\(\int (A+B x) (d+e x)^4 (a+b x+c x^2) \, dx\) [705]

Optimal result

Integrand size = 23, antiderivative size = 133 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (d+e x)^6}{6 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.89 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=a A d^4 x+\frac {1}{2} d^3 (A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 \left (A c d^2+4 a B d e+6 a A e^2+b d (B d+4 A e)\right ) x^3+\frac {1}{4} d \left (B d \left (c d^2+4 b d e+6 a e^2\right )+2 A e \left (2 c d^2+e (3 b d+2 a e)\right )\right ) x^4+\frac {1}{5} e \left (A e \left (6 c d^2+e (4 b d+a e)\right )+B \left (4 c d^3+2 d e (3 b d+2 a e)\right )\right ) x^5+\frac {1}{6} e^2 \left (6 B c d^2+B e (4 b d+a e)+A e (4 c d+b e)\right ) x^6+\frac {1}{7} e^3 (4 B c d+b B e+A c e) x^7+\frac {1}{8} B c e^4 x^8 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^6 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{6 e^4}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4}\)

Input:

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

Output:

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(125)=250\).

Time = 0.74 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.04

method	result	size
norman	\(\frac {B \,e^{4} c \,x^{8}}{8}+\left (\frac {1}{7} A c \,e^{4}+\frac {1}{7} B \,e^{4} b +\frac {4}{7} B c d \,e^{3}\right ) x^{7}+\left (\frac {1}{6} A b \,e^{4}+\frac {2}{3} A c d \,e^{3}+\frac {1}{6} B \,e^{4} a +\frac {2}{3} B b d \,e^{3}+B c \,d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} A a \,e^{4}+\frac {4}{5} A b d \,e^{3}+\frac {6}{5} A c \,d^{2} e^{2}+\frac {4}{5} B a d \,e^{3}+\frac {6}{5} B b \,d^{2} e^{2}+\frac {4}{5} B c \,d^{3} e \right ) x^{5}+\left (A a d \,e^{3}+\frac {3}{2} A b \,d^{2} e^{2}+A c \,d^{3} e +\frac {3}{2} B a \,d^{2} e^{2}+B b \,d^{3} e +\frac {1}{4} B c \,d^{4}\right ) x^{4}+\left (2 A a \,d^{2} e^{2}+\frac {4}{3} A b \,d^{3} e +\frac {1}{3} A c \,d^{4}+\frac {4}{3} B a \,d^{3} e +\frac {1}{3} B b \,d^{4}\right ) x^{3}+\left (2 A a \,d^{3} e +\frac {1}{2} A b \,d^{4}+\frac {1}{2} B a \,d^{4}\right ) x^{2}+A \,d^{4} a x\)	\(271\)
default	\(\frac {B \,e^{4} c \,x^{8}}{8}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) c +B \,e^{4} b \right ) x^{7}}{7}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c +\left (A \,e^{4}+4 B d \,e^{3}\right ) b +B \,e^{4} a \right ) x^{6}}{6}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) c +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b +\left (A \,e^{4}+4 B d \,e^{3}\right ) a \right ) x^{5}}{5}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c +\left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) b +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A c \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) b +\left (6 A \,d^{2} e^{2}+4 d^{3} B e \right ) a \right ) x^{3}}{3}+\frac {\left (A b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) a \right ) x^{2}}{2}+A \,d^{4} a x\)	\(283\)
gosper	\(\frac {1}{2} A \,d^{4} b \,x^{2}+x^{6} B c \,d^{2} e^{2}+\frac {1}{3} A c \,d^{4} x^{3}+\frac {1}{4} B c \,d^{4} x^{4}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {6}{5} x^{5} B b \,d^{2} e^{2}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {2}{3} x^{6} B b d \,e^{3}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {4}{5} x^{5} A b d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{6} x^{6} B \,e^{4} a +\frac {1}{5} x^{5} A a \,e^{4}+\frac {1}{8} B \,e^{4} c \,x^{8}+x^{4} A a d \,e^{3}+x^{4} B b \,d^{3} e +\frac {4}{5} x^{5} B a d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+A \,d^{4} a x +\frac {4}{5} x^{5} B c \,d^{3} e +\frac {4}{3} x^{3} A b \,d^{3} e +\frac {1}{7} x^{7} B \,e^{4} b +2 x^{2} A a \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{2} x^{2} B a \,d^{4}+\frac {1}{3} x^{3} B b \,d^{4}\)	\(325\)
risch	\(\frac {1}{2} A \,d^{4} b \,x^{2}+x^{6} B c \,d^{2} e^{2}+\frac {1}{3} A c \,d^{4} x^{3}+\frac {1}{4} B c \,d^{4} x^{4}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {6}{5} x^{5} B b \,d^{2} e^{2}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {2}{3} x^{6} B b d \,e^{3}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {4}{5} x^{5} A b d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{6} x^{6} B \,e^{4} a +\frac {1}{5} x^{5} A a \,e^{4}+\frac {1}{8} B \,e^{4} c \,x^{8}+x^{4} A a d \,e^{3}+x^{4} B b \,d^{3} e +\frac {4}{5} x^{5} B a d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+A \,d^{4} a x +\frac {4}{5} x^{5} B c \,d^{3} e +\frac {4}{3} x^{3} A b \,d^{3} e +\frac {1}{7} x^{7} B \,e^{4} b +2 x^{2} A a \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{2} x^{2} B a \,d^{4}+\frac {1}{3} x^{3} B b \,d^{4}\)	\(325\)
parallelrisch	\(\frac {1}{2} A \,d^{4} b \,x^{2}+x^{6} B c \,d^{2} e^{2}+\frac {1}{3} A c \,d^{4} x^{3}+\frac {1}{4} B c \,d^{4} x^{4}+\frac {3}{2} x^{4} A b \,d^{2} e^{2}+\frac {6}{5} x^{5} B b \,d^{2} e^{2}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {2}{3} x^{6} B b d \,e^{3}+\frac {4}{7} x^{7} B c d \,e^{3}+\frac {4}{5} x^{5} A b d \,e^{3}+x^{4} A c \,d^{3} e +\frac {3}{2} x^{4} B a \,d^{2} e^{2}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {4}{3} x^{3} B a \,d^{3} e +\frac {1}{6} x^{6} B \,e^{4} a +\frac {1}{5} x^{5} A a \,e^{4}+\frac {1}{8} B \,e^{4} c \,x^{8}+x^{4} A a d \,e^{3}+x^{4} B b \,d^{3} e +\frac {4}{5} x^{5} B a d \,e^{3}+\frac {2}{3} x^{6} A c d \,e^{3}+A \,d^{4} a x +\frac {4}{5} x^{5} B c \,d^{3} e +\frac {4}{3} x^{3} A b \,d^{3} e +\frac {1}{7} x^{7} B \,e^{4} b +2 x^{2} A a \,d^{3} e +2 x^{3} A a \,d^{2} e^{2}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {1}{2} x^{2} B a \,d^{4}+\frac {1}{3} x^{3} B b \,d^{4}\)	\(325\)
orering	\(\frac {x \left (105 B \,e^{4} c \,x^{7}+120 A c \,e^{4} x^{6}+120 B \,e^{4} b \,x^{6}+480 B c d \,e^{3} x^{6}+140 x^{5} A b \,e^{4}+560 A c d \,e^{3} x^{5}+140 x^{5} B \,e^{4} a +560 x^{5} B b d \,e^{3}+840 B c \,d^{2} e^{2} x^{5}+168 x^{4} A a \,e^{4}+672 x^{4} A b d \,e^{3}+1008 A c \,d^{2} e^{2} x^{4}+672 x^{4} B a d \,e^{3}+1008 x^{4} B b \,d^{2} e^{2}+672 B c \,d^{3} e \,x^{4}+840 x^{3} A a d \,e^{3}+1260 x^{3} A b \,d^{2} e^{2}+840 A c \,d^{3} e \,x^{3}+1260 x^{3} B a \,d^{2} e^{2}+840 x^{3} B b \,d^{3} e +210 B c \,d^{4} x^{3}+1680 x^{2} A a \,d^{2} e^{2}+1120 x^{2} A b \,d^{3} e +280 A c \,d^{4} x^{2}+1120 x^{2} B a \,d^{3} e +280 x^{2} B b \,d^{4}+1680 x A a \,d^{3} e +420 x A b \,d^{4}+420 x B a \,d^{4}+840 A a \,d^{4}\right )}{840}\)	\(326\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.83 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, {\left (B b + A c\right )} d e^{3} + {\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + A a e^{4} + 6 \, {\left (B b + A c\right )} d^{2} e^{2} + 4 \, {\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A a d e^{3} + 4 \, {\left (B b + A c\right )} d^{3} e + 6 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a d^{2} e^{2} + {\left (B b + A c\right )} d^{4} + 4 \, {\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a d^{3} e + {\left (B a + A b\right )} d^{4}\right )} x^{2} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (128) = 256\).

Time = 0.04 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.50 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=A a d^{4} x + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {B b e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac {A b e^{4}}{6} + \frac {2 A c d e^{3}}{3} + \frac {B a e^{4}}{6} + \frac {2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac {A a e^{4}}{5} + \frac {4 A b d e^{3}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {4 B a d e^{3}}{5} + \frac {6 B b d^{2} e^{2}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + \frac {3 A b d^{2} e^{2}}{2} + A c d^{3} e + \frac {3 B a d^{2} e^{2}}{2} + B b d^{3} e + \frac {B c d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a d^{2} e^{2} + \frac {4 A b d^{3} e}{3} + \frac {A c d^{4}}{3} + \frac {4 B a d^{3} e}{3} + \frac {B b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a d^{3} e + \frac {A b d^{4}}{2} + \frac {B a d^{4}}{2}\right ) \] Input:

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

Output:

A*a*d**4*x + B*c*e**4*x**8/8 + x**7*(A*c*e**4/7 + B*b*e**4/7 + 4*B*c*d*e** 
3/7) + x**6*(A*b*e**4/6 + 2*A*c*d*e**3/3 + B*a*e**4/6 + 2*B*b*d*e**3/3 + B 
*c*d**2*e**2) + x**5*(A*a*e**4/5 + 4*A*b*d*e**3/5 + 6*A*c*d**2*e**2/5 + 4* 
B*a*d*e**3/5 + 6*B*b*d**2*e**2/5 + 4*B*c*d**3*e/5) + x**4*(A*a*d*e**3 + 3* 
A*b*d**2*e**2/2 + A*c*d**3*e + 3*B*a*d**2*e**2/2 + B*b*d**3*e + B*c*d**4/4 
) + x**3*(2*A*a*d**2*e**2 + 4*A*b*d**3*e/3 + A*c*d**4/3 + 4*B*a*d**3*e/3 + 
 B*b*d**4/3) + x**2*(2*A*a*d**3*e + A*b*d**4/2 + B*a*d**4/2)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.83 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} x^{7} + A a d^{4} x + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + 4 \, {\left (B b + A c\right )} d e^{3} + {\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B c d^{3} e + A a e^{4} + 6 \, {\left (B b + A c\right )} d^{2} e^{2} + 4 \, {\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 4 \, A a d e^{3} + 4 \, {\left (B b + A c\right )} d^{3} e + 6 \, {\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a d^{2} e^{2} + {\left (B b + A c\right )} d^{4} + 4 \, {\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a d^{3} e + {\left (B a + A b\right )} d^{4}\right )} x^{2} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (125) = 250\).

Time = 0.19 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {4}{7} \, B c d e^{3} x^{7} + \frac {1}{7} \, B b e^{4} x^{7} + \frac {1}{7} \, A c e^{4} x^{7} + B c d^{2} e^{2} x^{6} + \frac {2}{3} \, B b d e^{3} x^{6} + \frac {2}{3} \, A c d e^{3} x^{6} + \frac {1}{6} \, B a e^{4} x^{6} + \frac {1}{6} \, A b e^{4} x^{6} + \frac {4}{5} \, B c d^{3} e x^{5} + \frac {6}{5} \, B b d^{2} e^{2} x^{5} + \frac {6}{5} \, A c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a d e^{3} x^{5} + \frac {4}{5} \, A b d e^{3} x^{5} + \frac {1}{5} \, A a e^{4} x^{5} + \frac {1}{4} \, B c d^{4} x^{4} + B b d^{3} e x^{4} + A c d^{3} e x^{4} + \frac {3}{2} \, B a d^{2} e^{2} x^{4} + \frac {3}{2} \, A b d^{2} e^{2} x^{4} + A a d e^{3} x^{4} + \frac {1}{3} \, B b d^{4} x^{3} + \frac {1}{3} \, A c d^{4} x^{3} + \frac {4}{3} \, B a d^{3} e x^{3} + \frac {4}{3} \, A b d^{3} e x^{3} + 2 \, A a d^{2} e^{2} x^{3} + \frac {1}{2} \, B a d^{4} x^{2} + \frac {1}{2} \, A b d^{4} x^{2} + 2 \, A a d^{3} e x^{2} + A a d^{4} x \] Input:

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

Output:

1/8*B*c*e^4*x^8 + 4/7*B*c*d*e^3*x^7 + 1/7*B*b*e^4*x^7 + 1/7*A*c*e^4*x^7 + 
B*c*d^2*e^2*x^6 + 2/3*B*b*d*e^3*x^6 + 2/3*A*c*d*e^3*x^6 + 1/6*B*a*e^4*x^6 
+ 1/6*A*b*e^4*x^6 + 4/5*B*c*d^3*e*x^5 + 6/5*B*b*d^2*e^2*x^5 + 6/5*A*c*d^2* 
e^2*x^5 + 4/5*B*a*d*e^3*x^5 + 4/5*A*b*d*e^3*x^5 + 1/5*A*a*e^4*x^5 + 1/4*B* 
c*d^4*x^4 + B*b*d^3*e*x^4 + A*c*d^3*e*x^4 + 3/2*B*a*d^2*e^2*x^4 + 3/2*A*b* 
d^2*e^2*x^4 + A*a*d*e^3*x^4 + 1/3*B*b*d^4*x^3 + 1/3*A*c*d^4*x^3 + 4/3*B*a* 
d^3*e*x^3 + 4/3*A*b*d^3*e*x^3 + 2*A*a*d^2*e^2*x^3 + 1/2*B*a*d^4*x^2 + 1/2* 
A*b*d^4*x^2 + 2*A*a*d^3*e*x^2 + A*a*d^4*x
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.03 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {B\,b\,d^4}{3}+\frac {4\,A\,b\,d^3\,e}{3}+\frac {4\,B\,a\,d^3\,e}{3}+2\,A\,a\,d^2\,e^2\right )+x^6\,\left (\frac {A\,b\,e^4}{6}+\frac {B\,a\,e^4}{6}+\frac {2\,A\,c\,d\,e^3}{3}+\frac {2\,B\,b\,d\,e^3}{3}+B\,c\,d^2\,e^2\right )+x^2\,\left (\frac {A\,b\,d^4}{2}+\frac {B\,a\,d^4}{2}+2\,A\,a\,d^3\,e\right )+x^7\,\left (\frac {A\,c\,e^4}{7}+\frac {B\,b\,e^4}{7}+\frac {4\,B\,c\,d\,e^3}{7}\right )+x^4\,\left (\frac {B\,c\,d^4}{4}+A\,a\,d\,e^3+A\,c\,d^3\,e+B\,b\,d^3\,e+\frac {3\,A\,b\,d^2\,e^2}{2}+\frac {3\,B\,a\,d^2\,e^2}{2}\right )+x^5\,\left (\frac {A\,a\,e^4}{5}+\frac {4\,A\,b\,d\,e^3}{5}+\frac {4\,B\,a\,d\,e^3}{5}+\frac {4\,B\,c\,d^3\,e}{5}+\frac {6\,A\,c\,d^2\,e^2}{5}+\frac {6\,B\,b\,d^2\,e^2}{5}\right )+A\,a\,d^4\,x+\frac {B\,c\,e^4\,x^8}{8} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.12 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {x \left (105 b c \,e^{4} x^{7}+120 a c \,e^{4} x^{6}+120 b^{2} e^{4} x^{6}+480 b c d \,e^{3} x^{6}+280 a b \,e^{4} x^{5}+560 a c d \,e^{3} x^{5}+560 b^{2} d \,e^{3} x^{5}+840 b c \,d^{2} e^{2} x^{5}+168 a^{2} e^{4} x^{4}+1344 a b d \,e^{3} x^{4}+1008 a c \,d^{2} e^{2} x^{4}+1008 b^{2} d^{2} e^{2} x^{4}+672 b c \,d^{3} e \,x^{4}+840 a^{2} d \,e^{3} x^{3}+2520 a b \,d^{2} e^{2} x^{3}+840 a c \,d^{3} e \,x^{3}+840 b^{2} d^{3} e \,x^{3}+210 b c \,d^{4} x^{3}+1680 a^{2} d^{2} e^{2} x^{2}+2240 a b \,d^{3} e \,x^{2}+280 a c \,d^{4} x^{2}+280 b^{2} d^{4} x^{2}+1680 a^{2} d^{3} e x +840 a b \,d^{4} x +840 a^{2} d^{4}\right )}{840} \] Input:

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

Output:

(x*(840*a**2*d**4 + 1680*a**2*d**3*e*x + 1680*a**2*d**2*e**2*x**2 + 840*a* 
*2*d*e**3*x**3 + 168*a**2*e**4*x**4 + 840*a*b*d**4*x + 2240*a*b*d**3*e*x** 
2 + 2520*a*b*d**2*e**2*x**3 + 1344*a*b*d*e**3*x**4 + 280*a*b*e**4*x**5 + 2 
80*a*c*d**4*x**2 + 840*a*c*d**3*e*x**3 + 1008*a*c*d**2*e**2*x**4 + 560*a*c 
*d*e**3*x**5 + 120*a*c*e**4*x**6 + 280*b**2*d**4*x**2 + 840*b**2*d**3*e*x* 
*3 + 1008*b**2*d**2*e**2*x**4 + 560*b**2*d*e**3*x**5 + 120*b**2*e**4*x**6 
+ 210*b*c*d**4*x**3 + 672*b*c*d**3*e*x**4 + 840*b*c*d**2*e**2*x**5 + 480*b 
*c*d*e**3*x**6 + 105*b*c*e**4*x**7))/840