\(\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx\) [1083]
Optimal result
Integrand size = 20, antiderivative size = 243 \[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\frac {(A b-a B) (b d-a e)^5 (a+b x)^{11}}{11 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^{12}}{12 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^{13}}{13 b^7}+\frac {5 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{14}}{7 b^7}+\frac {e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{15}}{3 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{16}}{16 b^7}+\frac {B e^5 (a+b x)^{17}}{17 b^7}
\]
[Out]
1/11*(A*b-B*a)*(-a*e+b*d)^5*(b*x+a)^11/b^7+1/12*(-a*e+b*d)^4*(5*A*b*e-6*B*a*e+B*b*d)*(b*x+a)^12/b^7+5/13*e*(-a
*e+b*d)^3*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^13/b^7+5/7*e^2*(-a*e+b*d)^2*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^14/b^7+1/3
*e^3*(-a*e+b*d)*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^15/b^7+1/16*e^4*(A*b*e-6*B*a*e+5*B*b*d)*(b*x+a)^16/b^7+1/17*B*
e^5*(b*x+a)^17/b^7
Rubi [A] (verified)
Time = 0.95 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78}
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\frac {e^4 (a+b x)^{16} (-6 a B e+A b e+5 b B d)}{16 b^7}+\frac {e^3 (a+b x)^{15} (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac {5 e^2 (a+b x)^{14} (b d-a e)^2 (-2 a B e+A b e+b B d)}{7 b^7}+\frac {5 e (a+b x)^{13} (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{13 b^7}+\frac {(a+b x)^{12} (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{12 b^7}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^5}{11 b^7}+\frac {B e^5 (a+b x)^{17}}{17 b^7}
\]
[In]
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^5,x]
[Out]
((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^11)/(11*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*b*e - 6*a*B*e)*(a + b*x)^12)/(
12*b^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^13)/(13*b^7) + (5*e^2*(b*d - a*e)^2*(b*B*d
+ A*b*e - 2*a*B*e)*(a + b*x)^14)/(7*b^7) + (e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^15)/(3*b^7)
+ (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^16)/(16*b^7) + (B*e^5*(a + b*x)^17)/(17*b^7)
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))
Rubi steps \begin{align*}
\text {integral}& = \int \left (\frac {(A b-a B) (b d-a e)^5 (a+b x)^{10}}{b^6}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^{11}}{b^6}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^{12}}{b^6}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{13}}{b^6}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{14}}{b^6}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{15}}{b^6}+\frac {B e^5 (a+b x)^{16}}{b^6}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^5 (a+b x)^{11}}{11 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^{12}}{12 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^{13}}{13 b^7}+\frac {5 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{14}}{7 b^7}+\frac {e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{15}}{3 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{16}}{16 b^7}+\frac {B e^5 (a+b x)^{17}}{17 b^7} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1509\) vs. \(2(243)=486\).
Time = 0.34 (sec) , antiderivative size = 1509, normalized size of antiderivative = 6.21
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=a^{10} A d^5 x+\frac {1}{2} a^9 d^4 (a B d+5 A (2 b d+a e)) x^2+\frac {5}{3} a^8 d^3 \left (a B d (2 b d+a e)+A \left (9 b^2 d^2+10 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {5}{4} a^7 d^2 \left (a B d \left (9 b^2 d^2+10 a b d e+2 a^2 e^2\right )+A \left (24 b^3 d^3+45 a b^2 d^2 e+20 a^2 b d e^2+2 a^3 e^3\right )\right ) x^4+a^6 d \left (a B d \left (24 b^3 d^3+45 a b^2 d^2 e+20 a^2 b d e^2+2 a^3 e^3\right )+A \left (42 b^4 d^4+120 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{6} a^5 \left (5 a B d \left (42 b^4 d^4+120 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4\right )+A \left (252 b^5 d^5+1050 a b^4 d^4 e+1200 a^2 b^3 d^3 e^2+450 a^3 b^2 d^2 e^3+50 a^4 b d e^4+a^5 e^5\right )\right ) x^6+\frac {1}{7} a^4 \left (a B \left (252 b^5 d^5+1050 a b^4 d^4 e+1200 a^2 b^3 d^3 e^2+450 a^3 b^2 d^2 e^3+50 a^4 b d e^4+a^5 e^5\right )+5 A b \left (42 b^5 d^5+252 a b^4 d^4 e+420 a^2 b^3 d^3 e^2+240 a^3 b^2 d^2 e^3+45 a^4 b d e^4+2 a^5 e^5\right )\right ) x^7+\frac {5}{8} a^3 b \left (a B \left (42 b^5 d^5+252 a b^4 d^4 e+420 a^2 b^3 d^3 e^2+240 a^3 b^2 d^2 e^3+45 a^4 b d e^4+2 a^5 e^5\right )+3 A b \left (8 b^5 d^5+70 a b^4 d^4 e+168 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+40 a^4 b d e^4+3 a^5 e^5\right )\right ) x^8+\frac {5}{3} a^2 b^2 \left (a B \left (8 b^5 d^5+70 a b^4 d^4 e+168 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+40 a^4 b d e^4+3 a^5 e^5\right )+A b \left (3 b^5 d^5+40 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+168 a^3 b^2 d^2 e^3+70 a^4 b d e^4+8 a^5 e^5\right )\right ) x^9+\frac {1}{2} a b^3 \left (3 a B \left (3 b^5 d^5+40 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+168 a^3 b^2 d^2 e^3+70 a^4 b d e^4+8 a^5 e^5\right )+A b \left (2 b^5 d^5+45 a b^4 d^4 e+240 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+252 a^4 b d e^4+42 a^5 e^5\right )\right ) x^{10}+\frac {1}{11} b^4 \left (5 a B \left (2 b^5 d^5+45 a b^4 d^4 e+240 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+252 a^4 b d e^4+42 a^5 e^5\right )+A b \left (b^5 d^5+50 a b^4 d^4 e+450 a^2 b^3 d^3 e^2+1200 a^3 b^2 d^2 e^3+1050 a^4 b d e^4+252 a^5 e^5\right )\right ) x^{11}+\frac {1}{12} b^5 \left (252 a^5 B e^5+450 a^2 b^3 d^2 e^2 (B d+A e)+600 a^3 b^2 d e^3 (2 B d+A e)+210 a^4 b e^4 (5 B d+A e)+50 a b^4 d^3 e (B d+2 A e)+b^5 d^4 (B d+5 A e)\right ) x^{12}+\frac {5}{13} b^6 e \left (42 a^4 B e^4+20 a b^3 d^2 e (B d+A e)+45 a^2 b^2 d e^2 (2 B d+A e)+24 a^3 b e^3 (5 B d+A e)+b^4 d^3 (B d+2 A e)\right ) x^{13}+\frac {5}{14} b^7 e^2 \left (24 a^3 B e^3+2 b^3 d^2 (B d+A e)+10 a b^2 d e (2 B d+A e)+9 a^2 b e^2 (5 B d+A e)\right ) x^{14}+\frac {1}{3} b^8 e^3 \left (9 a^2 B e^2+b^2 d (2 B d+A e)+2 a b e (5 B d+A e)\right ) x^{15}+\frac {1}{16} b^9 e^4 (5 b B d+A b e+10 a B e) x^{16}+\frac {1}{17} b^{10} B e^5 x^{17}
\]
[In]
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^5,x]
[Out]
a^10*A*d^5*x + (a^9*d^4*(a*B*d + 5*A*(2*b*d + a*e))*x^2)/2 + (5*a^8*d^3*(a*B*d*(2*b*d + a*e) + A*(9*b^2*d^2 +
10*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (5*a^7*d^2*(a*B*d*(9*b^2*d^2 + 10*a*b*d*e + 2*a^2*e^2) + A*(24*b^3*d^3 + 45*
a*b^2*d^2*e + 20*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^6*d*(a*B*d*(24*b^3*d^3 + 45*a*b^2*d^2*e + 20*a^2*b*d*e^2
+ 2*a^3*e^3) + A*(42*b^4*d^4 + 120*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a^5*(
5*a*B*d*(42*b^4*d^4 + 120*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4) + A*(252*b^5*d^5 + 1050
*a*b^4*d^4*e + 1200*a^2*b^3*d^3*e^2 + 450*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + a^5*e^5))*x^6)/6 + (a^4*(a*B*(252
*b^5*d^5 + 1050*a*b^4*d^4*e + 1200*a^2*b^3*d^3*e^2 + 450*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + a^5*e^5) + 5*A*b*(
42*b^5*d^5 + 252*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 240*a^3*b^2*d^2*e^3 + 45*a^4*b*d*e^4 + 2*a^5*e^5))*x^7)/7
+ (5*a^3*b*(a*B*(42*b^5*d^5 + 252*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 240*a^3*b^2*d^2*e^3 + 45*a^4*b*d*e^4 +
2*a^5*e^5) + 3*A*b*(8*b^5*d^5 + 70*a*b^4*d^4*e + 168*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 + 40*a^4*b*d*e^4 +
3*a^5*e^5))*x^8)/8 + (5*a^2*b^2*(a*B*(8*b^5*d^5 + 70*a*b^4*d^4*e + 168*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 +
40*a^4*b*d*e^4 + 3*a^5*e^5) + A*b*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 168*a^3*b^2*d^2*e^3 + 7
0*a^4*b*d*e^4 + 8*a^5*e^5))*x^9)/3 + (a*b^3*(3*a*B*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 168*a^3
*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 8*a^5*e^5) + A*b*(2*b^5*d^5 + 45*a*b^4*d^4*e + 240*a^2*b^3*d^3*e^2 + 420*a^3*b
^2*d^2*e^3 + 252*a^4*b*d*e^4 + 42*a^5*e^5))*x^10)/2 + (b^4*(5*a*B*(2*b^5*d^5 + 45*a*b^4*d^4*e + 240*a^2*b^3*d^
3*e^2 + 420*a^3*b^2*d^2*e^3 + 252*a^4*b*d*e^4 + 42*a^5*e^5) + A*b*(b^5*d^5 + 50*a*b^4*d^4*e + 450*a^2*b^3*d^3*
e^2 + 1200*a^3*b^2*d^2*e^3 + 1050*a^4*b*d*e^4 + 252*a^5*e^5))*x^11)/11 + (b^5*(252*a^5*B*e^5 + 450*a^2*b^3*d^2
*e^2*(B*d + A*e) + 600*a^3*b^2*d*e^3*(2*B*d + A*e) + 210*a^4*b*e^4*(5*B*d + A*e) + 50*a*b^4*d^3*e*(B*d + 2*A*e
) + b^5*d^4*(B*d + 5*A*e))*x^12)/12 + (5*b^6*e*(42*a^4*B*e^4 + 20*a*b^3*d^2*e*(B*d + A*e) + 45*a^2*b^2*d*e^2*(
2*B*d + A*e) + 24*a^3*b*e^3*(5*B*d + A*e) + b^4*d^3*(B*d + 2*A*e))*x^13)/13 + (5*b^7*e^2*(24*a^3*B*e^3 + 2*b^3
*d^2*(B*d + A*e) + 10*a*b^2*d*e*(2*B*d + A*e) + 9*a^2*b*e^2*(5*B*d + A*e))*x^14)/14 + (b^8*e^3*(9*a^2*B*e^2 +
b^2*d*(2*B*d + A*e) + 2*a*b*e*(5*B*d + A*e))*x^15)/3 + (b^9*e^4*(5*b*B*d + A*b*e + 10*a*B*e)*x^16)/16 + (b^10*
B*e^5*x^17)/17
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1620\) vs. \(2(229)=458\).
Time = 0.70 (sec) , antiderivative size = 1621, normalized size of antiderivative =
6.67
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(1621\) |
| norman |
\(\text {Expression too large to display}\) |
\(1718\) |
| gosper |
\(\text {Expression too large to display}\) |
\(2033\) |
| risch |
\(\text {Expression too large to display}\) |
\(2033\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2033\) |
| | |
|
|
|
|
[In]
int((b*x+a)^10*(B*x+A)*(e*x+d)^5,x,method=_RETURNVERBOSE)
[Out]
1/17*b^10*B*e^5*x^17+1/16*((A*b^10+10*B*a*b^9)*e^5+5*b^10*B*d*e^4)*x^16+1/15*((10*A*a*b^9+45*B*a^2*b^8)*e^5+5*
(A*b^10+10*B*a*b^9)*d*e^4+10*b^10*B*d^2*e^3)*x^15+1/14*((45*A*a^2*b^8+120*B*a^3*b^7)*e^5+5*(10*A*a*b^9+45*B*a^
2*b^8)*d*e^4+10*(A*b^10+10*B*a*b^9)*d^2*e^3+10*b^10*B*d^3*e^2)*x^14+1/13*((120*A*a^3*b^7+210*B*a^4*b^6)*e^5+5*
(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^4+10*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^3+10*(A*b^10+10*B*a*b^9)*d^3*e^2+5*b^10*
B*d^4*e)*x^13+1/12*((210*A*a^4*b^6+252*B*a^5*b^5)*e^5+5*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^4+10*(45*A*a^2*b^8+1
20*B*a^3*b^7)*d^2*e^3+10*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^2+5*(A*b^10+10*B*a*b^9)*d^4*e+b^10*B*d^5)*x^12+1/11*(
(252*A*a^5*b^5+210*B*a^6*b^4)*e^5+5*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e^4+10*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e
^3+10*(45*A*a^2*b^8+120*B*a^3*b^7)*d^3*e^2+5*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e+(A*b^10+10*B*a*b^9)*d^5)*x^11+1/1
0*((210*A*a^6*b^4+120*B*a^7*b^3)*e^5+5*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^4+10*(210*A*a^4*b^6+252*B*a^5*b^5)*d^
2*e^3+10*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^2+5*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e+(10*A*a*b^9+45*B*a^2*b^8)*
d^5)*x^10+1/9*((120*A*a^7*b^3+45*B*a^8*b^2)*e^5+5*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^4+10*(252*A*a^5*b^5+210*B*
a^6*b^4)*d^2*e^3+10*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^2+5*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e+(45*A*a^2*b^8+
120*B*a^3*b^7)*d^5)*x^9+1/8*((45*A*a^8*b^2+10*B*a^9*b)*e^5+5*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^4+10*(210*A*a^6*
b^4+120*B*a^7*b^3)*d^2*e^3+10*(252*A*a^5*b^5+210*B*a^6*b^4)*d^3*e^2+5*(210*A*a^4*b^6+252*B*a^5*b^5)*d^4*e+(120
*A*a^3*b^7+210*B*a^4*b^6)*d^5)*x^8+1/7*((10*A*a^9*b+B*a^10)*e^5+5*(45*A*a^8*b^2+10*B*a^9*b)*d*e^4+10*(120*A*a^
7*b^3+45*B*a^8*b^2)*d^2*e^3+10*(210*A*a^6*b^4+120*B*a^7*b^3)*d^3*e^2+5*(252*A*a^5*b^5+210*B*a^6*b^4)*d^4*e+(21
0*A*a^4*b^6+252*B*a^5*b^5)*d^5)*x^7+1/6*(a^10*A*e^5+5*(10*A*a^9*b+B*a^10)*d*e^4+10*(45*A*a^8*b^2+10*B*a^9*b)*d
^2*e^3+10*(120*A*a^7*b^3+45*B*a^8*b^2)*d^3*e^2+5*(210*A*a^6*b^4+120*B*a^7*b^3)*d^4*e+(252*A*a^5*b^5+210*B*a^6*
b^4)*d^5)*x^6+1/5*(5*a^10*A*d*e^4+10*(10*A*a^9*b+B*a^10)*d^2*e^3+10*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^2+5*(120*A
*a^7*b^3+45*B*a^8*b^2)*d^4*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^5)*x^5+1/4*(10*a^10*A*d^2*e^3+10*(10*A*a^9*b+B*a^
10)*d^3*e^2+5*(45*A*a^8*b^2+10*B*a^9*b)*d^4*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^5)*x^4+1/3*(10*a^10*A*d^3*e^2+5*(
10*A*a^9*b+B*a^10)*d^4*e+(45*A*a^8*b^2+10*B*a^9*b)*d^5)*x^3+1/2*(5*a^10*A*d^4*e+(10*A*a^9*b+B*a^10)*d^5)*x^2+a
^10*A*d^5*x
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (229) = 458\).
Time = 0.23 (sec) , antiderivative size = 1625, normalized size of antiderivative = 6.69
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^5,x, algorithm="fricas")
[Out]
1/17*B*b^10*e^5*x^17 + A*a^10*d^5*x + 1/16*(5*B*b^10*d*e^4 + (10*B*a*b^9 + A*b^10)*e^5)*x^16 + 1/3*(2*B*b^10*d
^2*e^3 + (10*B*a*b^9 + A*b^10)*d*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^15 + 5/14*(2*B*b^10*d^3*e^2 + 2*(10*B*
a*b^9 + A*b^10)*d^2*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^5)*x^14 + 5/13*(
B*b^10*d^4*e + 2*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^3 + 15*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^4 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^13 + 1/12*(B*b^10*d^5 + 5*(10*B*a*b^9 + A*b^10)*d^4*e +
50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 + 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d*e^4 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^5 + 25*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^4*e + 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^2 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^3 + 210*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d*e^4 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^5)*x^11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^5 + 1
5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^2 + 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^2*e^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^5)*x^10 + 5/3*((8*B*a^3*b^7 +
3*A*a^2*b^8)*d^5 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e + 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^2 + 28*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d^2*e^3 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^4 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^5)*x^9 + 5
/8*(6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^3*e^2 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^4 + (2*B*a^9*b + 9*A*a^8
*b^2)*e^5)*x^8 + 1/7*(42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e + 300*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^3*e^2 + 150*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^3 + 25*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^4 +
(B*a^10 + 10*A*a^9*b)*e^5)*x^7 + 1/6*(A*a^10*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5 + 150*(4*B*a^7*b^3 + 7*A
*a^6*b^4)*d^4*e + 150*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^2 + 50*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^3 + 5*(B*a^10 +
10*A*a^9*b)*d*e^4)*x^6 + (A*a^10*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d
^4*e + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*e^2 + 2*(B*a^10 + 10*A*a^9*b)*d^2*e^3)*x^5 + 5/4*(2*A*a^10*d^2*e^3 + 3
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^4*e + 2*(B*a^10 + 10*A*a^9*b)*d^3*e^2)*x^4 +
5/3*(2*A*a^10*d^3*e^2 + (2*B*a^9*b + 9*A*a^8*b^2)*d^5 + (B*a^10 + 10*A*a^9*b)*d^4*e)*x^3 + 1/2*(5*A*a^10*d^4*e
+ (B*a^10 + 10*A*a^9*b)*d^5)*x^2
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2076 vs. \(2 (243) = 486\).
Time = 0.14 (sec) , antiderivative size = 2076, normalized size of antiderivative = 8.54
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**5,x)
[Out]
A*a**10*d**5*x + B*b**10*e**5*x**17/17 + x**16*(A*b**10*e**5/16 + 5*B*a*b**9*e**5/8 + 5*B*b**10*d*e**4/16) + x
**15*(2*A*a*b**9*e**5/3 + A*b**10*d*e**4/3 + 3*B*a**2*b**8*e**5 + 10*B*a*b**9*d*e**4/3 + 2*B*b**10*d**2*e**3/3
) + x**14*(45*A*a**2*b**8*e**5/14 + 25*A*a*b**9*d*e**4/7 + 5*A*b**10*d**2*e**3/7 + 60*B*a**3*b**7*e**5/7 + 225
*B*a**2*b**8*d*e**4/14 + 50*B*a*b**9*d**2*e**3/7 + 5*B*b**10*d**3*e**2/7) + x**13*(120*A*a**3*b**7*e**5/13 + 2
25*A*a**2*b**8*d*e**4/13 + 100*A*a*b**9*d**2*e**3/13 + 10*A*b**10*d**3*e**2/13 + 210*B*a**4*b**6*e**5/13 + 600
*B*a**3*b**7*d*e**4/13 + 450*B*a**2*b**8*d**2*e**3/13 + 100*B*a*b**9*d**3*e**2/13 + 5*B*b**10*d**4*e/13) + x**
12*(35*A*a**4*b**6*e**5/2 + 50*A*a**3*b**7*d*e**4 + 75*A*a**2*b**8*d**2*e**3/2 + 25*A*a*b**9*d**3*e**2/3 + 5*A
*b**10*d**4*e/12 + 21*B*a**5*b**5*e**5 + 175*B*a**4*b**6*d*e**4/2 + 100*B*a**3*b**7*d**2*e**3 + 75*B*a**2*b**8
*d**3*e**2/2 + 25*B*a*b**9*d**4*e/6 + B*b**10*d**5/12) + x**11*(252*A*a**5*b**5*e**5/11 + 1050*A*a**4*b**6*d*e
**4/11 + 1200*A*a**3*b**7*d**2*e**3/11 + 450*A*a**2*b**8*d**3*e**2/11 + 50*A*a*b**9*d**4*e/11 + A*b**10*d**5/1
1 + 210*B*a**6*b**4*e**5/11 + 1260*B*a**5*b**5*d*e**4/11 + 2100*B*a**4*b**6*d**2*e**3/11 + 1200*B*a**3*b**7*d*
*3*e**2/11 + 225*B*a**2*b**8*d**4*e/11 + 10*B*a*b**9*d**5/11) + x**10*(21*A*a**6*b**4*e**5 + 126*A*a**5*b**5*d
*e**4 + 210*A*a**4*b**6*d**2*e**3 + 120*A*a**3*b**7*d**3*e**2 + 45*A*a**2*b**8*d**4*e/2 + A*a*b**9*d**5 + 12*B
*a**7*b**3*e**5 + 105*B*a**6*b**4*d*e**4 + 252*B*a**5*b**5*d**2*e**3 + 210*B*a**4*b**6*d**3*e**2 + 60*B*a**3*b
**7*d**4*e + 9*B*a**2*b**8*d**5/2) + x**9*(40*A*a**7*b**3*e**5/3 + 350*A*a**6*b**4*d*e**4/3 + 280*A*a**5*b**5*
d**2*e**3 + 700*A*a**4*b**6*d**3*e**2/3 + 200*A*a**3*b**7*d**4*e/3 + 5*A*a**2*b**8*d**5 + 5*B*a**8*b**2*e**5 +
200*B*a**7*b**3*d*e**4/3 + 700*B*a**6*b**4*d**2*e**3/3 + 280*B*a**5*b**5*d**3*e**2 + 350*B*a**4*b**6*d**4*e/3
+ 40*B*a**3*b**7*d**5/3) + x**8*(45*A*a**8*b**2*e**5/8 + 75*A*a**7*b**3*d*e**4 + 525*A*a**6*b**4*d**2*e**3/2
+ 315*A*a**5*b**5*d**3*e**2 + 525*A*a**4*b**6*d**4*e/4 + 15*A*a**3*b**7*d**5 + 5*B*a**9*b*e**5/4 + 225*B*a**8*
b**2*d*e**4/8 + 150*B*a**7*b**3*d**2*e**3 + 525*B*a**6*b**4*d**3*e**2/2 + 315*B*a**5*b**5*d**4*e/2 + 105*B*a**
4*b**6*d**5/4) + x**7*(10*A*a**9*b*e**5/7 + 225*A*a**8*b**2*d*e**4/7 + 1200*A*a**7*b**3*d**2*e**3/7 + 300*A*a*
*6*b**4*d**3*e**2 + 180*A*a**5*b**5*d**4*e + 30*A*a**4*b**6*d**5 + B*a**10*e**5/7 + 50*B*a**9*b*d*e**4/7 + 450
*B*a**8*b**2*d**2*e**3/7 + 1200*B*a**7*b**3*d**3*e**2/7 + 150*B*a**6*b**4*d**4*e + 36*B*a**5*b**5*d**5) + x**6
*(A*a**10*e**5/6 + 25*A*a**9*b*d*e**4/3 + 75*A*a**8*b**2*d**2*e**3 + 200*A*a**7*b**3*d**3*e**2 + 175*A*a**6*b*
*4*d**4*e + 42*A*a**5*b**5*d**5 + 5*B*a**10*d*e**4/6 + 50*B*a**9*b*d**2*e**3/3 + 75*B*a**8*b**2*d**3*e**2 + 10
0*B*a**7*b**3*d**4*e + 35*B*a**6*b**4*d**5) + x**5*(A*a**10*d*e**4 + 20*A*a**9*b*d**2*e**3 + 90*A*a**8*b**2*d*
*3*e**2 + 120*A*a**7*b**3*d**4*e + 42*A*a**6*b**4*d**5 + 2*B*a**10*d**2*e**3 + 20*B*a**9*b*d**3*e**2 + 45*B*a*
*8*b**2*d**4*e + 24*B*a**7*b**3*d**5) + x**4*(5*A*a**10*d**2*e**3/2 + 25*A*a**9*b*d**3*e**2 + 225*A*a**8*b**2*
d**4*e/4 + 30*A*a**7*b**3*d**5 + 5*B*a**10*d**3*e**2/2 + 25*B*a**9*b*d**4*e/2 + 45*B*a**8*b**2*d**5/4) + x**3*
(10*A*a**10*d**3*e**2/3 + 50*A*a**9*b*d**4*e/3 + 15*A*a**8*b**2*d**5 + 5*B*a**10*d**4*e/3 + 10*B*a**9*b*d**5/3
) + x**2*(5*A*a**10*d**4*e/2 + 5*A*a**9*b*d**5 + B*a**10*d**5/2)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (229) = 458\).
Time = 0.22 (sec) , antiderivative size = 1625, normalized size of antiderivative = 6.69
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^5,x, algorithm="maxima")
[Out]
1/17*B*b^10*e^5*x^17 + A*a^10*d^5*x + 1/16*(5*B*b^10*d*e^4 + (10*B*a*b^9 + A*b^10)*e^5)*x^16 + 1/3*(2*B*b^10*d
^2*e^3 + (10*B*a*b^9 + A*b^10)*d*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^15 + 5/14*(2*B*b^10*d^3*e^2 + 2*(10*B*
a*b^9 + A*b^10)*d^2*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^5)*x^14 + 5/13*(
B*b^10*d^4*e + 2*(10*B*a*b^9 + A*b^10)*d^3*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^3 + 15*(8*B*a^3*b^7 + 3*A*
a^2*b^8)*d*e^4 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^13 + 1/12*(B*b^10*d^5 + 5*(10*B*a*b^9 + A*b^10)*d^4*e +
50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 + 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d*e^4 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^5 + 25*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^4*e + 150*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^2 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^3 + 210*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d*e^4 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^5)*x^11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^5 + 1
5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e + 60*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^2 + 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^2*e^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^5)*x^10 + 5/3*((8*B*a^3*b^7 +
3*A*a^2*b^8)*d^5 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e + 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^2 + 28*(5*B*a^
6*b^4 + 6*A*a^5*b^5)*d^2*e^3 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^4 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^5)*x^9 + 5
/8*(6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^3*e^2 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^4 + (2*B*a^9*b + 9*A*a^8
*b^2)*e^5)*x^8 + 1/7*(42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5 + 210*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e + 300*(4*B*a^
7*b^3 + 7*A*a^6*b^4)*d^3*e^2 + 150*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^3 + 25*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^4 +
(B*a^10 + 10*A*a^9*b)*e^5)*x^7 + 1/6*(A*a^10*e^5 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5 + 150*(4*B*a^7*b^3 + 7*A
*a^6*b^4)*d^4*e + 150*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^2 + 50*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^3 + 5*(B*a^10 +
10*A*a^9*b)*d*e^4)*x^6 + (A*a^10*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d
^4*e + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d^3*e^2 + 2*(B*a^10 + 10*A*a^9*b)*d^2*e^3)*x^5 + 5/4*(2*A*a^10*d^2*e^3 + 3
*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^4*e + 2*(B*a^10 + 10*A*a^9*b)*d^3*e^2)*x^4 +
5/3*(2*A*a^10*d^3*e^2 + (2*B*a^9*b + 9*A*a^8*b^2)*d^5 + (B*a^10 + 10*A*a^9*b)*d^4*e)*x^3 + 1/2*(5*A*a^10*d^4*e
+ (B*a^10 + 10*A*a^9*b)*d^5)*x^2
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (229) = 458\).
Time = 0.29 (sec) , antiderivative size = 2032, normalized size of antiderivative = 8.36
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^5,x, algorithm="giac")
[Out]
1/17*B*b^10*e^5*x^17 + 5/16*B*b^10*d*e^4*x^16 + 5/8*B*a*b^9*e^5*x^16 + 1/16*A*b^10*e^5*x^16 + 2/3*B*b^10*d^2*e
^3*x^15 + 10/3*B*a*b^9*d*e^4*x^15 + 1/3*A*b^10*d*e^4*x^15 + 3*B*a^2*b^8*e^5*x^15 + 2/3*A*a*b^9*e^5*x^15 + 5/7*
B*b^10*d^3*e^2*x^14 + 50/7*B*a*b^9*d^2*e^3*x^14 + 5/7*A*b^10*d^2*e^3*x^14 + 225/14*B*a^2*b^8*d*e^4*x^14 + 25/7
*A*a*b^9*d*e^4*x^14 + 60/7*B*a^3*b^7*e^5*x^14 + 45/14*A*a^2*b^8*e^5*x^14 + 5/13*B*b^10*d^4*e*x^13 + 100/13*B*a
*b^9*d^3*e^2*x^13 + 10/13*A*b^10*d^3*e^2*x^13 + 450/13*B*a^2*b^8*d^2*e^3*x^13 + 100/13*A*a*b^9*d^2*e^3*x^13 +
600/13*B*a^3*b^7*d*e^4*x^13 + 225/13*A*a^2*b^8*d*e^4*x^13 + 210/13*B*a^4*b^6*e^5*x^13 + 120/13*A*a^3*b^7*e^5*x
^13 + 1/12*B*b^10*d^5*x^12 + 25/6*B*a*b^9*d^4*e*x^12 + 5/12*A*b^10*d^4*e*x^12 + 75/2*B*a^2*b^8*d^3*e^2*x^12 +
25/3*A*a*b^9*d^3*e^2*x^12 + 100*B*a^3*b^7*d^2*e^3*x^12 + 75/2*A*a^2*b^8*d^2*e^3*x^12 + 175/2*B*a^4*b^6*d*e^4*x
^12 + 50*A*a^3*b^7*d*e^4*x^12 + 21*B*a^5*b^5*e^5*x^12 + 35/2*A*a^4*b^6*e^5*x^12 + 10/11*B*a*b^9*d^5*x^11 + 1/1
1*A*b^10*d^5*x^11 + 225/11*B*a^2*b^8*d^4*e*x^11 + 50/11*A*a*b^9*d^4*e*x^11 + 1200/11*B*a^3*b^7*d^3*e^2*x^11 +
450/11*A*a^2*b^8*d^3*e^2*x^11 + 2100/11*B*a^4*b^6*d^2*e^3*x^11 + 1200/11*A*a^3*b^7*d^2*e^3*x^11 + 1260/11*B*a^
5*b^5*d*e^4*x^11 + 1050/11*A*a^4*b^6*d*e^4*x^11 + 210/11*B*a^6*b^4*e^5*x^11 + 252/11*A*a^5*b^5*e^5*x^11 + 9/2*
B*a^2*b^8*d^5*x^10 + A*a*b^9*d^5*x^10 + 60*B*a^3*b^7*d^4*e*x^10 + 45/2*A*a^2*b^8*d^4*e*x^10 + 210*B*a^4*b^6*d^
3*e^2*x^10 + 120*A*a^3*b^7*d^3*e^2*x^10 + 252*B*a^5*b^5*d^2*e^3*x^10 + 210*A*a^4*b^6*d^2*e^3*x^10 + 105*B*a^6*
b^4*d*e^4*x^10 + 126*A*a^5*b^5*d*e^4*x^10 + 12*B*a^7*b^3*e^5*x^10 + 21*A*a^6*b^4*e^5*x^10 + 40/3*B*a^3*b^7*d^5
*x^9 + 5*A*a^2*b^8*d^5*x^9 + 350/3*B*a^4*b^6*d^4*e*x^9 + 200/3*A*a^3*b^7*d^4*e*x^9 + 280*B*a^5*b^5*d^3*e^2*x^9
+ 700/3*A*a^4*b^6*d^3*e^2*x^9 + 700/3*B*a^6*b^4*d^2*e^3*x^9 + 280*A*a^5*b^5*d^2*e^3*x^9 + 200/3*B*a^7*b^3*d*e
^4*x^9 + 350/3*A*a^6*b^4*d*e^4*x^9 + 5*B*a^8*b^2*e^5*x^9 + 40/3*A*a^7*b^3*e^5*x^9 + 105/4*B*a^4*b^6*d^5*x^8 +
15*A*a^3*b^7*d^5*x^8 + 315/2*B*a^5*b^5*d^4*e*x^8 + 525/4*A*a^4*b^6*d^4*e*x^8 + 525/2*B*a^6*b^4*d^3*e^2*x^8 + 3
15*A*a^5*b^5*d^3*e^2*x^8 + 150*B*a^7*b^3*d^2*e^3*x^8 + 525/2*A*a^6*b^4*d^2*e^3*x^8 + 225/8*B*a^8*b^2*d*e^4*x^8
+ 75*A*a^7*b^3*d*e^4*x^8 + 5/4*B*a^9*b*e^5*x^8 + 45/8*A*a^8*b^2*e^5*x^8 + 36*B*a^5*b^5*d^5*x^7 + 30*A*a^4*b^6
*d^5*x^7 + 150*B*a^6*b^4*d^4*e*x^7 + 180*A*a^5*b^5*d^4*e*x^7 + 1200/7*B*a^7*b^3*d^3*e^2*x^7 + 300*A*a^6*b^4*d^
3*e^2*x^7 + 450/7*B*a^8*b^2*d^2*e^3*x^7 + 1200/7*A*a^7*b^3*d^2*e^3*x^7 + 50/7*B*a^9*b*d*e^4*x^7 + 225/7*A*a^8*
b^2*d*e^4*x^7 + 1/7*B*a^10*e^5*x^7 + 10/7*A*a^9*b*e^5*x^7 + 35*B*a^6*b^4*d^5*x^6 + 42*A*a^5*b^5*d^5*x^6 + 100*
B*a^7*b^3*d^4*e*x^6 + 175*A*a^6*b^4*d^4*e*x^6 + 75*B*a^8*b^2*d^3*e^2*x^6 + 200*A*a^7*b^3*d^3*e^2*x^6 + 50/3*B*
a^9*b*d^2*e^3*x^6 + 75*A*a^8*b^2*d^2*e^3*x^6 + 5/6*B*a^10*d*e^4*x^6 + 25/3*A*a^9*b*d*e^4*x^6 + 1/6*A*a^10*e^5*
x^6 + 24*B*a^7*b^3*d^5*x^5 + 42*A*a^6*b^4*d^5*x^5 + 45*B*a^8*b^2*d^4*e*x^5 + 120*A*a^7*b^3*d^4*e*x^5 + 20*B*a^
9*b*d^3*e^2*x^5 + 90*A*a^8*b^2*d^3*e^2*x^5 + 2*B*a^10*d^2*e^3*x^5 + 20*A*a^9*b*d^2*e^3*x^5 + A*a^10*d*e^4*x^5
+ 45/4*B*a^8*b^2*d^5*x^4 + 30*A*a^7*b^3*d^5*x^4 + 25/2*B*a^9*b*d^4*e*x^4 + 225/4*A*a^8*b^2*d^4*e*x^4 + 5/2*B*a
^10*d^3*e^2*x^4 + 25*A*a^9*b*d^3*e^2*x^4 + 5/2*A*a^10*d^2*e^3*x^4 + 10/3*B*a^9*b*d^5*x^3 + 15*A*a^8*b^2*d^5*x^
3 + 5/3*B*a^10*d^4*e*x^3 + 50/3*A*a^9*b*d^4*e*x^3 + 10/3*A*a^10*d^3*e^2*x^3 + 1/2*B*a^10*d^5*x^2 + 5*A*a^9*b*d
^5*x^2 + 5/2*A*a^10*d^4*e*x^2 + A*a^10*d^5*x
Mupad [B] (verification not implemented)
Time = 1.76 (sec) , antiderivative size = 1685, normalized size of antiderivative = 6.93
\[
\int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display}
\]
[In]
int((A + B*x)*(a + b*x)^10*(d + e*x)^5,x)
[Out]
x^9*(5*A*a^2*b^8*d^5 + (40*A*a^7*b^3*e^5)/3 + (40*B*a^3*b^7*d^5)/3 + 5*B*a^8*b^2*e^5 + (200*A*a^3*b^7*d^4*e)/3
+ (350*A*a^6*b^4*d*e^4)/3 + (350*B*a^4*b^6*d^4*e)/3 + (200*B*a^7*b^3*d*e^4)/3 + (700*A*a^4*b^6*d^3*e^2)/3 + 2
80*A*a^5*b^5*d^2*e^3 + 280*B*a^5*b^5*d^3*e^2 + (700*B*a^6*b^4*d^2*e^3)/3) + x^7*((B*a^10*e^5)/7 + (10*A*a^9*b*
e^5)/7 + 30*A*a^4*b^6*d^5 + 36*B*a^5*b^5*d^5 + 180*A*a^5*b^5*d^4*e + (225*A*a^8*b^2*d*e^4)/7 + 150*B*a^6*b^4*d
^4*e + 300*A*a^6*b^4*d^3*e^2 + (1200*A*a^7*b^3*d^2*e^3)/7 + (1200*B*a^7*b^3*d^3*e^2)/7 + (450*B*a^8*b^2*d^2*e^
3)/7 + (50*B*a^9*b*d*e^4)/7) + x^11*((A*b^10*d^5)/11 + (10*B*a*b^9*d^5)/11 + (252*A*a^5*b^5*e^5)/11 + (210*B*a
^6*b^4*e^5)/11 + (1050*A*a^4*b^6*d*e^4)/11 + (225*B*a^2*b^8*d^4*e)/11 + (1260*B*a^5*b^5*d*e^4)/11 + (450*A*a^2
*b^8*d^3*e^2)/11 + (1200*A*a^3*b^7*d^2*e^3)/11 + (1200*B*a^3*b^7*d^3*e^2)/11 + (2100*B*a^4*b^6*d^2*e^3)/11 + (
50*A*a*b^9*d^4*e)/11) + x^10*(A*a*b^9*d^5 + 21*A*a^6*b^4*e^5 + (9*B*a^2*b^8*d^5)/2 + 12*B*a^7*b^3*e^5 + (45*A*
a^2*b^8*d^4*e)/2 + 126*A*a^5*b^5*d*e^4 + 60*B*a^3*b^7*d^4*e + 105*B*a^6*b^4*d*e^4 + 120*A*a^3*b^7*d^3*e^2 + 21
0*A*a^4*b^6*d^2*e^3 + 210*B*a^4*b^6*d^3*e^2 + 252*B*a^5*b^5*d^2*e^3) + x^8*((5*B*a^9*b*e^5)/4 + 15*A*a^3*b^7*d
^5 + (45*A*a^8*b^2*e^5)/8 + (105*B*a^4*b^6*d^5)/4 + (525*A*a^4*b^6*d^4*e)/4 + 75*A*a^7*b^3*d*e^4 + (315*B*a^5*
b^5*d^4*e)/2 + (225*B*a^8*b^2*d*e^4)/8 + 315*A*a^5*b^5*d^3*e^2 + (525*A*a^6*b^4*d^2*e^3)/2 + (525*B*a^6*b^4*d^
3*e^2)/2 + 150*B*a^7*b^3*d^2*e^3) + x^5*(A*a^10*d*e^4 + 42*A*a^6*b^4*d^5 + 24*B*a^7*b^3*d^5 + 2*B*a^10*d^2*e^3
+ 120*A*a^7*b^3*d^4*e + 20*A*a^9*b*d^2*e^3 + 45*B*a^8*b^2*d^4*e + 20*B*a^9*b*d^3*e^2 + 90*A*a^8*b^2*d^3*e^2)
+ x^13*((5*B*b^10*d^4*e)/13 + (120*A*a^3*b^7*e^5)/13 + (210*B*a^4*b^6*e^5)/13 + (10*A*b^10*d^3*e^2)/13 + (100*
A*a*b^9*d^2*e^3)/13 + (225*A*a^2*b^8*d*e^4)/13 + (100*B*a*b^9*d^3*e^2)/13 + (600*B*a^3*b^7*d*e^4)/13 + (450*B*
a^2*b^8*d^2*e^3)/13) + x^4*(30*A*a^7*b^3*d^5 + (45*B*a^8*b^2*d^5)/4 + (5*A*a^10*d^2*e^3)/2 + (5*B*a^10*d^3*e^2
)/2 + (225*A*a^8*b^2*d^4*e)/4 + 25*A*a^9*b*d^3*e^2 + (25*B*a^9*b*d^4*e)/2) + x^14*((45*A*a^2*b^8*e^5)/14 + (60
*B*a^3*b^7*e^5)/7 + (5*A*b^10*d^2*e^3)/7 + (5*B*b^10*d^3*e^2)/7 + (50*B*a*b^9*d^2*e^3)/7 + (225*B*a^2*b^8*d*e^
4)/14 + (25*A*a*b^9*d*e^4)/7) + x^6*((A*a^10*e^5)/6 + (5*B*a^10*d*e^4)/6 + 42*A*a^5*b^5*d^5 + 35*B*a^6*b^4*d^5
+ 175*A*a^6*b^4*d^4*e + 100*B*a^7*b^3*d^4*e + (50*B*a^9*b*d^2*e^3)/3 + 200*A*a^7*b^3*d^3*e^2 + 75*A*a^8*b^2*d
^2*e^3 + 75*B*a^8*b^2*d^3*e^2 + (25*A*a^9*b*d*e^4)/3) + x^12*((B*b^10*d^5)/12 + (5*A*b^10*d^4*e)/12 + (35*A*a^
4*b^6*e^5)/2 + 21*B*a^5*b^5*e^5 + (25*A*a*b^9*d^3*e^2)/3 + 50*A*a^3*b^7*d*e^4 + (175*B*a^4*b^6*d*e^4)/2 + (75*
A*a^2*b^8*d^2*e^3)/2 + (75*B*a^2*b^8*d^3*e^2)/2 + 100*B*a^3*b^7*d^2*e^3 + (25*B*a*b^9*d^4*e)/6) + (a^9*d^4*x^2
*(5*A*a*e + 10*A*b*d + B*a*d))/2 + (b^9*e^4*x^16*(A*b*e + 10*B*a*e + 5*B*b*d))/16 + A*a^10*d^5*x + (5*a^8*d^3*
x^3*(2*A*a^2*e^2 + 9*A*b^2*d^2 + 2*B*a*b*d^2 + B*a^2*d*e + 10*A*a*b*d*e))/3 + (b^8*e^3*x^15*(9*B*a^2*e^2 + 2*B
*b^2*d^2 + 2*A*a*b*e^2 + A*b^2*d*e + 10*B*a*b*d*e))/3 + (B*b^10*e^5*x^17)/17