\(\int (A+B x) (d+e x)^4 (a+b x+c x^2)^3 \, dx\) [733]

Optimal result

Integrand size = 25, antiderivative size = 553 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) (d+e x)^6}{6 e^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^7}{7 e^8}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^8}{8 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^9}{9 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{10}}{10 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \] Output:

-1/5*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^8+1/6*(a*e^2-b*d*e+c*d^2 
)^2*(7*B*c*d^2-B*e*(-a*e+4*b*d)-3*A*e*(-b*e+2*c*d))*(e*x+d)^6/e^8-3/7*(a*e 
^2-b*d*e+c*d^2)*(B*(7*c^2*d^3-c*d*e*(-3*a*e+8*b*d)+b*e^2*(-a*e+2*b*d))-A*e 
*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))*(e*x+d)^7/e^8-1/8*(A*e*(-b*e+2*c*d) 
*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))-B*(35*c^3*d^4-b^2*e^3*(-3*a*e+4 
*b*d)-30*c^2*d^2*e*(-a*e+2*b*d)+3*c*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2)))*( 
e*x+d)^8/e^8-1/9*(B*(35*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+5*b*d)-15*c^2*d* 
e*(-a*e+3*b*d))-3*A*c*e*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))*(e*x+d)^9/e^ 
8-3/10*c*(A*c*e*(-b*e+2*c*d)-B*(7*c^2*d^2+b^2*e^2-c*e*(-a*e+6*b*d)))*(e*x+ 
d)^10/e^8-1/11*c^2*(-A*c*e-3*B*b*e+7*B*c*d)*(e*x+d)^11/e^8+1/12*B*c^3*(e*x 
+d)^12/e^8
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.73 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^3 A d^4 x+\frac {1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+\frac {1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )\right ) x^3+\frac {1}{4} d \left (3 a B d \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )+A \left (b^3 d^3+12 a b^2 d^2 e+4 a^2 e \left (3 c d^2+a e^2\right )+6 a b d \left (c d^2+3 a e^2\right )\right )\right ) x^4+\frac {1}{5} \left (b^3 d^3 (B d+4 A e)+3 b^2 d^2 \left (A c d^2+4 a B d e+6 a A e^2\right )+6 a b d \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right )+a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right )\right ) x^5+\frac {1}{6} \left (2 b^3 d^2 e (2 B d+3 A e)+12 a A c d e \left (c d^2+a e^2\right )+12 a b B d e \left (2 c d^2+a e^2\right )+3 b^2 d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right )+3 A b \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+a B \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} \left (2 b^3 d e^2 (3 B d+2 A e)+12 a B c d e \left (c d^2+a e^2\right )+12 A b c d e \left (c d^2+2 a e^2\right )+3 b^2 e \left (4 B c d^3+6 A c d^2 e+4 a B d e^2+a A e^3\right )+3 b B \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+A c \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )+B \left (c^3 d^4+6 c^2 d^2 e (2 b d+3 a e)+b^2 e^3 (4 b d+3 a e)+3 c e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )+B \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \] Input:

Integrate[(A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x]
 

Output:

a^3*A*d^4*x + (a^2*d^3*(3*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a*d^2*(a*B*d* 
(3*b*d + 4*a*e) + 3*A*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2)))*x^3)/3 
+ (d*(3*a*B*d*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2)) + A*(b^3*d^3 + 1 
2*a*b^2*d^2*e + 4*a^2*e*(3*c*d^2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2)))*x^ 
4)/4 + ((b^3*d^3*(B*d + 4*A*e) + 3*b^2*d^2*(A*c*d^2 + 4*a*B*d*e + 6*a*A*e^ 
2) + 6*a*b*d*(B*c*d^3 + 4*A*c*d^2*e + 3*a*B*d*e^2 + 2*a*A*e^3) + a*(4*a*B* 
d*e*(3*c*d^2 + a*e^2) + A*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4)))*x^5)/5 
+ ((2*b^3*d^2*e*(2*B*d + 3*A*e) + 12*a*A*c*d*e*(c*d^2 + a*e^2) + 12*a*b*B* 
d*e*(2*c*d^2 + a*e^2) + 3*b^2*d*(B*c*d^3 + 4*A*c*d^2*e + 6*a*B*d*e^2 + 4*a 
*A*e^3) + 3*A*b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + a*B*(3*c^2*d^4 + 18 
*a*c*d^2*e^2 + a^2*e^4))*x^6)/6 + ((2*b^3*d*e^2*(3*B*d + 2*A*e) + 12*a*B*c 
*d*e*(c*d^2 + a*e^2) + 12*A*b*c*d*e*(c*d^2 + 2*a*e^2) + 3*b^2*e*(4*B*c*d^3 
 + 6*A*c*d^2*e + 4*a*B*d*e^2 + a*A*e^3) + 3*b*B*(c^2*d^4 + 12*a*c*d^2*e^2 
+ a^2*e^4) + A*c*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^7)/7 + ((A*e*(4 
*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e)) 
+ B*(c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3*c 
*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 
 b^2*e^2 + c*e*(4*b*d + a*e)) + B*(4*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d 
+ a*e) + 6*c^2*d*e*(3*b*d + 2*a*e)))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e 
) + 3*B*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e)))*x^10)/10 + (c^2*e^3*...
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.

\(\displaystyle \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {(d+e x)^7 \left (B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^7}+\frac {3 c (d+e x)^9 \left (B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7}+\frac {3 (d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7}+\frac {(d+e x)^8 \left (3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )\right )}{e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}+\frac {(d+e x)^4 (A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7}+\frac {c^2 (d+e x)^{10} (A c e+3 b B e-7 B c d)}{e^7}+\frac {B c^3 (d+e x)^{11}}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(d+e x)^8 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{8 e^8}-\frac {3 c (d+e x)^{10} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{10 e^8}-\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac {(d+e x)^9 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{9 e^8}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {c^2 (d+e x)^{11} (-A c e-3 b B e+7 B c d)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8}\)

Input:

Int[(A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x]
 

Output:

-1/5*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/e^8 + ((c*d^2 - b 
*d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e))*(d + 
 e*x)^6)/(6*e^8) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d 
 - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - 
 a*e)))*(d + e*x)^7)/(7*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 
 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2 
*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e 
*x)^8)/(8*e^8) - ((B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 1 
5*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e 
)))*(d + e*x)^9)/(9*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2* 
e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^10)/(10*e^8) - (c^2*(7*B*c*d - 3*b*B*e 
 - A*c*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*e^8)
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 1041, normalized size of antiderivative = 1.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1041\)
norman	\(\text {Expression too large to display}\)	\(1110\)
gosper	\(\text {Expression too large to display}\)	\(1354\)
risch	\(\text {Expression too large to display}\)	\(1354\)
parallelrisch	\(\text {Expression too large to display}\)	\(1354\)
orering	\(\text {Expression too large to display}\)	\(1356\)

Input:

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/12*B*e^4*c^3*x^12+1/11*((A*e^4+4*B*d*e^3)*c^3+3*B*e^4*b*c^2)*x^11+1/10*( 
(4*A*d*e^3+6*B*d^2*e^2)*c^3+3*(A*e^4+4*B*d*e^3)*b*c^2+B*e^4*(a*c^2+2*b^2*c 
+c*(2*a*c+b^2)))*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(4*A*d*e^3+6*B*d^ 
2*e^2)*b*c^2+(A*e^4+4*B*d*e^3)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*e^4*(4*a*b* 
c+b*(2*a*c+b^2)))*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(6*A*d^2*e^2+4*B*d^3*e) 
*b*c^2+(4*A*d*e^3+6*B*d^2*e^2)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(A*e^4+4*B*d* 
e^3)*(4*a*b*c+b*(2*a*c+b^2))+B*e^4*(a*(2*a*c+b^2)+2*a*b^2+a^2*c))*x^8+1/7* 
(A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b*c^2+(6*A*d^2*e^2+4*B*d^3*e)*(a*c^2+2*b^2* 
c+c*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(4*a*b*c+b*(2*a*c+b^2))+(A*e^4+4* 
B*d*e^3)*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*B*e^4*a^2*b)*x^7+1/6*(3*A*d^4*b*c 
^2+(4*A*d^3*e+B*d^4)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^3*e) 
*(4*a*b*c+b*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(a*(2*a*c+b^2)+2*a*b^2+a^ 
2*c)+3*(A*e^4+4*B*d*e^3)*a^2*b+B*e^4*a^3)*x^6+1/5*(A*d^4*(a*c^2+2*b^2*c+c* 
(2*a*c+b^2))+(4*A*d^3*e+B*d^4)*(4*a*b*c+b*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^ 
3*e)*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*(4*A*d*e^3+6*B*d^2*e^2)*a^2*b+(A*e^4+ 
4*B*d*e^3)*a^3)*x^5+1/4*(A*d^4*(4*a*b*c+b*(2*a*c+b^2))+(4*A*d^3*e+B*d^4)*( 
a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*(6*A*d^2*e^2+4*B*d^3*e)*a^2*b+(4*A*d*e^3+6* 
B*d^2*e^2)*a^3)*x^4+1/3*(A*d^4*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*(4*A*d^3*e+ 
B*d^4)*a^2*b+(6*A*d^2*e^2+4*B*d^3*e)*a^3)*x^3+1/2*(3*A*a^2*b*d^4+(4*A*d^3* 
e+B*d^4)*a^3)*x^2+A*d^4*a^3*x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 
+ 1/10*(6*B*c^3*d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + (B*a 
+ A*b)*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 
 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b 
+ A*b^2)*c)*e^4)*x^9 + A*a^3*d^4*x + 1/8*(B*c^3*d^4 + 4*(3*B*b*c^2 + A*c^3 
)*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 
3*(2*B*a*b + A*b^2)*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c) 
*e^4)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + (B*a + A*b)*c^2)* 
d^3*e + 6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^2 + 4*(3*B*a*b 
^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^3 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c 
)*e^4)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d^4 + 4*(B*b^3 + 3*A*a*c^2 
 + 3*(2*B*a*b + A*b^2)*c)*d^3*e + 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a* 
b)*c)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^3 + (B*a^3 + 3*A*a^2* 
b)*e^4)*x^6 + 1/5*(A*a^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c) 
*d^4 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e + 18*(B*a^2*b + 
 A*a*b^2 + A*a^2*c)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A* 
a^3*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^4 + 12*(B*a^2*b 
+ A*a*b^2 + A*a^2*c)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A 
*a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^4 + 4*(B*a^3 + 3*A*a^2*b) 
*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (544) = 1088\).

Time = 0.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.53 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x+a)**3,x)
 

Output:

A*a**3*d**4*x + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 3*B*b*c**2* 
e**4/11 + 4*B*c**3*d*e**3/11) + x**10*(3*A*b*c**2*e**4/10 + 2*A*c**3*d*e** 
3/5 + 3*B*a*c**2*e**4/10 + 3*B*b**2*c*e**4/10 + 6*B*b*c**2*d*e**3/5 + 3*B* 
c**3*d**2*e**2/5) + x**9*(A*a*c**2*e**4/3 + A*b**2*c*e**4/3 + 4*A*b*c**2*d 
*e**3/3 + 2*A*c**3*d**2*e**2/3 + 2*B*a*b*c*e**4/3 + 4*B*a*c**2*d*e**3/3 + 
B*b**3*e**4/9 + 4*B*b**2*c*d*e**3/3 + 2*B*b*c**2*d**2*e**2 + 4*B*c**3*d**3 
*e/9) + x**8*(3*A*a*b*c*e**4/4 + 3*A*a*c**2*d*e**3/2 + A*b**3*e**4/8 + 3*A 
*b**2*c*d*e**3/2 + 9*A*b*c**2*d**2*e**2/4 + A*c**3*d**3*e/2 + 3*B*a**2*c*e 
**4/8 + 3*B*a*b**2*e**4/8 + 3*B*a*b*c*d*e**3 + 9*B*a*c**2*d**2*e**2/4 + B* 
b**3*d*e**3/2 + 9*B*b**2*c*d**2*e**2/4 + 3*B*b*c**2*d**3*e/2 + B*c**3*d**4 
/8) + x**7*(3*A*a**2*c*e**4/7 + 3*A*a*b**2*e**4/7 + 24*A*a*b*c*d*e**3/7 + 
18*A*a*c**2*d**2*e**2/7 + 4*A*b**3*d*e**3/7 + 18*A*b**2*c*d**2*e**2/7 + 12 
*A*b*c**2*d**3*e/7 + A*c**3*d**4/7 + 3*B*a**2*b*e**4/7 + 12*B*a**2*c*d*e** 
3/7 + 12*B*a*b**2*d*e**3/7 + 36*B*a*b*c*d**2*e**2/7 + 12*B*a*c**2*d**3*e/7 
 + 6*B*b**3*d**2*e**2/7 + 12*B*b**2*c*d**3*e/7 + 3*B*b*c**2*d**4/7) + x**6 
*(A*a**2*b*e**4/2 + 2*A*a**2*c*d*e**3 + 2*A*a*b**2*d*e**3 + 6*A*a*b*c*d**2 
*e**2 + 2*A*a*c**2*d**3*e + A*b**3*d**2*e**2 + 2*A*b**2*c*d**3*e + A*b*c** 
2*d**4/2 + B*a**3*e**4/6 + 2*B*a**2*b*d*e**3 + 3*B*a**2*c*d**2*e**2 + 3*B* 
a*b**2*d**2*e**2 + 4*B*a*b*c*d**3*e + B*a*c**2*d**4/2 + 2*B*b**3*d**3*e/3 
+ B*b**2*c*d**4/2) + x**5*(A*a**3*e**4/5 + 12*A*a**2*b*d*e**3/5 + 18*A*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 
+ 1/10*(6*B*c^3*d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + (B*a 
+ A*b)*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 
 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b 
+ A*b^2)*c)*e^4)*x^9 + A*a^3*d^4*x + 1/8*(B*c^3*d^4 + 4*(3*B*b*c^2 + A*c^3 
)*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 
3*(2*B*a*b + A*b^2)*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c) 
*e^4)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + (B*a + A*b)*c^2)* 
d^3*e + 6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^2 + 4*(3*B*a*b 
^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^3 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c 
)*e^4)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d^4 + 4*(B*b^3 + 3*A*a*c^2 
 + 3*(2*B*a*b + A*b^2)*c)*d^3*e + 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a* 
b)*c)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^3 + (B*a^3 + 3*A*a^2* 
b)*e^4)*x^6 + 1/5*(A*a^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c) 
*d^4 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e + 18*(B*a^2*b + 
 A*a*b^2 + A*a^2*c)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A* 
a^3*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^4 + 12*(B*a^2*b 
+ A*a*b^2 + A*a^2*c)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A 
*a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^4 + 4*(B*a^3 + 3*A*a^2*b) 
*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1353 vs. \(2 (538) = 1076\).

Time = 0.14 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.45 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/12*B*c^3*e^4*x^12 + 4/11*B*c^3*d*e^3*x^11 + 3/11*B*b*c^2*e^4*x^11 + 1/11 
*A*c^3*e^4*x^11 + 3/5*B*c^3*d^2*e^2*x^10 + 6/5*B*b*c^2*d*e^3*x^10 + 2/5*A* 
c^3*d*e^3*x^10 + 3/10*B*b^2*c*e^4*x^10 + 3/10*B*a*c^2*e^4*x^10 + 3/10*A*b* 
c^2*e^4*x^10 + 4/9*B*c^3*d^3*e*x^9 + 2*B*b*c^2*d^2*e^2*x^9 + 2/3*A*c^3*d^2 
*e^2*x^9 + 4/3*B*b^2*c*d*e^3*x^9 + 4/3*B*a*c^2*d*e^3*x^9 + 4/3*A*b*c^2*d*e 
^3*x^9 + 1/9*B*b^3*e^4*x^9 + 2/3*B*a*b*c*e^4*x^9 + 1/3*A*b^2*c*e^4*x^9 + 1 
/3*A*a*c^2*e^4*x^9 + 1/8*B*c^3*d^4*x^8 + 3/2*B*b*c^2*d^3*e*x^8 + 1/2*A*c^3 
*d^3*e*x^8 + 9/4*B*b^2*c*d^2*e^2*x^8 + 9/4*B*a*c^2*d^2*e^2*x^8 + 9/4*A*b*c 
^2*d^2*e^2*x^8 + 1/2*B*b^3*d*e^3*x^8 + 3*B*a*b*c*d*e^3*x^8 + 3/2*A*b^2*c*d 
*e^3*x^8 + 3/2*A*a*c^2*d*e^3*x^8 + 3/8*B*a*b^2*e^4*x^8 + 1/8*A*b^3*e^4*x^8 
 + 3/8*B*a^2*c*e^4*x^8 + 3/4*A*a*b*c*e^4*x^8 + 3/7*B*b*c^2*d^4*x^7 + 1/7*A 
*c^3*d^4*x^7 + 12/7*B*b^2*c*d^3*e*x^7 + 12/7*B*a*c^2*d^3*e*x^7 + 12/7*A*b* 
c^2*d^3*e*x^7 + 6/7*B*b^3*d^2*e^2*x^7 + 36/7*B*a*b*c*d^2*e^2*x^7 + 18/7*A* 
b^2*c*d^2*e^2*x^7 + 18/7*A*a*c^2*d^2*e^2*x^7 + 12/7*B*a*b^2*d*e^3*x^7 + 4/ 
7*A*b^3*d*e^3*x^7 + 12/7*B*a^2*c*d*e^3*x^7 + 24/7*A*a*b*c*d*e^3*x^7 + 3/7* 
B*a^2*b*e^4*x^7 + 3/7*A*a*b^2*e^4*x^7 + 3/7*A*a^2*c*e^4*x^7 + 1/2*B*b^2*c* 
d^4*x^6 + 1/2*B*a*c^2*d^4*x^6 + 1/2*A*b*c^2*d^4*x^6 + 2/3*B*b^3*d^3*e*x^6 
+ 4*B*a*b*c*d^3*e*x^6 + 2*A*b^2*c*d^3*e*x^6 + 2*A*a*c^2*d^3*e*x^6 + 3*B*a* 
b^2*d^2*e^2*x^6 + A*b^3*d^2*e^2*x^6 + 3*B*a^2*c*d^2*e^2*x^6 + 6*A*a*b*c*d^ 
2*e^2*x^6 + 2*B*a^2*b*d*e^3*x^6 + 2*A*a*b^2*d*e^3*x^6 + 2*A*a^2*c*d*e^3...
 

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.98 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x)
 

Output:

x^5*((A*a^3*e^4)/5 + (B*b^3*d^4)/5 + (3*A*a*c^2*d^4)/5 + (3*A*b^2*c*d^4)/5 
 + (4*A*b^3*d^3*e)/5 + (4*B*a^3*d*e^3)/5 + (18*A*a*b^2*d^2*e^2)/5 + (18*A* 
a^2*c*d^2*e^2)/5 + (18*B*a^2*b*d^2*e^2)/5 + (6*B*a*b*c*d^4)/5 + (12*A*a^2* 
b*d*e^3)/5 + (12*B*a*b^2*d^3*e)/5 + (12*B*a^2*c*d^3*e)/5 + (24*A*a*b*c*d^3 
*e)/5) + x^8*((A*b^3*e^4)/8 + (B*c^3*d^4)/8 + (3*B*a*b^2*e^4)/8 + (3*B*a^2 
*c*e^4)/8 + (A*c^3*d^3*e)/2 + (B*b^3*d*e^3)/2 + (9*A*b*c^2*d^2*e^2)/4 + (9 
*B*a*c^2*d^2*e^2)/4 + (9*B*b^2*c*d^2*e^2)/4 + (3*A*a*b*c*e^4)/4 + (3*A*a*c 
^2*d*e^3)/2 + (3*A*b^2*c*d*e^3)/2 + (3*B*b*c^2*d^3*e)/2 + 3*B*a*b*c*d*e^3) 
 + x^3*(A*a*b^2*d^4 + A*a^2*c*d^4 + B*a^2*b*d^4 + (4*B*a^3*d^3*e)/3 + 2*A* 
a^3*d^2*e^2 + 4*A*a^2*b*d^3*e) + x^10*((3*A*b*c^2*e^4)/10 + (3*B*a*c^2*e^4 
)/10 + (3*B*b^2*c*e^4)/10 + (2*A*c^3*d*e^3)/5 + (3*B*c^3*d^2*e^2)/5 + (6*B 
*b*c^2*d*e^3)/5) + x^6*((B*a^3*e^4)/6 + (A*a^2*b*e^4)/2 + (A*b*c^2*d^4)/2 
+ (B*a*c^2*d^4)/2 + (B*b^2*c*d^4)/2 + (2*B*b^3*d^3*e)/3 + A*b^3*d^2*e^2 + 
3*B*a*b^2*d^2*e^2 + 3*B*a^2*c*d^2*e^2 + 2*A*a*b^2*d*e^3 + 2*A*a*c^2*d^3*e 
+ 2*A*a^2*c*d*e^3 + 2*B*a^2*b*d*e^3 + 2*A*b^2*c*d^3*e + 6*A*a*b*c*d^2*e^2 
+ 4*B*a*b*c*d^3*e) + x^7*((A*c^3*d^4)/7 + (3*A*a*b^2*e^4)/7 + (3*A*a^2*c*e 
^4)/7 + (3*B*a^2*b*e^4)/7 + (3*B*b*c^2*d^4)/7 + (4*A*b^3*d*e^3)/7 + (6*B*b 
^3*d^2*e^2)/7 + (18*A*a*c^2*d^2*e^2)/7 + (18*A*b^2*c*d^2*e^2)/7 + (12*B*a* 
b^2*d*e^3)/7 + (12*A*b*c^2*d^3*e)/7 + (12*B*a*c^2*d^3*e)/7 + (12*B*a^2*c*d 
*e^3)/7 + (12*B*b^2*c*d^3*e)/7 + (36*B*a*b*c*d^2*e^2)/7 + (24*A*a*b*c*d...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.70 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^3,x)
 

Output:

(x*(27720*a**4*d**4 + 55440*a**4*d**3*e*x + 55440*a**4*d**2*e**2*x**2 + 27 
720*a**4*d*e**3*x**3 + 5544*a**4*e**4*x**4 + 55440*a**3*b*d**4*x + 147840* 
a**3*b*d**3*e*x**2 + 166320*a**3*b*d**2*e**2*x**3 + 88704*a**3*b*d*e**3*x* 
*4 + 18480*a**3*b*e**4*x**5 + 27720*a**3*c*d**4*x**2 + 83160*a**3*c*d**3*e 
*x**3 + 99792*a**3*c*d**2*e**2*x**4 + 55440*a**3*c*d*e**3*x**5 + 11880*a** 
3*c*e**4*x**6 + 55440*a**2*b**2*d**4*x**2 + 166320*a**2*b**2*d**3*e*x**3 + 
 199584*a**2*b**2*d**2*e**2*x**4 + 110880*a**2*b**2*d*e**3*x**5 + 23760*a* 
*2*b**2*e**4*x**6 + 62370*a**2*b*c*d**4*x**3 + 199584*a**2*b*c*d**3*e*x**4 
 + 249480*a**2*b*c*d**2*e**2*x**5 + 142560*a**2*b*c*d*e**3*x**6 + 31185*a* 
*2*b*c*e**4*x**7 + 16632*a**2*c**2*d**4*x**4 + 55440*a**2*c**2*d**3*e*x**5 
 + 71280*a**2*c**2*d**2*e**2*x**6 + 41580*a**2*c**2*d*e**3*x**7 + 9240*a** 
2*c**2*e**4*x**8 + 27720*a*b**3*d**4*x**3 + 88704*a*b**3*d**3*e*x**4 + 110 
880*a*b**3*d**2*e**2*x**5 + 63360*a*b**3*d*e**3*x**6 + 13860*a*b**3*e**4*x 
**7 + 49896*a*b**2*c*d**4*x**4 + 166320*a*b**2*c*d**3*e*x**5 + 213840*a*b* 
*2*c*d**2*e**2*x**6 + 124740*a*b**2*c*d*e**3*x**7 + 27720*a*b**2*c*e**4*x* 
*8 + 27720*a*b*c**2*d**4*x**5 + 95040*a*b*c**2*d**3*e*x**6 + 124740*a*b*c* 
*2*d**2*e**2*x**7 + 73920*a*b*c**2*d*e**3*x**8 + 16632*a*b*c**2*e**4*x**9 
+ 3960*a*c**3*d**4*x**6 + 13860*a*c**3*d**3*e*x**7 + 18480*a*c**3*d**2*e** 
2*x**8 + 11088*a*c**3*d*e**3*x**9 + 2520*a*c**3*e**4*x**10 + 5544*b**4*d** 
4*x**4 + 18480*b**4*d**3*e*x**5 + 23760*b**4*d**2*e**2*x**6 + 13860*b**...