b^2\right )}{4} \] Input:

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

Output:

(19683*a^8*x^2*(3*A*b + B*a))/2 - (64*B*b^9*x^11)/11 - (16*C*b^9*x^12)/3 - 
 x^10*((32*A*b^9)/5 - (648*C*a^2*b^7)/5) + 6561*a^7*x^3*(3*A*b^2 + C*a^2 + 
 3*B*a*b) + 19683*A*a^9*x + 144*a^2*b^6*x^9*(B*b + C*a) - (729*a^4*b^3*x^6 
*(4*A*b^2 - 5*C*a^2 + 8*B*a*b))/2 + (81*a^2*b^5*x^8*(4*A*b^2 - 27*C*a^2 + 
4*B*a*b))/2 + (2187*a^5*b^2*x^5*(27*C*a^2 - 8*A*b^2 + 5*B*a*b))/5 - (324*a 
^3*b^4*x^7*(54*C*a^2 - 4*A*b^2 + 27*B*a*b))/7 + (2187*a^6*b*x^4*(5*A*b^2 + 
 27*C*a^2 + 27*B*a*b))/4
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.09 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3 \, dx=\frac {x \left (-24640 b^{9} c \,x^{11}-26880 b^{10} x^{10}+598752 a^{2} b^{7} c \,x^{9}-29568 a \,b^{9} x^{9}+665280 a^{3} b^{6} c \,x^{8}+665280 a^{2} b^{8} x^{8}-5051970 a^{4} b^{5} c \,x^{7}+1496880 a^{3} b^{7} x^{7}-11547360 a^{5} b^{4} c \,x^{6}-4918320 a^{4} b^{6} x^{6}+8419950 a^{6} b^{3} c \,x^{5}-20207880 a^{5} b^{5} x^{5}[phone] a^{7} b^{2} c \,x^{4}-6062364 a^{6} b^{4} x^{4}[phone] a^{8} b c \,x^{3}[phone] a^{7} b^{3} x^{3}[phone] a^{9} c \,x^{2}[phone] a^{8} b^{2} x^{2}[phone] a^{9} b x [phone] a^{10}\right )}{4620} \] Input:

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

Output:

(x*(90935460*a**10 + 181870920*a**9*b*x + 30311820*a**9*c*x**2 + 181870920 
*a**8*b**2*x**2 + 68201595*a**8*b*c*x**3 + 80831520*a**7*b**3*x**3 + 54561 
276*a**7*b**2*c*x**4 - 6062364*a**6*b**4*x**4 + 8419950*a**6*b**3*c*x**5 - 
 20207880*a**5*b**5*x**5 - 11547360*a**5*b**4*c*x**6 - 4918320*a**4*b**6*x 
**6 - 5051970*a**4*b**5*c*x**7 + 1496880*a**3*b**7*x**7 + 665280*a**3*b**6 
*c*x**8 + 665280*a**2*b**8*x**8 + 598752*a**2*b**7*c*x**9 - 29568*a*b**9*x 
**9 - 26880*b**10*x**10 - 24640*b**9*c*x**11))/4620