\(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx\) [294]

Optimal result
Mathematica [B] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [B] (verification not implemented)
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 31, antiderivative size = 206 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {(b d-a e)^4 (B d-A e) (d+e x)^6}{6 e^6}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^7}{7 e^6}-\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^8}{4 e^6}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^9}{9 e^6}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{10}}{10 e^6}+\frac {b^4 B (d+e x)^{11}}{11 e^6} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(615\) vs. \(2(206)=412\).

Time = 0.24 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.99 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 A d^5 x+\frac {1}{2} a^3 d^4 (4 A b d+a B d+5 a A e) x^2+\frac {1}{3} a^2 d^3 \left (a B d (4 b d+5 a e)+2 A \left (3 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x^3+\frac {1}{2} a d^2 \left (a B d \left (3 b^2 d^2+10 a b d e+5 a^2 e^2\right )+A \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )\right ) x^4+\frac {1}{5} d \left (2 a B d \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )+A \left (b^4 d^4+20 a b^3 d^3 e+60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4\right )\right ) x^5+\frac {1}{6} \left (60 a^2 b^2 d^2 e^2 (B d+A e)+20 a^3 b d e^3 (2 B d+A e)+a^4 e^4 (5 B d+A e)+20 a b^3 d^3 e (B d+2 A e)+b^4 d^4 (B d+5 A e)\right ) x^6+\frac {1}{7} e \left (a^4 B e^4+40 a b^3 d^2 e (B d+A e)+30 a^2 b^2 d e^2 (2 B d+A e)+4 a^3 b e^3 (5 B d+A e)+5 b^4 d^3 (B d+2 A e)\right ) x^7+\frac {1}{4} b e^2 \left (2 a^3 B e^3+5 b^3 d^2 (B d+A e)+10 a b^2 d e (2 B d+A e)+3 a^2 b e^2 (5 B d+A e)\right ) x^8+\frac {1}{9} b^2 e^3 \left (6 a^2 B e^2+5 b^2 d (2 B d+A e)+4 a b e (5 B d+A e)\right ) x^9+\frac {1}{10} b^3 e^4 (5 b B d+A b e+4 a B e) x^{10}+\frac {1}{11} b^4 B e^5 x^{11} \] Input:

Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
 

Output:

a^4*A*d^5*x + (a^3*d^4*(4*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^2*d^3*(a*B* 
d*(4*b*d + 5*a*e) + 2*A*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a* 
d^2*(a*B*d*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2) + A*(2*b^3*d^3 + 15*a*b^2* 
d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3))*x^4)/2 + (d*(2*a*B*d*(2*b^3*d^3 + 15* 
a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3) + A*(b^4*d^4 + 20*a*b^3*d^3*e + 
60*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4))*x^5)/5 + ((60*a^2*b^2*d^ 
2*e^2*(B*d + A*e) + 20*a^3*b*d*e^3*(2*B*d + A*e) + a^4*e^4*(5*B*d + A*e) + 
 20*a*b^3*d^3*e*(B*d + 2*A*e) + b^4*d^4*(B*d + 5*A*e))*x^6)/6 + (e*(a^4*B* 
e^4 + 40*a*b^3*d^2*e*(B*d + A*e) + 30*a^2*b^2*d*e^2*(2*B*d + A*e) + 4*a^3* 
b*e^3*(5*B*d + A*e) + 5*b^4*d^3*(B*d + 2*A*e))*x^7)/7 + (b*e^2*(2*a^3*B*e^ 
3 + 5*b^3*d^2*(B*d + A*e) + 10*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*(5*B* 
d + A*e))*x^8)/4 + (b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) + 4*a*b*e 
*(5*B*d + A*e))*x^9)/9 + (b^3*e^4*(5*b*B*d + A*b*e + 4*a*B*e)*x^10)/10 + ( 
b^4*B*e^5*x^11)/11
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 86, 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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x) (d+e x)^5 \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int b^4 (a+b x)^4 (A+B x) (d+e x)^5dx}{b^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (a+b x)^4 (A+B x) (d+e x)^5dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac {2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac {(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac {(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac {b^4 B (d+e x)^{11}}{11 e^6}\)

Input:

Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 1184
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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. \(691\) vs. \(2(194)=388\).

Time = 1.15 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.36

method result size
default \(\frac {B \,b^{4} e^{5} x^{11}}{11}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) b^{4}+4 B a \,b^{3} e^{5}\right ) x^{10}}{10}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b^{4}+4 \left (A \,e^{5}+5 B d \,e^{4}\right ) a \,b^{3}+6 B \,a^{2} b^{2} e^{5}\right ) x^{9}}{9}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b^{4}+4 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a \,b^{3}+6 \left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2} b^{2}+4 B \,a^{3} b \,e^{5}\right ) x^{8}}{8}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 d^{4} B e \right ) b^{4}+4 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a \,b^{3}+6 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2} b^{2}+4 \left (A \,e^{5}+5 B d \,e^{4}\right ) a^{3} b +B \,e^{5} a^{4}\right ) x^{7}}{7}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) b^{4}+4 \left (10 A \,d^{3} e^{2}+5 d^{4} B e \right ) a \,b^{3}+6 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2} b^{2}+4 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{3} b +\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{5}+4 \left (5 A \,d^{4} e +B \,d^{5}\right ) a \,b^{3}+6 \left (10 A \,d^{3} e^{2}+5 d^{4} B e \right ) a^{2} b^{2}+4 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{3} b +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{4}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{5}+6 \left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2} b^{2}+4 \left (10 A \,d^{3} e^{2}+5 d^{4} B e \right ) a^{3} b +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{4}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{5}+4 \left (5 A \,d^{4} e +B \,d^{5}\right ) a^{3} b +\left (10 A \,d^{3} e^{2}+5 d^{4} B e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{5}+\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{4}\right ) x^{2}}{2}+a^{4} A \,d^{5} x\) \(692\)
norman \(\frac {B \,b^{4} e^{5} x^{11}}{11}+\left (\frac {1}{10} A \,b^{4} e^{5}+\frac {2}{5} B a \,b^{3} e^{5}+\frac {1}{2} B \,b^{4} d \,e^{4}\right ) x^{10}+\left (\frac {4}{9} A a \,b^{3} e^{5}+\frac {5}{9} A \,b^{4} d \,e^{4}+\frac {2}{3} B \,a^{2} b^{2} e^{5}+\frac {20}{9} B a \,b^{3} d \,e^{4}+\frac {10}{9} B \,b^{4} d^{2} e^{3}\right ) x^{9}+\left (\frac {3}{4} A \,a^{2} b^{2} e^{5}+\frac {5}{2} A a \,b^{3} d \,e^{4}+\frac {5}{4} A \,b^{4} d^{2} e^{3}+\frac {1}{2} B \,a^{3} b \,e^{5}+\frac {15}{4} B \,a^{2} b^{2} d \,e^{4}+5 B a \,b^{3} d^{2} e^{3}+\frac {5}{4} B \,b^{4} d^{3} e^{2}\right ) x^{8}+\left (\frac {4}{7} A \,a^{3} b \,e^{5}+\frac {30}{7} A \,a^{2} b^{2} d \,e^{4}+\frac {40}{7} A a \,b^{3} d^{2} e^{3}+\frac {10}{7} A \,b^{4} d^{3} e^{2}+\frac {1}{7} B \,e^{5} a^{4}+\frac {20}{7} B \,a^{3} b d \,e^{4}+\frac {60}{7} B \,a^{2} b^{2} d^{2} e^{3}+\frac {40}{7} B a \,b^{3} d^{3} e^{2}+\frac {5}{7} B \,b^{4} d^{4} e \right ) x^{7}+\left (\frac {1}{6} a^{4} A \,e^{5}+\frac {10}{3} A \,a^{3} b d \,e^{4}+10 A \,a^{2} b^{2} d^{2} e^{3}+\frac {20}{3} A a \,b^{3} d^{3} e^{2}+\frac {5}{6} A \,b^{4} d^{4} e +\frac {5}{6} B \,a^{4} d \,e^{4}+\frac {20}{3} B \,a^{3} b \,d^{2} e^{3}+10 B \,a^{2} b^{2} d^{3} e^{2}+\frac {10}{3} B a \,b^{3} d^{4} e +\frac {1}{6} B \,b^{4} d^{5}\right ) x^{6}+\left (a^{4} A d \,e^{4}+8 A \,a^{3} b \,d^{2} e^{3}+12 A \,a^{2} b^{2} d^{3} e^{2}+4 A a \,b^{3} d^{4} e +\frac {1}{5} A \,b^{4} d^{5}+2 B \,a^{4} d^{2} e^{3}+8 B \,a^{3} b \,d^{3} e^{2}+6 B \,a^{2} b^{2} d^{4} e +\frac {4}{5} B a \,b^{3} d^{5}\right ) x^{5}+\left (\frac {5}{2} a^{4} A \,d^{2} e^{3}+10 A \,a^{3} b \,d^{3} e^{2}+\frac {15}{2} A \,a^{2} b^{2} d^{4} e +A a \,b^{3} d^{5}+\frac {5}{2} B \,a^{4} d^{3} e^{2}+5 B \,a^{3} b \,d^{4} e +\frac {3}{2} B \,a^{2} b^{2} d^{5}\right ) x^{4}+\left (\frac {10}{3} a^{4} A \,d^{3} e^{2}+\frac {20}{3} A \,a^{3} b \,d^{4} e +2 A \,a^{2} b^{2} d^{5}+\frac {5}{3} B \,a^{4} d^{4} e +\frac {4}{3} B \,a^{3} b \,d^{5}\right ) x^{3}+\left (\frac {5}{2} a^{4} A \,d^{4} e +2 A \,a^{3} b \,d^{5}+\frac {1}{2} B \,a^{4} d^{5}\right ) x^{2}+a^{4} A \,d^{5} x\) \(728\)
gosper \(\frac {x \left (1260 B \,b^{4} e^{5} x^{10}+1386 x^{9} A \,b^{4} e^{5}+5544 x^{9} B a \,b^{3} e^{5}+6930 x^{9} B \,b^{4} d \,e^{4}+6160 x^{8} A a \,b^{3} e^{5}+7700 x^{8} A \,b^{4} d \,e^{4}+9240 x^{8} B \,a^{2} b^{2} e^{5}+30800 x^{8} B a \,b^{3} d \,e^{4}+15400 x^{8} B \,b^{4} d^{2} e^{3}+10395 x^{7} A \,a^{2} b^{2} e^{5}+34650 x^{7} A a \,b^{3} d \,e^{4}+17325 x^{7} A \,b^{4} d^{2} e^{3}+6930 x^{7} B \,a^{3} b \,e^{5}+51975 x^{7} B \,a^{2} b^{2} d \,e^{4}+69300 x^{7} B a \,b^{3} d^{2} e^{3}+17325 x^{7} B \,b^{4} d^{3} e^{2}+7920 x^{6} A \,a^{3} b \,e^{5}+59400 x^{6} A \,a^{2} b^{2} d \,e^{4}+79200 x^{6} A a \,b^{3} d^{2} e^{3}+19800 x^{6} A \,b^{4} d^{3} e^{2}+1980 x^{6} B \,e^{5} a^{4}+39600 x^{6} B \,a^{3} b d \,e^{4}+118800 x^{6} B \,a^{2} b^{2} d^{2} e^{3}+79200 x^{6} B a \,b^{3} d^{3} e^{2}+9900 x^{6} B \,b^{4} d^{4} e +2310 x^{5} a^{4} A \,e^{5}+46200 x^{5} A \,a^{3} b d \,e^{4}+138600 x^{5} A \,a^{2} b^{2} d^{2} e^{3}+92400 x^{5} A a \,b^{3} d^{3} e^{2}+11550 x^{5} A \,b^{4} d^{4} e +11550 x^{5} B \,a^{4} d \,e^{4}+92400 x^{5} B \,a^{3} b \,d^{2} e^{3}+138600 x^{5} B \,a^{2} b^{2} d^{3} e^{2}+46200 x^{5} B a \,b^{3} d^{4} e +2310 x^{5} B \,b^{4} d^{5}+13860 x^{4} a^{4} A d \,e^{4}+110880 x^{4} A \,a^{3} b \,d^{2} e^{3}+166320 x^{4} A \,a^{2} b^{2} d^{3} e^{2}+55440 x^{4} A a \,b^{3} d^{4} e +2772 x^{4} A \,b^{4} d^{5}+27720 x^{4} B \,a^{4} d^{2} e^{3}+110880 x^{4} B \,a^{3} b \,d^{3} e^{2}+83160 x^{4} B \,a^{2} b^{2} d^{4} e +11088 x^{4} B a \,b^{3} d^{5}+34650 x^{3} a^{4} A \,d^{2} e^{3}+138600 x^{3} A \,a^{3} b \,d^{3} e^{2}+103950 x^{3} A \,a^{2} b^{2} d^{4} e +13860 x^{3} A a \,b^{3} d^{5}+34650 x^{3} B \,a^{4} d^{3} e^{2}+69300 x^{3} B \,a^{3} b \,d^{4} e +20790 x^{3} B \,a^{2} b^{2} d^{5}+46200 x^{2} a^{4} A \,d^{3} e^{2}+92400 x^{2} A \,a^{3} b \,d^{4} e +27720 x^{2} A \,a^{2} b^{2} d^{5}+23100 x^{2} B \,a^{4} d^{4} e +18480 x^{2} B \,a^{3} b \,d^{5}+34650 x \,a^{4} A \,d^{4} e +27720 x A \,a^{3} b \,d^{5}+6930 x B \,a^{4} d^{5}+13860 a^{4} A \,d^{5}\right )}{13860}\) \(856\)
risch \(\frac {1}{2} x^{2} B \,a^{4} d^{5}+\frac {1}{2} x^{8} B \,a^{3} b \,e^{5}+\frac {5}{4} x^{8} B \,b^{4} d^{3} e^{2}+\frac {4}{7} x^{7} A \,a^{3} b \,e^{5}+\frac {10}{7} x^{7} A \,b^{4} d^{3} e^{2}+\frac {4}{9} x^{9} A a \,b^{3} e^{5}+\frac {5}{9} x^{9} A \,b^{4} d \,e^{4}+\frac {2}{3} x^{9} B \,a^{2} b^{2} e^{5}+\frac {10}{9} x^{9} B \,b^{4} d^{2} e^{3}+\frac {3}{4} x^{8} A \,a^{2} b^{2} e^{5}+\frac {2}{5} x^{10} B a \,b^{3} e^{5}+\frac {1}{2} x^{10} B \,b^{4} d \,e^{4}+\frac {1}{10} x^{10} A \,b^{4} e^{5}+\frac {1}{7} x^{7} B \,e^{5} a^{4}+\frac {1}{6} x^{6} a^{4} A \,e^{5}+\frac {1}{6} x^{6} B \,b^{4} d^{5}+\frac {1}{5} x^{5} A \,b^{4} d^{5}+\frac {5}{2} x^{2} a^{4} A \,d^{4} e +2 x^{2} A \,a^{3} b \,d^{5}+a^{4} A \,d^{5} x +\frac {4}{5} x^{5} B a \,b^{3} d^{5}+\frac {5}{2} x^{4} a^{4} A \,d^{2} e^{3}+x^{4} A a \,b^{3} d^{5}+\frac {5}{2} x^{4} B \,a^{4} d^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b^{2} d^{5}+\frac {10}{3} x^{3} a^{4} A \,d^{3} e^{2}+2 x^{3} A \,a^{2} b^{2} d^{5}+\frac {5}{3} x^{3} B \,a^{4} d^{4} e +\frac {4}{3} x^{3} B \,a^{3} b \,d^{5}+\frac {5}{4} x^{8} A \,b^{4} d^{2} e^{3}+\frac {60}{7} x^{7} B \,a^{2} b^{2} d^{2} e^{3}+\frac {40}{7} x^{7} B a \,b^{3} d^{3} e^{2}+\frac {10}{3} x^{6} A \,a^{3} b d \,e^{4}+10 x^{6} A \,a^{2} b^{2} d^{2} e^{3}+10 x^{6} B \,a^{2} b^{2} d^{3} e^{2}+\frac {10}{3} x^{6} B a \,b^{3} d^{4} e +8 x^{5} A \,a^{3} b \,d^{2} e^{3}+12 x^{5} A \,a^{2} b^{2} d^{3} e^{2}+4 x^{5} A a \,b^{3} d^{4} e +8 x^{5} B \,a^{3} b \,d^{3} e^{2}+6 x^{5} B \,a^{2} b^{2} d^{4} e +10 x^{4} A \,a^{3} b \,d^{3} e^{2}+\frac {15}{2} x^{4} A \,a^{2} b^{2} d^{4} e +\frac {20}{3} x^{6} B \,a^{3} b \,d^{2} e^{3}+\frac {5}{7} x^{7} B \,b^{4} d^{4} e +\frac {5}{6} x^{6} A \,b^{4} d^{4} e +\frac {5}{6} x^{6} B \,a^{4} d \,e^{4}+x^{5} a^{4} A d \,e^{4}+2 x^{5} B \,a^{4} d^{2} e^{3}+\frac {1}{11} B \,b^{4} e^{5} x^{11}+\frac {20}{3} x^{6} A a \,b^{3} d^{3} e^{2}+\frac {20}{9} x^{9} B a \,b^{3} d \,e^{4}+\frac {5}{2} x^{8} A a \,b^{3} d \,e^{4}+\frac {15}{4} x^{8} B \,a^{2} b^{2} d \,e^{4}+5 x^{8} B a \,b^{3} d^{2} e^{3}+5 x^{4} B \,a^{3} b \,d^{4} e +\frac {20}{3} x^{3} A \,a^{3} b \,d^{4} e +\frac {30}{7} x^{7} A \,a^{2} b^{2} d \,e^{4}+\frac {40}{7} x^{7} A a \,b^{3} d^{2} e^{3}+\frac {20}{7} x^{7} B \,a^{3} b d \,e^{4}\) \(857\)
parallelrisch \(\frac {1}{2} x^{2} B \,a^{4} d^{5}+\frac {1}{2} x^{8} B \,a^{3} b \,e^{5}+\frac {5}{4} x^{8} B \,b^{4} d^{3} e^{2}+\frac {4}{7} x^{7} A \,a^{3} b \,e^{5}+\frac {10}{7} x^{7} A \,b^{4} d^{3} e^{2}+\frac {4}{9} x^{9} A a \,b^{3} e^{5}+\frac {5}{9} x^{9} A \,b^{4} d \,e^{4}+\frac {2}{3} x^{9} B \,a^{2} b^{2} e^{5}+\frac {10}{9} x^{9} B \,b^{4} d^{2} e^{3}+\frac {3}{4} x^{8} A \,a^{2} b^{2} e^{5}+\frac {2}{5} x^{10} B a \,b^{3} e^{5}+\frac {1}{2} x^{10} B \,b^{4} d \,e^{4}+\frac {1}{10} x^{10} A \,b^{4} e^{5}+\frac {1}{7} x^{7} B \,e^{5} a^{4}+\frac {1}{6} x^{6} a^{4} A \,e^{5}+\frac {1}{6} x^{6} B \,b^{4} d^{5}+\frac {1}{5} x^{5} A \,b^{4} d^{5}+\frac {5}{2} x^{2} a^{4} A \,d^{4} e +2 x^{2} A \,a^{3} b \,d^{5}+a^{4} A \,d^{5} x +\frac {4}{5} x^{5} B a \,b^{3} d^{5}+\frac {5}{2} x^{4} a^{4} A \,d^{2} e^{3}+x^{4} A a \,b^{3} d^{5}+\frac {5}{2} x^{4} B \,a^{4} d^{3} e^{2}+\frac {3}{2} x^{4} B \,a^{2} b^{2} d^{5}+\frac {10}{3} x^{3} a^{4} A \,d^{3} e^{2}+2 x^{3} A \,a^{2} b^{2} d^{5}+\frac {5}{3} x^{3} B \,a^{4} d^{4} e +\frac {4}{3} x^{3} B \,a^{3} b \,d^{5}+\frac {5}{4} x^{8} A \,b^{4} d^{2} e^{3}+\frac {60}{7} x^{7} B \,a^{2} b^{2} d^{2} e^{3}+\frac {40}{7} x^{7} B a \,b^{3} d^{3} e^{2}+\frac {10}{3} x^{6} A \,a^{3} b d \,e^{4}+10 x^{6} A \,a^{2} b^{2} d^{2} e^{3}+10 x^{6} B \,a^{2} b^{2} d^{3} e^{2}+\frac {10}{3} x^{6} B a \,b^{3} d^{4} e +8 x^{5} A \,a^{3} b \,d^{2} e^{3}+12 x^{5} A \,a^{2} b^{2} d^{3} e^{2}+4 x^{5} A a \,b^{3} d^{4} e +8 x^{5} B \,a^{3} b \,d^{3} e^{2}+6 x^{5} B \,a^{2} b^{2} d^{4} e +10 x^{4} A \,a^{3} b \,d^{3} e^{2}+\frac {15}{2} x^{4} A \,a^{2} b^{2} d^{4} e +\frac {20}{3} x^{6} B \,a^{3} b \,d^{2} e^{3}+\frac {5}{7} x^{7} B \,b^{4} d^{4} e +\frac {5}{6} x^{6} A \,b^{4} d^{4} e +\frac {5}{6} x^{6} B \,a^{4} d \,e^{4}+x^{5} a^{4} A d \,e^{4}+2 x^{5} B \,a^{4} d^{2} e^{3}+\frac {1}{11} B \,b^{4} e^{5} x^{11}+\frac {20}{3} x^{6} A a \,b^{3} d^{3} e^{2}+\frac {20}{9} x^{9} B a \,b^{3} d \,e^{4}+\frac {5}{2} x^{8} A a \,b^{3} d \,e^{4}+\frac {15}{4} x^{8} B \,a^{2} b^{2} d \,e^{4}+5 x^{8} B a \,b^{3} d^{2} e^{3}+5 x^{4} B \,a^{3} b \,d^{4} e +\frac {20}{3} x^{3} A \,a^{3} b \,d^{4} e +\frac {30}{7} x^{7} A \,a^{2} b^{2} d \,e^{4}+\frac {40}{7} x^{7} A a \,b^{3} d^{2} e^{3}+\frac {20}{7} x^{7} B \,a^{3} b d \,e^{4}\) \(857\)
orering \(\frac {x \left (1260 B \,b^{4} e^{5} x^{10}+1386 x^{9} A \,b^{4} e^{5}+5544 x^{9} B a \,b^{3} e^{5}+6930 x^{9} B \,b^{4} d \,e^{4}+6160 x^{8} A a \,b^{3} e^{5}+7700 x^{8} A \,b^{4} d \,e^{4}+9240 x^{8} B \,a^{2} b^{2} e^{5}+30800 x^{8} B a \,b^{3} d \,e^{4}+15400 x^{8} B \,b^{4} d^{2} e^{3}+10395 x^{7} A \,a^{2} b^{2} e^{5}+34650 x^{7} A a \,b^{3} d \,e^{4}+17325 x^{7} A \,b^{4} d^{2} e^{3}+6930 x^{7} B \,a^{3} b \,e^{5}+51975 x^{7} B \,a^{2} b^{2} d \,e^{4}+69300 x^{7} B a \,b^{3} d^{2} e^{3}+17325 x^{7} B \,b^{4} d^{3} e^{2}+7920 x^{6} A \,a^{3} b \,e^{5}+59400 x^{6} A \,a^{2} b^{2} d \,e^{4}+79200 x^{6} A a \,b^{3} d^{2} e^{3}+19800 x^{6} A \,b^{4} d^{3} e^{2}+1980 x^{6} B \,e^{5} a^{4}+39600 x^{6} B \,a^{3} b d \,e^{4}+118800 x^{6} B \,a^{2} b^{2} d^{2} e^{3}+79200 x^{6} B a \,b^{3} d^{3} e^{2}+9900 x^{6} B \,b^{4} d^{4} e +2310 x^{5} a^{4} A \,e^{5}+46200 x^{5} A \,a^{3} b d \,e^{4}+138600 x^{5} A \,a^{2} b^{2} d^{2} e^{3}+92400 x^{5} A a \,b^{3} d^{3} e^{2}+11550 x^{5} A \,b^{4} d^{4} e +11550 x^{5} B \,a^{4} d \,e^{4}+92400 x^{5} B \,a^{3} b \,d^{2} e^{3}+138600 x^{5} B \,a^{2} b^{2} d^{3} e^{2}+46200 x^{5} B a \,b^{3} d^{4} e +2310 x^{5} B \,b^{4} d^{5}+13860 x^{4} a^{4} A d \,e^{4}+110880 x^{4} A \,a^{3} b \,d^{2} e^{3}+166320 x^{4} A \,a^{2} b^{2} d^{3} e^{2}+55440 x^{4} A a \,b^{3} d^{4} e +2772 x^{4} A \,b^{4} d^{5}+27720 x^{4} B \,a^{4} d^{2} e^{3}+110880 x^{4} B \,a^{3} b \,d^{3} e^{2}+83160 x^{4} B \,a^{2} b^{2} d^{4} e +11088 x^{4} B a \,b^{3} d^{5}+34650 x^{3} a^{4} A \,d^{2} e^{3}+138600 x^{3} A \,a^{3} b \,d^{3} e^{2}+103950 x^{3} A \,a^{2} b^{2} d^{4} e +13860 x^{3} A a \,b^{3} d^{5}+34650 x^{3} B \,a^{4} d^{3} e^{2}+69300 x^{3} B \,a^{3} b \,d^{4} e +20790 x^{3} B \,a^{2} b^{2} d^{5}+46200 x^{2} a^{4} A \,d^{3} e^{2}+92400 x^{2} A \,a^{3} b \,d^{4} e +27720 x^{2} A \,a^{2} b^{2} d^{5}+23100 x^{2} B \,a^{4} d^{4} e +18480 x^{2} B \,a^{3} b \,d^{5}+34650 x \,a^{4} A \,d^{4} e +27720 x A \,a^{3} b \,d^{5}+6930 x B \,a^{4} d^{5}+13860 a^{4} A \,d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{13860 \left (b x +a \right )^{4}}\) \(881\)

Input:

int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (194) = 388\).

Time = 0.10 (sec) , antiderivative size = 686, normalized size of antiderivative = 3.33 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
 

Output:

1/11*B*b^4*e^5*x^11 + A*a^4*d^5*x + 1/10*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b 
^4)*e^5)*x^10 + 1/9*(10*B*b^4*d^2*e^3 + 5*(4*B*a*b^3 + A*b^4)*d*e^4 + 2*(3 
*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^9 + 1/4*(5*B*b^4*d^3*e^2 + 5*(4*B*a*b^3 + A 
*b^4)*d^2*e^3 + 5*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b 
^2)*e^5)*x^8 + 1/7*(5*B*b^4*d^4*e + 10*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 20*(3 
*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B* 
a^4 + 4*A*a^3*b)*e^5)*x^7 + 1/6*(B*b^4*d^5 + A*a^4*e^5 + 5*(4*B*a*b^3 + A* 
b^4)*d^4*e + 20*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A*a^ 
2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4)*x^6 + 1/5*(5*A*a^4*d*e^4 + ( 
4*B*a*b^3 + A*b^4)*d^5 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e + 20*(2*B*a^3* 
b + 3*A*a^2*b^2)*d^3*e^2 + 10*(B*a^4 + 4*A*a^3*b)*d^2*e^3)*x^5 + 1/2*(5*A* 
a^4*d^2*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^5 + 5*(2*B*a^3*b + 3*A*a^2*b^2)* 
d^4*e + 5*(B*a^4 + 4*A*a^3*b)*d^3*e^2)*x^4 + 1/3*(10*A*a^4*d^3*e^2 + 2*(2* 
B*a^3*b + 3*A*a^2*b^2)*d^5 + 5*(B*a^4 + 4*A*a^3*b)*d^4*e)*x^3 + 1/2*(5*A*a 
^4*d^4*e + (B*a^4 + 4*A*a^3*b)*d^5)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (202) = 404\).

Time = 0.06 (sec) , antiderivative size = 884, normalized size of antiderivative = 4.29 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)
 

Output:

A*a**4*d**5*x + B*b**4*e**5*x**11/11 + x**10*(A*b**4*e**5/10 + 2*B*a*b**3* 
e**5/5 + B*b**4*d*e**4/2) + x**9*(4*A*a*b**3*e**5/9 + 5*A*b**4*d*e**4/9 + 
2*B*a**2*b**2*e**5/3 + 20*B*a*b**3*d*e**4/9 + 10*B*b**4*d**2*e**3/9) + x** 
8*(3*A*a**2*b**2*e**5/4 + 5*A*a*b**3*d*e**4/2 + 5*A*b**4*d**2*e**3/4 + B*a 
**3*b*e**5/2 + 15*B*a**2*b**2*d*e**4/4 + 5*B*a*b**3*d**2*e**3 + 5*B*b**4*d 
**3*e**2/4) + x**7*(4*A*a**3*b*e**5/7 + 30*A*a**2*b**2*d*e**4/7 + 40*A*a*b 
**3*d**2*e**3/7 + 10*A*b**4*d**3*e**2/7 + B*a**4*e**5/7 + 20*B*a**3*b*d*e* 
*4/7 + 60*B*a**2*b**2*d**2*e**3/7 + 40*B*a*b**3*d**3*e**2/7 + 5*B*b**4*d** 
4*e/7) + x**6*(A*a**4*e**5/6 + 10*A*a**3*b*d*e**4/3 + 10*A*a**2*b**2*d**2* 
e**3 + 20*A*a*b**3*d**3*e**2/3 + 5*A*b**4*d**4*e/6 + 5*B*a**4*d*e**4/6 + 2 
0*B*a**3*b*d**2*e**3/3 + 10*B*a**2*b**2*d**3*e**2 + 10*B*a*b**3*d**4*e/3 + 
 B*b**4*d**5/6) + x**5*(A*a**4*d*e**4 + 8*A*a**3*b*d**2*e**3 + 12*A*a**2*b 
**2*d**3*e**2 + 4*A*a*b**3*d**4*e + A*b**4*d**5/5 + 2*B*a**4*d**2*e**3 + 8 
*B*a**3*b*d**3*e**2 + 6*B*a**2*b**2*d**4*e + 4*B*a*b**3*d**5/5) + x**4*(5* 
A*a**4*d**2*e**3/2 + 10*A*a**3*b*d**3*e**2 + 15*A*a**2*b**2*d**4*e/2 + A*a 
*b**3*d**5 + 5*B*a**4*d**3*e**2/2 + 5*B*a**3*b*d**4*e + 3*B*a**2*b**2*d**5 
/2) + x**3*(10*A*a**4*d**3*e**2/3 + 20*A*a**3*b*d**4*e/3 + 2*A*a**2*b**2*d 
**5 + 5*B*a**4*d**4*e/3 + 4*B*a**3*b*d**5/3) + x**2*(5*A*a**4*d**4*e/2 + 2 
*A*a**3*b*d**5 + B*a**4*d**5/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (194) = 388\).

Time = 0.04 (sec) , antiderivative size = 686, normalized size of antiderivative = 3.33 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
 

Output:

1/11*B*b^4*e^5*x^11 + A*a^4*d^5*x + 1/10*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b 
^4)*e^5)*x^10 + 1/9*(10*B*b^4*d^2*e^3 + 5*(4*B*a*b^3 + A*b^4)*d*e^4 + 2*(3 
*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^9 + 1/4*(5*B*b^4*d^3*e^2 + 5*(4*B*a*b^3 + A 
*b^4)*d^2*e^3 + 5*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b 
^2)*e^5)*x^8 + 1/7*(5*B*b^4*d^4*e + 10*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 20*(3 
*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B* 
a^4 + 4*A*a^3*b)*e^5)*x^7 + 1/6*(B*b^4*d^5 + A*a^4*e^5 + 5*(4*B*a*b^3 + A* 
b^4)*d^4*e + 20*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A*a^ 
2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4)*x^6 + 1/5*(5*A*a^4*d*e^4 + ( 
4*B*a*b^3 + A*b^4)*d^5 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e + 20*(2*B*a^3* 
b + 3*A*a^2*b^2)*d^3*e^2 + 10*(B*a^4 + 4*A*a^3*b)*d^2*e^3)*x^5 + 1/2*(5*A* 
a^4*d^2*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^5 + 5*(2*B*a^3*b + 3*A*a^2*b^2)* 
d^4*e + 5*(B*a^4 + 4*A*a^3*b)*d^3*e^2)*x^4 + 1/3*(10*A*a^4*d^3*e^2 + 2*(2* 
B*a^3*b + 3*A*a^2*b^2)*d^5 + 5*(B*a^4 + 4*A*a^3*b)*d^4*e)*x^3 + 1/2*(5*A*a 
^4*d^4*e + (B*a^4 + 4*A*a^3*b)*d^5)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (194) = 388\).

Time = 0.20 (sec) , antiderivative size = 856, normalized size of antiderivative = 4.16 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
 

Output:

1/11*B*b^4*e^5*x^11 + 1/2*B*b^4*d*e^4*x^10 + 2/5*B*a*b^3*e^5*x^10 + 1/10*A 
*b^4*e^5*x^10 + 10/9*B*b^4*d^2*e^3*x^9 + 20/9*B*a*b^3*d*e^4*x^9 + 5/9*A*b^ 
4*d*e^4*x^9 + 2/3*B*a^2*b^2*e^5*x^9 + 4/9*A*a*b^3*e^5*x^9 + 5/4*B*b^4*d^3* 
e^2*x^8 + 5*B*a*b^3*d^2*e^3*x^8 + 5/4*A*b^4*d^2*e^3*x^8 + 15/4*B*a^2*b^2*d 
*e^4*x^8 + 5/2*A*a*b^3*d*e^4*x^8 + 1/2*B*a^3*b*e^5*x^8 + 3/4*A*a^2*b^2*e^5 
*x^8 + 5/7*B*b^4*d^4*e*x^7 + 40/7*B*a*b^3*d^3*e^2*x^7 + 10/7*A*b^4*d^3*e^2 
*x^7 + 60/7*B*a^2*b^2*d^2*e^3*x^7 + 40/7*A*a*b^3*d^2*e^3*x^7 + 20/7*B*a^3* 
b*d*e^4*x^7 + 30/7*A*a^2*b^2*d*e^4*x^7 + 1/7*B*a^4*e^5*x^7 + 4/7*A*a^3*b*e 
^5*x^7 + 1/6*B*b^4*d^5*x^6 + 10/3*B*a*b^3*d^4*e*x^6 + 5/6*A*b^4*d^4*e*x^6 
+ 10*B*a^2*b^2*d^3*e^2*x^6 + 20/3*A*a*b^3*d^3*e^2*x^6 + 20/3*B*a^3*b*d^2*e 
^3*x^6 + 10*A*a^2*b^2*d^2*e^3*x^6 + 5/6*B*a^4*d*e^4*x^6 + 10/3*A*a^3*b*d*e 
^4*x^6 + 1/6*A*a^4*e^5*x^6 + 4/5*B*a*b^3*d^5*x^5 + 1/5*A*b^4*d^5*x^5 + 6*B 
*a^2*b^2*d^4*e*x^5 + 4*A*a*b^3*d^4*e*x^5 + 8*B*a^3*b*d^3*e^2*x^5 + 12*A*a^ 
2*b^2*d^3*e^2*x^5 + 2*B*a^4*d^2*e^3*x^5 + 8*A*a^3*b*d^2*e^3*x^5 + A*a^4*d* 
e^4*x^5 + 3/2*B*a^2*b^2*d^5*x^4 + A*a*b^3*d^5*x^4 + 5*B*a^3*b*d^4*e*x^4 + 
15/2*A*a^2*b^2*d^4*e*x^4 + 5/2*B*a^4*d^3*e^2*x^4 + 10*A*a^3*b*d^3*e^2*x^4 
+ 5/2*A*a^4*d^2*e^3*x^4 + 4/3*B*a^3*b*d^5*x^3 + 2*A*a^2*b^2*d^5*x^3 + 5/3* 
B*a^4*d^4*e*x^3 + 20/3*A*a^3*b*d^4*e*x^3 + 10/3*A*a^4*d^3*e^2*x^3 + 1/2*B* 
a^4*d^5*x^2 + 2*A*a^3*b*d^5*x^2 + 5/2*A*a^4*d^4*e*x^2 + A*a^4*d^5*x
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.45 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^5\,\left (2\,B\,a^4\,d^2\,e^3+A\,a^4\,d\,e^4+8\,B\,a^3\,b\,d^3\,e^2+8\,A\,a^3\,b\,d^2\,e^3+6\,B\,a^2\,b^2\,d^4\,e+12\,A\,a^2\,b^2\,d^3\,e^2+\frac {4\,B\,a\,b^3\,d^5}{5}+4\,A\,a\,b^3\,d^4\,e+\frac {A\,b^4\,d^5}{5}\right )+x^7\,\left (\frac {B\,a^4\,e^5}{7}+\frac {20\,B\,a^3\,b\,d\,e^4}{7}+\frac {4\,A\,a^3\,b\,e^5}{7}+\frac {60\,B\,a^2\,b^2\,d^2\,e^3}{7}+\frac {30\,A\,a^2\,b^2\,d\,e^4}{7}+\frac {40\,B\,a\,b^3\,d^3\,e^2}{7}+\frac {40\,A\,a\,b^3\,d^2\,e^3}{7}+\frac {5\,B\,b^4\,d^4\,e}{7}+\frac {10\,A\,b^4\,d^3\,e^2}{7}\right )+x^4\,\left (\frac {5\,B\,a^4\,d^3\,e^2}{2}+\frac {5\,A\,a^4\,d^2\,e^3}{2}+5\,B\,a^3\,b\,d^4\,e+10\,A\,a^3\,b\,d^3\,e^2+\frac {3\,B\,a^2\,b^2\,d^5}{2}+\frac {15\,A\,a^2\,b^2\,d^4\,e}{2}+A\,a\,b^3\,d^5\right )+x^8\,\left (\frac {B\,a^3\,b\,e^5}{2}+\frac {15\,B\,a^2\,b^2\,d\,e^4}{4}+\frac {3\,A\,a^2\,b^2\,e^5}{4}+5\,B\,a\,b^3\,d^2\,e^3+\frac {5\,A\,a\,b^3\,d\,e^4}{2}+\frac {5\,B\,b^4\,d^3\,e^2}{4}+\frac {5\,A\,b^4\,d^2\,e^3}{4}\right )+x^3\,\left (\frac {5\,B\,a^4\,d^4\,e}{3}+\frac {10\,A\,a^4\,d^3\,e^2}{3}+\frac {4\,B\,a^3\,b\,d^5}{3}+\frac {20\,A\,a^3\,b\,d^4\,e}{3}+2\,A\,a^2\,b^2\,d^5\right )+x^9\,\left (\frac {2\,B\,a^2\,b^2\,e^5}{3}+\frac {20\,B\,a\,b^3\,d\,e^4}{9}+\frac {4\,A\,a\,b^3\,e^5}{9}+\frac {10\,B\,b^4\,d^2\,e^3}{9}+\frac {5\,A\,b^4\,d\,e^4}{9}\right )+x^6\,\left (\frac {5\,B\,a^4\,d\,e^4}{6}+\frac {A\,a^4\,e^5}{6}+\frac {20\,B\,a^3\,b\,d^2\,e^3}{3}+\frac {10\,A\,a^3\,b\,d\,e^4}{3}+10\,B\,a^2\,b^2\,d^3\,e^2+10\,A\,a^2\,b^2\,d^2\,e^3+\frac {10\,B\,a\,b^3\,d^4\,e}{3}+\frac {20\,A\,a\,b^3\,d^3\,e^2}{3}+\frac {B\,b^4\,d^5}{6}+\frac {5\,A\,b^4\,d^4\,e}{6}\right )+\frac {a^3\,d^4\,x^2\,\left (5\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^4\,x^{10}\,\left (A\,b\,e+4\,B\,a\,e+5\,B\,b\,d\right )}{10}+A\,a^4\,d^5\,x+\frac {B\,b^4\,e^5\,x^{11}}{11} \] Input:

int((A + B*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
 

Output:

x^5*((A*b^4*d^5)/5 + (4*B*a*b^3*d^5)/5 + A*a^4*d*e^4 + 2*B*a^4*d^2*e^3 + 8 
*A*a^3*b*d^2*e^3 + 6*B*a^2*b^2*d^4*e + 8*B*a^3*b*d^3*e^2 + 12*A*a^2*b^2*d^ 
3*e^2 + 4*A*a*b^3*d^4*e) + x^7*((B*a^4*e^5)/7 + (4*A*a^3*b*e^5)/7 + (5*B*b 
^4*d^4*e)/7 + (10*A*b^4*d^3*e^2)/7 + (40*A*a*b^3*d^2*e^3)/7 + (30*A*a^2*b^ 
2*d*e^4)/7 + (40*B*a*b^3*d^3*e^2)/7 + (60*B*a^2*b^2*d^2*e^3)/7 + (20*B*a^3 
*b*d*e^4)/7) + x^4*(A*a*b^3*d^5 + (3*B*a^2*b^2*d^5)/2 + (5*A*a^4*d^2*e^3)/ 
2 + (5*B*a^4*d^3*e^2)/2 + (15*A*a^2*b^2*d^4*e)/2 + 10*A*a^3*b*d^3*e^2 + 5* 
B*a^3*b*d^4*e) + x^8*((B*a^3*b*e^5)/2 + (3*A*a^2*b^2*e^5)/4 + (5*A*b^4*d^2 
*e^3)/4 + (5*B*b^4*d^3*e^2)/4 + 5*B*a*b^3*d^2*e^3 + (15*B*a^2*b^2*d*e^4)/4 
 + (5*A*a*b^3*d*e^4)/2) + x^3*((4*B*a^3*b*d^5)/3 + (5*B*a^4*d^4*e)/3 + 2*A 
*a^2*b^2*d^5 + (10*A*a^4*d^3*e^2)/3 + (20*A*a^3*b*d^4*e)/3) + x^9*((4*A*a* 
b^3*e^5)/9 + (5*A*b^4*d*e^4)/9 + (2*B*a^2*b^2*e^5)/3 + (10*B*b^4*d^2*e^3)/ 
9 + (20*B*a*b^3*d*e^4)/9) + x^6*((A*a^4*e^5)/6 + (B*b^4*d^5)/6 + (5*A*b^4* 
d^4*e)/6 + (5*B*a^4*d*e^4)/6 + (20*A*a*b^3*d^3*e^2)/3 + (20*B*a^3*b*d^2*e^ 
3)/3 + 10*A*a^2*b^2*d^2*e^3 + 10*B*a^2*b^2*d^3*e^2 + (10*A*a^3*b*d*e^4)/3 
+ (10*B*a*b^3*d^4*e)/3) + (a^3*d^4*x^2*(5*A*a*e + 4*A*b*d + B*a*d))/2 + (b 
^3*e^4*x^10*(A*b*e + 4*B*a*e + 5*B*b*d))/10 + A*a^4*d^5*x + (B*b^4*e^5*x^1 
1)/11
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.37 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {x \left (252 b^{5} e^{5} x^{10}+1386 a \,b^{4} e^{5} x^{9}+1386 b^{5} d \,e^{4} x^{9}+3080 a^{2} b^{3} e^{5} x^{8}+7700 a \,b^{4} d \,e^{4} x^{8}+3080 b^{5} d^{2} e^{3} x^{8}+3465 a^{3} b^{2} e^{5} x^{7}+17325 a^{2} b^{3} d \,e^{4} x^{7}+17325 a \,b^{4} d^{2} e^{3} x^{7}+3465 b^{5} d^{3} e^{2} x^{7}+1980 a^{4} b \,e^{5} x^{6}+19800 a^{3} b^{2} d \,e^{4} x^{6}+39600 a^{2} b^{3} d^{2} e^{3} x^{6}+19800 a \,b^{4} d^{3} e^{2} x^{6}+1980 b^{5} d^{4} e \,x^{6}+462 a^{5} e^{5} x^{5}+11550 a^{4} b d \,e^{4} x^{5}+46200 a^{3} b^{2} d^{2} e^{3} x^{5}+46200 a^{2} b^{3} d^{3} e^{2} x^{5}+11550 a \,b^{4} d^{4} e \,x^{5}+462 b^{5} d^{5} x^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 a^{5} d^{2} e^{3} x^{3}+34650 a^{4} b \,d^{3} e^{2} x^{3}+34650 a^{3} b^{2} d^{4} e \,x^{3}+6930 a^{2} b^{3} d^{5} x^{3}+9240 a^{5} d^{3} e^{2} x^{2}+23100 a^{4} b \,d^{4} e \,x^{2}+9240 a^{3} b^{2} d^{5} x^{2}+6930 a^{5} d^{4} e x +6930 a^{4} b \,d^{5} x +2772 a^{5} d^{5}\right )}{2772} \] Input:

int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)
 

Output:

(x*(2772*a**5*d**5 + 6930*a**5*d**4*e*x + 9240*a**5*d**3*e**2*x**2 + 6930* 
a**5*d**2*e**3*x**3 + 2772*a**5*d*e**4*x**4 + 462*a**5*e**5*x**5 + 6930*a* 
*4*b*d**5*x + 23100*a**4*b*d**4*e*x**2 + 34650*a**4*b*d**3*e**2*x**3 + 277 
20*a**4*b*d**2*e**3*x**4 + 11550*a**4*b*d*e**4*x**5 + 1980*a**4*b*e**5*x** 
6 + 9240*a**3*b**2*d**5*x**2 + 34650*a**3*b**2*d**4*e*x**3 + 55440*a**3*b* 
*2*d**3*e**2*x**4 + 46200*a**3*b**2*d**2*e**3*x**5 + 19800*a**3*b**2*d*e** 
4*x**6 + 3465*a**3*b**2*e**5*x**7 + 6930*a**2*b**3*d**5*x**3 + 27720*a**2* 
b**3*d**4*e*x**4 + 46200*a**2*b**3*d**3*e**2*x**5 + 39600*a**2*b**3*d**2*e 
**3*x**6 + 17325*a**2*b**3*d*e**4*x**7 + 3080*a**2*b**3*e**5*x**8 + 2772*a 
*b**4*d**5*x**4 + 11550*a*b**4*d**4*e*x**5 + 19800*a*b**4*d**3*e**2*x**6 + 
 17325*a*b**4*d**2*e**3*x**7 + 7700*a*b**4*d*e**4*x**8 + 1386*a*b**4*e**5* 
x**9 + 462*b**5*d**5*x**5 + 1980*b**5*d**4*e*x**6 + 3465*b**5*d**3*e**2*x* 
*7 + 3080*b**5*d**2*e**3*x**8 + 1386*b**5*d*e**4*x**9 + 252*b**5*e**5*x**1 
0))/2772