\(\int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx\) [1081]

Optimal result

Integrand size = 20, antiderivative size = 329 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\frac {(A b-a B) (b d-a e)^7 (a+b x)^{11}}{11 b^9}+\frac {(b d-a e)^6 (b B d+7 A b e-8 a B e) (a+b x)^{12}}{12 b^9}+\frac {7 e (b d-a e)^5 (b B d+3 A b e-4 a B e) (a+b x)^{13}}{13 b^9}+\frac {e^2 (b d-a e)^4 (3 b B d+5 A b e-8 a B e) (a+b x)^{14}}{2 b^9}+\frac {7 e^3 (b d-a e)^3 (b B d+A b e-2 a B e) (a+b x)^{15}}{3 b^9}+\frac {7 e^4 (b d-a e)^2 (5 b B d+3 A b e-8 a B e) (a+b x)^{16}}{16 b^9}+\frac {7 e^5 (b d-a e) (3 b B d+A b e-4 a B e) (a+b x)^{17}}{17 b^9}+\frac {e^6 (7 b B d+A b e-8 a B e) (a+b x)^{18}}{18 b^9}+\frac {B e^7 (a+b x)^{19}}{19 b^9} \]

[Out]

1/11*(A*b-B*a)*(-a*e+b*d)^7*(b*x+a)^11/b^9+1/12*(-a*e+b*d)^6*(7*A*b*e-8*B*a*e+B*b*d)*(b*x+a)^12/b^9+7/13*e*(-a
*e+b*d)^5*(3*A*b*e-4*B*a*e+B*b*d)*(b*x+a)^13/b^9+1/2*e^2*(-a*e+b*d)^4*(5*A*b*e-8*B*a*e+3*B*b*d)*(b*x+a)^14/b^9
+7/3*e^3*(-a*e+b*d)^3*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^15/b^9+7/16*e^4*(-a*e+b*d)^2*(3*A*b*e-8*B*a*e+5*B*b*d)*(b*
x+a)^16/b^9+7/17*e^5*(-a*e+b*d)*(A*b*e-4*B*a*e+3*B*b*d)*(b*x+a)^17/b^9+1/18*e^6*(A*b*e-8*B*a*e+7*B*b*d)*(b*x+a
)^18/b^9+1/19*B*e^7*(b*x+a)^19/b^9

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 329, 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} \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\frac {e^6 (a+b x)^{18} (-8 a B e+A b e+7 b B d)}{18 b^9}+\frac {7 e^5 (a+b x)^{17} (b d-a e) (-4 a B e+A b e+3 b B d)}{17 b^9}+\frac {7 e^4 (a+b x)^{16} (b d-a e)^2 (-8 a B e+3 A b e+5 b B d)}{16 b^9}+\frac {7 e^3 (a+b x)^{15} (b d-a e)^3 (-2 a B e+A b e+b B d)}{3 b^9}+\frac {e^2 (a+b x)^{14} (b d-a e)^4 (-8 a B e+5 A b e+3 b B d)}{2 b^9}+\frac {7 e (a+b x)^{13} (b d-a e)^5 (-4 a B e+3 A b e+b B d)}{13 b^9}+\frac {(a+b x)^{12} (b d-a e)^6 (-8 a B e+7 A b e+b B d)}{12 b^9}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^7}{11 b^9}+\frac {B e^7 (a+b x)^{19}}{19 b^9} \]

[In]

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]

[Out]

((A*b - a*B)*(b*d - a*e)^7*(a + b*x)^11)/(11*b^9) + ((b*d - a*e)^6*(b*B*d + 7*A*b*e - 8*a*B*e)*(a + b*x)^12)/(
12*b^9) + (7*e*(b*d - a*e)^5*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^13)/(13*b^9) + (e^2*(b*d - a*e)^4*(3*b*B*d
+ 5*A*b*e - 8*a*B*e)*(a + b*x)^14)/(2*b^9) + (7*e^3*(b*d - a*e)^3*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^15)/(3*b
^9) + (7*e^4*(b*d - a*e)^2*(5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^16)/(16*b^9) + (7*e^5*(b*d - a*e)*(3*b*B*d
+ A*b*e - 4*a*B*e)*(a + b*x)^17)/(17*b^9) + (e^6*(7*b*B*d + A*b*e - 8*a*B*e)*(a + b*x)^18)/(18*b^9) + (B*e^7*(
a + b*x)^19)/(19*b^9)

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)^7 (a+b x)^{10}}{b^8}+\frac {(b d-a e)^6 (b B d+7 A b e-8 a B e) (a+b x)^{11}}{b^8}+\frac {7 e (b d-a e)^5 (b B d+3 A b e-4 a B e) (a+b x)^{12}}{b^8}+\frac {7 e^2 (b d-a e)^4 (3 b B d+5 A b e-8 a B e) (a+b x)^{13}}{b^8}+\frac {35 e^3 (b d-a e)^3 (b B d+A b e-2 a B e) (a+b x)^{14}}{b^8}+\frac {7 e^4 (b d-a e)^2 (5 b B d+3 A b e-8 a B e) (a+b x)^{15}}{b^8}+\frac {7 e^5 (b d-a e) (3 b B d+A b e-4 a B e) (a+b x)^{16}}{b^8}+\frac {e^6 (7 b B d+A b e-8 a B e) (a+b x)^{17}}{b^8}+\frac {B e^7 (a+b x)^{18}}{b^8}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^7 (a+b x)^{11}}{11 b^9}+\frac {(b d-a e)^6 (b B d+7 A b e-8 a B e) (a+b x)^{12}}{12 b^9}+\frac {7 e (b d-a e)^5 (b B d+3 A b e-4 a B e) (a+b x)^{13}}{13 b^9}+\frac {e^2 (b d-a e)^4 (3 b B d+5 A b e-8 a B e) (a+b x)^{14}}{2 b^9}+\frac {7 e^3 (b d-a e)^3 (b B d+A b e-2 a B e) (a+b x)^{15}}{3 b^9}+\frac {7 e^4 (b d-a e)^2 (5 b B d+3 A b e-8 a B e) (a+b x)^{16}}{16 b^9}+\frac {7 e^5 (b d-a e) (3 b B d+A b e-4 a B e) (a+b x)^{17}}{17 b^9}+\frac {e^6 (7 b B d+A b e-8 a B e) (a+b x)^{18}}{18 b^9}+\frac {B e^7 (a+b x)^{19}}{19 b^9} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2034\) vs. \(2(329)=658\).

Time = 0.49 (sec) , antiderivative size = 2034, normalized size of antiderivative = 6.18 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Result too large to show} \]

[In]

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]

[Out]

a^10*A*d^7*x + (a^9*d^6*(10*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^8*d^5*(a*B*d*(10*b*d + 7*a*e) + A*(45*b^2*d^2
+ 70*a*b*d*e + 21*a^2*e^2))*x^3)/3 + (a^7*d^4*(a*B*d*(45*b^2*d^2 + 70*a*b*d*e + 21*a^2*e^2) + 5*A*(24*b^3*d^3
+ 63*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 7*a^3*e^3))*x^4)/4 + a^6*d^3*(a*B*d*(24*b^3*d^3 + 63*a*b^2*d^2*e + 42*a^2
*b*d*e^2 + 7*a^3*e^3) + 7*A*(6*b^4*d^4 + 24*a*b^3*d^3*e + 27*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4))*x^5
+ (7*a^5*d^2*(5*a*B*d*(6*b^4*d^4 + 24*a*b^3*d^3*e + 27*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4) + A*(36*b^5
*d^5 + 210*a*b^4*d^4*e + 360*a^2*b^3*d^3*e^2 + 225*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + 3*a^5*e^5))*x^6)/6 + a^4
*d*(a*B*d*(36*b^5*d^5 + 210*a*b^4*d^4*e + 360*a^2*b^3*d^3*e^2 + 225*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + 3*a^5*e
^5) + A*(30*b^6*d^6 + 252*a*b^5*d^5*e + 630*a^2*b^4*d^4*e^2 + 600*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a
^5*b*d*e^5 + a^6*e^6))*x^7 + (a^3*(7*a*B*d*(30*b^6*d^6 + 252*a*b^5*d^5*e + 630*a^2*b^4*d^4*e^2 + 600*a^3*b^3*d
^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a^5*b*d*e^5 + a^6*e^6) + A*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 5292*a^2*b^5*d^
5*e^2 + 7350*a^3*b^4*d^4*e^3 + 4200*a^4*b^3*d^3*e^4 + 945*a^5*b^2*d^2*e^5 + 70*a^6*b*d*e^6 + a^7*e^7))*x^8)/8
+ (a^2*(a*B*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 5292*a^2*b^5*d^5*e^2 + 7350*a^3*b^4*d^4*e^3 + 4200*a^4*b^3*d^3*e
^4 + 945*a^5*b^2*d^2*e^5 + 70*a^6*b*d*e^6 + a^7*e^7) + 5*A*b*(9*b^7*d^7 + 168*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^
2 + 1764*a^3*b^4*d^4*e^3 + 1470*a^4*b^3*d^3*e^4 + 504*a^5*b^2*d^2*e^5 + 63*a^6*b*d*e^6 + 2*a^7*e^7))*x^9)/9 +
(a*b*(a*B*(9*b^7*d^7 + 168*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^2 + 1764*a^3*b^4*d^4*e^3 + 1470*a^4*b^3*d^3*e^4 + 5
04*a^5*b^2*d^2*e^5 + 63*a^6*b*d*e^6 + 2*a^7*e^7) + A*b*(2*b^7*d^7 + 63*a*b^6*d^6*e + 504*a^2*b^5*d^5*e^2 + 147
0*a^3*b^4*d^4*e^3 + 1764*a^4*b^3*d^3*e^4 + 882*a^5*b^2*d^2*e^5 + 168*a^6*b*d*e^6 + 9*a^7*e^7))*x^10)/2 + (b^2*
(5*a*B*(2*b^7*d^7 + 63*a*b^6*d^6*e + 504*a^2*b^5*d^5*e^2 + 1470*a^3*b^4*d^4*e^3 + 1764*a^4*b^3*d^3*e^4 + 882*a
^5*b^2*d^2*e^5 + 168*a^6*b*d*e^6 + 9*a^7*e^7) + A*b*(b^7*d^7 + 70*a*b^6*d^6*e + 945*a^2*b^5*d^5*e^2 + 4200*a^3
*b^4*d^4*e^3 + 7350*a^4*b^3*d^3*e^4 + 5292*a^5*b^2*d^2*e^5 + 1470*a^6*b*d*e^6 + 120*a^7*e^7))*x^11)/11 + (b^3*
(120*a^7*B*e^7 + 4200*a^3*b^4*d^3*e^3*(B*d + A*e) + 1764*a^5*b^2*d*e^5*(3*B*d + A*e) + 210*a^6*b*e^6*(7*B*d +
A*e) + 70*a*b^6*d^5*e*(B*d + 3*A*e) + 1470*a^4*b^3*d^2*e^4*(5*B*d + 3*A*e) + 315*a^2*b^5*d^4*e^2*(3*B*d + 5*A*
e) + b^7*d^6*(B*d + 7*A*e))*x^12)/12 + (7*b^4*e*(30*a^6*B*e^6 + 225*a^2*b^4*d^3*e^2*(B*d + A*e) + 210*a^4*b^2*
d*e^4*(3*B*d + A*e) + 36*a^5*b*e^5*(7*B*d + A*e) + b^6*d^5*(B*d + 3*A*e) + 120*a^3*b^3*d^2*e^3*(5*B*d + 3*A*e)
+ 10*a*b^5*d^4*e*(3*B*d + 5*A*e))*x^13)/13 + (b^5*e^2*(36*a^5*B*e^5 + 50*a*b^4*d^3*e*(B*d + A*e) + 120*a^3*b^
2*d*e^3*(3*B*d + A*e) + 30*a^4*b*e^4*(7*B*d + A*e) + 45*a^2*b^3*d^2*e^2*(5*B*d + 3*A*e) + b^5*d^4*(3*B*d + 5*A
*e))*x^14)/2 + (b^6*e^3*(42*a^4*B*e^4 + 7*b^4*d^3*(B*d + A*e) + 63*a^2*b^2*d*e^2*(3*B*d + A*e) + 24*a^3*b*e^3*
(7*B*d + A*e) + 14*a*b^3*d^2*e*(5*B*d + 3*A*e))*x^15)/3 + (b^7*e^4*(120*a^3*B*e^3 + 70*a*b^2*d*e*(3*B*d + A*e)
+ 45*a^2*b*e^2*(7*B*d + A*e) + 7*b^3*d^2*(5*B*d + 3*A*e))*x^16)/16 + (b^8*e^5*(45*a^2*B*e^2 + 7*b^2*d*(3*B*d
+ A*e) + 10*a*b*e*(7*B*d + A*e))*x^17)/17 + (b^9*e^6*(7*b*B*d + A*b*e + 10*a*B*e)*x^18)/18 + (b^10*B*e^7*x^19)
/19

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2188\) vs. \(2(311)=622\).

Time = 0.69 (sec) , antiderivative size = 2189, normalized size of antiderivative = 6.65

method	result	size
default	\(\text {Expression too large to display}\)	\(2189\)
norman	\(\text {Expression too large to display}\)	\(2347\)
gosper	\(\text {Expression too large to display}\)	\(2784\)
risch	\(\text {Expression too large to display}\)	\(2784\)
parallelrisch	\(\text {Expression too large to display}\)	\(2784\)

[In]

int((b*x+a)^10*(B*x+A)*(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

1/19*b^10*B*e^7*x^19+1/18*((A*b^10+10*B*a*b^9)*e^7+7*b^10*B*d*e^6)*x^18+1/17*((10*A*a*b^9+45*B*a^2*b^8)*e^7+7*
(A*b^10+10*B*a*b^9)*d*e^6+21*b^10*B*d^2*e^5)*x^17+1/16*((45*A*a^2*b^8+120*B*a^3*b^7)*e^7+7*(10*A*a*b^9+45*B*a^
2*b^8)*d*e^6+21*(A*b^10+10*B*a*b^9)*d^2*e^5+35*b^10*B*d^3*e^4)*x^16+1/15*((120*A*a^3*b^7+210*B*a^4*b^6)*e^7+7*
(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^6+21*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^5+35*(A*b^10+10*B*a*b^9)*d^3*e^4+35*b^10
*B*d^4*e^3)*x^15+1/14*((210*A*a^4*b^6+252*B*a^5*b^5)*e^7+7*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^6+21*(45*A*a^2*b^
8+120*B*a^3*b^7)*d^2*e^5+35*(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^4+35*(A*b^10+10*B*a*b^9)*d^4*e^3+21*b^10*B*d^5*e^2
)*x^14+1/13*((252*A*a^5*b^5+210*B*a^6*b^4)*e^7+7*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e^6+21*(120*A*a^3*b^7+210*B*a
^4*b^6)*d^2*e^5+35*(45*A*a^2*b^8+120*B*a^3*b^7)*d^3*e^4+35*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^3+21*(A*b^10+10*B*a
*b^9)*d^5*e^2+7*b^10*B*d^6*e)*x^13+1/12*((210*A*a^6*b^4+120*B*a^7*b^3)*e^7+7*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e
^6+21*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^5+35*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^4+35*(45*A*a^2*b^8+120*B*a^
3*b^7)*d^4*e^3+21*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^2+7*(A*b^10+10*B*a*b^9)*d^6*e+b^10*B*d^7)*x^12+1/11*((120*A*
a^7*b^3+45*B*a^8*b^2)*e^7+7*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^6+21*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^5+35*(2
10*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^4+35*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^3+21*(45*A*a^2*b^8+120*B*a^3*b^7)*d
^5*e^2+7*(10*A*a*b^9+45*B*a^2*b^8)*d^6*e+(A*b^10+10*B*a*b^9)*d^7)*x^11+1/10*((45*A*a^8*b^2+10*B*a^9*b)*e^7+7*(
120*A*a^7*b^3+45*B*a^8*b^2)*d*e^6+21*(210*A*a^6*b^4+120*B*a^7*b^3)*d^2*e^5+35*(252*A*a^5*b^5+210*B*a^6*b^4)*d^
3*e^4+35*(210*A*a^4*b^6+252*B*a^5*b^5)*d^4*e^3+21*(120*A*a^3*b^7+210*B*a^4*b^6)*d^5*e^2+7*(45*A*a^2*b^8+120*B*
a^3*b^7)*d^6*e+(10*A*a*b^9+45*B*a^2*b^8)*d^7)*x^10+1/9*((10*A*a^9*b+B*a^10)*e^7+7*(45*A*a^8*b^2+10*B*a^9*b)*d*
e^6+21*(120*A*a^7*b^3+45*B*a^8*b^2)*d^2*e^5+35*(210*A*a^6*b^4+120*B*a^7*b^3)*d^3*e^4+35*(252*A*a^5*b^5+210*B*a
^6*b^4)*d^4*e^3+21*(210*A*a^4*b^6+252*B*a^5*b^5)*d^5*e^2+7*(120*A*a^3*b^7+210*B*a^4*b^6)*d^6*e+(45*A*a^2*b^8+1
20*B*a^3*b^7)*d^7)*x^9+1/8*(a^10*A*e^7+7*(10*A*a^9*b+B*a^10)*d*e^6+21*(45*A*a^8*b^2+10*B*a^9*b)*d^2*e^5+35*(12
0*A*a^7*b^3+45*B*a^8*b^2)*d^3*e^4+35*(210*A*a^6*b^4+120*B*a^7*b^3)*d^4*e^3+21*(252*A*a^5*b^5+210*B*a^6*b^4)*d^
5*e^2+7*(210*A*a^4*b^6+252*B*a^5*b^5)*d^6*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^7)*x^8+1/7*(7*a^10*A*d*e^6+21*(10*
A*a^9*b+B*a^10)*d^2*e^5+35*(45*A*a^8*b^2+10*B*a^9*b)*d^3*e^4+35*(120*A*a^7*b^3+45*B*a^8*b^2)*d^4*e^3+21*(210*A
*a^6*b^4+120*B*a^7*b^3)*d^5*e^2+7*(252*A*a^5*b^5+210*B*a^6*b^4)*d^6*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^7)*x^7+1
/6*(21*a^10*A*d^2*e^5+35*(10*A*a^9*b+B*a^10)*d^3*e^4+35*(45*A*a^8*b^2+10*B*a^9*b)*d^4*e^3+21*(120*A*a^7*b^3+45
*B*a^8*b^2)*d^5*e^2+7*(210*A*a^6*b^4+120*B*a^7*b^3)*d^6*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^7)*x^6+1/5*(35*a^10*
A*d^3*e^4+35*(10*A*a^9*b+B*a^10)*d^4*e^3+21*(45*A*a^8*b^2+10*B*a^9*b)*d^5*e^2+7*(120*A*a^7*b^3+45*B*a^8*b^2)*d
^6*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^7)*x^5+1/4*(35*a^10*A*d^4*e^3+21*(10*A*a^9*b+B*a^10)*d^5*e^2+7*(45*A*a^8*
b^2+10*B*a^9*b)*d^6*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^7)*x^4+1/3*(21*a^10*A*d^5*e^2+7*(10*A*a^9*b+B*a^10)*d^6*e
+(45*A*a^8*b^2+10*B*a^9*b)*d^7)*x^3+1/2*(7*a^10*A*d^6*e+(10*A*a^9*b+B*a^10)*d^7)*x^2+a^10*A*d^7*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2198 vs. \(2 (311) = 622\).

Time = 0.24 (sec) , antiderivative size = 2198, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="fricas")

[Out]

1/19*B*b^10*e^7*x^19 + A*a^10*d^7*x + 1/18*(7*B*b^10*d*e^6 + (10*B*a*b^9 + A*b^10)*e^7)*x^18 + 1/17*(21*B*b^10
*d^2*e^5 + 7*(10*B*a*b^9 + A*b^10)*d*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^17 + 1/16*(35*B*b^10*d^3*e^4 + 2
1*(10*B*a*b^9 + A*b^10)*d^2*e^5 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^6 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^1
6 + 1/3*(7*B*b^10*d^4*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 + 21*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^7)*x^15 + 1/2*(3*B*b^10*d^5*e^2 + 5*(10*B*a*b^9
+ A*b^10)*d^4*e^3 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^4 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 30*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^7)*x^14 + 7/13*(B*b^10*d^6*e + 3*(10*B*a*b^9 + A*
b^10)*d^5*e^2 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^3 + 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 90*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*d^2*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^13 + 1/
12*(B*b^10*d^7 + 7*(10*B*a*b^9 + A*b^10)*d^6*e + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 + 525*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^4*e^3 + 1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 294*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^6 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^7
+ 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^2 + 1050*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^4*e^3 + 1470*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^4 + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^5 + 210*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^6 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^7)*x^11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^7 +
21*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^2 + 294*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^4*e^3 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^4 + 126*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^5 + 21*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*d*e^6 + (2*B*a^9*b + 9*A*a^8*b^2)*e^7)*x^10 + 1/9*(15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7 + 210*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^2 + 1470*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^3 + 1050*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^4 + 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^5 + 35*(2*B*a^9*b
+ 9*A*a^8*b^2)*d*e^6 + (B*a^10 + 10*A*a^9*b)*e^7)*x^9 + 1/8*(A*a^10*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7 +
294*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^2 + 1050*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*d^4*e^3 + 525*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^4 + 105*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^5 + 7*(B*a^10 +
10*A*a^9*b)*d*e^6)*x^8 + (A*a^10*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
6*e + 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5*e^2 + 75*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^4*e^3 + 25*(2*B*a^9*b + 9*A*a^
8*b^2)*d^3*e^4 + 3*(B*a^10 + 10*A*a^9*b)*d^2*e^5)*x^7 + 7/6*(3*A*a^10*d^2*e^5 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^7 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^6*e + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5*e^2 + 25*(2*B*a^9*b + 9*A*a^8*
b^2)*d^4*e^3 + 5*(B*a^10 + 10*A*a^9*b)*d^3*e^4)*x^6 + (7*A*a^10*d^3*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^7 +
21*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^6*e + 21*(2*B*a^9*b + 9*A*a^8*b^2)*d^5*e^2 + 7*(B*a^10 + 10*A*a^9*b)*d^4*e^3)
*x^5 + 1/4*(35*A*a^10*d^4*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^7 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^6*e + 21*(
B*a^10 + 10*A*a^9*b)*d^5*e^2)*x^4 + 1/3*(21*A*a^10*d^5*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^7 + 7*(B*a^10 + 10*
A*a^9*b)*d^6*e)*x^3 + 1/2*(7*A*a^10*d^6*e + (B*a^10 + 10*A*a^9*b)*d^7)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2824 vs. \(2 (335) = 670\).

Time = 0.16 (sec) , antiderivative size = 2824, normalized size of antiderivative = 8.58 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)**10*(B*x+A)*(e*x+d)**7,x)

[Out]

A*a**10*d**7*x + B*b**10*e**7*x**19/19 + x**18*(A*b**10*e**7/18 + 5*B*a*b**9*e**7/9 + 7*B*b**10*d*e**6/18) + x
**17*(10*A*a*b**9*e**7/17 + 7*A*b**10*d*e**6/17 + 45*B*a**2*b**8*e**7/17 + 70*B*a*b**9*d*e**6/17 + 21*B*b**10*
d**2*e**5/17) + x**16*(45*A*a**2*b**8*e**7/16 + 35*A*a*b**9*d*e**6/8 + 21*A*b**10*d**2*e**5/16 + 15*B*a**3*b**
7*e**7/2 + 315*B*a**2*b**8*d*e**6/16 + 105*B*a*b**9*d**2*e**5/8 + 35*B*b**10*d**3*e**4/16) + x**15*(8*A*a**3*b
**7*e**7 + 21*A*a**2*b**8*d*e**6 + 14*A*a*b**9*d**2*e**5 + 7*A*b**10*d**3*e**4/3 + 14*B*a**4*b**6*e**7 + 56*B*
a**3*b**7*d*e**6 + 63*B*a**2*b**8*d**2*e**5 + 70*B*a*b**9*d**3*e**4/3 + 7*B*b**10*d**4*e**3/3) + x**14*(15*A*a
**4*b**6*e**7 + 60*A*a**3*b**7*d*e**6 + 135*A*a**2*b**8*d**2*e**5/2 + 25*A*a*b**9*d**3*e**4 + 5*A*b**10*d**4*e
**3/2 + 18*B*a**5*b**5*e**7 + 105*B*a**4*b**6*d*e**6 + 180*B*a**3*b**7*d**2*e**5 + 225*B*a**2*b**8*d**3*e**4/2
+ 25*B*a*b**9*d**4*e**3 + 3*B*b**10*d**5*e**2/2) + x**13*(252*A*a**5*b**5*e**7/13 + 1470*A*a**4*b**6*d*e**6/1
3 + 2520*A*a**3*b**7*d**2*e**5/13 + 1575*A*a**2*b**8*d**3*e**4/13 + 350*A*a*b**9*d**4*e**3/13 + 21*A*b**10*d**
5*e**2/13 + 210*B*a**6*b**4*e**7/13 + 1764*B*a**5*b**5*d*e**6/13 + 4410*B*a**4*b**6*d**2*e**5/13 + 4200*B*a**3
*b**7*d**3*e**4/13 + 1575*B*a**2*b**8*d**4*e**3/13 + 210*B*a*b**9*d**5*e**2/13 + 7*B*b**10*d**6*e/13) + x**12*
(35*A*a**6*b**4*e**7/2 + 147*A*a**5*b**5*d*e**6 + 735*A*a**4*b**6*d**2*e**5/2 + 350*A*a**3*b**7*d**3*e**4 + 52
5*A*a**2*b**8*d**4*e**3/4 + 35*A*a*b**9*d**5*e**2/2 + 7*A*b**10*d**6*e/12 + 10*B*a**7*b**3*e**7 + 245*B*a**6*b
**4*d*e**6/2 + 441*B*a**5*b**5*d**2*e**5 + 1225*B*a**4*b**6*d**3*e**4/2 + 350*B*a**3*b**7*d**4*e**3 + 315*B*a*
*2*b**8*d**5*e**2/4 + 35*B*a*b**9*d**6*e/6 + B*b**10*d**7/12) + x**11*(120*A*a**7*b**3*e**7/11 + 1470*A*a**6*b
**4*d*e**6/11 + 5292*A*a**5*b**5*d**2*e**5/11 + 7350*A*a**4*b**6*d**3*e**4/11 + 4200*A*a**3*b**7*d**4*e**3/11
+ 945*A*a**2*b**8*d**5*e**2/11 + 70*A*a*b**9*d**6*e/11 + A*b**10*d**7/11 + 45*B*a**8*b**2*e**7/11 + 840*B*a**7
*b**3*d*e**6/11 + 4410*B*a**6*b**4*d**2*e**5/11 + 8820*B*a**5*b**5*d**3*e**4/11 + 7350*B*a**4*b**6*d**4*e**3/1
1 + 2520*B*a**3*b**7*d**5*e**2/11 + 315*B*a**2*b**8*d**6*e/11 + 10*B*a*b**9*d**7/11) + x**10*(9*A*a**8*b**2*e*
*7/2 + 84*A*a**7*b**3*d*e**6 + 441*A*a**6*b**4*d**2*e**5 + 882*A*a**5*b**5*d**3*e**4 + 735*A*a**4*b**6*d**4*e*
*3 + 252*A*a**3*b**7*d**5*e**2 + 63*A*a**2*b**8*d**6*e/2 + A*a*b**9*d**7 + B*a**9*b*e**7 + 63*B*a**8*b**2*d*e*
*6/2 + 252*B*a**7*b**3*d**2*e**5 + 735*B*a**6*b**4*d**3*e**4 + 882*B*a**5*b**5*d**4*e**3 + 441*B*a**4*b**6*d**
5*e**2 + 84*B*a**3*b**7*d**6*e + 9*B*a**2*b**8*d**7/2) + x**9*(10*A*a**9*b*e**7/9 + 35*A*a**8*b**2*d*e**6 + 28
0*A*a**7*b**3*d**2*e**5 + 2450*A*a**6*b**4*d**3*e**4/3 + 980*A*a**5*b**5*d**4*e**3 + 490*A*a**4*b**6*d**5*e**2
+ 280*A*a**3*b**7*d**6*e/3 + 5*A*a**2*b**8*d**7 + B*a**10*e**7/9 + 70*B*a**9*b*d*e**6/9 + 105*B*a**8*b**2*d**
2*e**5 + 1400*B*a**7*b**3*d**3*e**4/3 + 2450*B*a**6*b**4*d**4*e**3/3 + 588*B*a**5*b**5*d**5*e**2 + 490*B*a**4*
b**6*d**6*e/3 + 40*B*a**3*b**7*d**7/3) + x**8*(A*a**10*e**7/8 + 35*A*a**9*b*d*e**6/4 + 945*A*a**8*b**2*d**2*e*
*5/8 + 525*A*a**7*b**3*d**3*e**4 + 3675*A*a**6*b**4*d**4*e**3/4 + 1323*A*a**5*b**5*d**5*e**2/2 + 735*A*a**4*b*
*6*d**6*e/4 + 15*A*a**3*b**7*d**7 + 7*B*a**10*d*e**6/8 + 105*B*a**9*b*d**2*e**5/4 + 1575*B*a**8*b**2*d**3*e**4
/8 + 525*B*a**7*b**3*d**4*e**3 + 2205*B*a**6*b**4*d**5*e**2/4 + 441*B*a**5*b**5*d**6*e/2 + 105*B*a**4*b**6*d**
7/4) + x**7*(A*a**10*d*e**6 + 30*A*a**9*b*d**2*e**5 + 225*A*a**8*b**2*d**3*e**4 + 600*A*a**7*b**3*d**4*e**3 +
630*A*a**6*b**4*d**5*e**2 + 252*A*a**5*b**5*d**6*e + 30*A*a**4*b**6*d**7 + 3*B*a**10*d**2*e**5 + 50*B*a**9*b*d
**3*e**4 + 225*B*a**8*b**2*d**4*e**3 + 360*B*a**7*b**3*d**5*e**2 + 210*B*a**6*b**4*d**6*e + 36*B*a**5*b**5*d**
7) + x**6*(7*A*a**10*d**2*e**5/2 + 175*A*a**9*b*d**3*e**4/3 + 525*A*a**8*b**2*d**4*e**3/2 + 420*A*a**7*b**3*d*
*5*e**2 + 245*A*a**6*b**4*d**6*e + 42*A*a**5*b**5*d**7 + 35*B*a**10*d**3*e**4/6 + 175*B*a**9*b*d**4*e**3/3 + 3
15*B*a**8*b**2*d**5*e**2/2 + 140*B*a**7*b**3*d**6*e + 35*B*a**6*b**4*d**7) + x**5*(7*A*a**10*d**3*e**4 + 70*A*
a**9*b*d**4*e**3 + 189*A*a**8*b**2*d**5*e**2 + 168*A*a**7*b**3*d**6*e + 42*A*a**6*b**4*d**7 + 7*B*a**10*d**4*e
**3 + 42*B*a**9*b*d**5*e**2 + 63*B*a**8*b**2*d**6*e + 24*B*a**7*b**3*d**7) + x**4*(35*A*a**10*d**4*e**3/4 + 10
5*A*a**9*b*d**5*e**2/2 + 315*A*a**8*b**2*d**6*e/4 + 30*A*a**7*b**3*d**7 + 21*B*a**10*d**5*e**2/4 + 35*B*a**9*b
*d**6*e/2 + 45*B*a**8*b**2*d**7/4) + x**3*(7*A*a**10*d**5*e**2 + 70*A*a**9*b*d**6*e/3 + 15*A*a**8*b**2*d**7 +
7*B*a**10*d**6*e/3 + 10*B*a**9*b*d**7/3) + x**2*(7*A*a**10*d**6*e/2 + 5*A*a**9*b*d**7 + B*a**10*d**7/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2198 vs. \(2 (311) = 622\).

Time = 0.21 (sec) , antiderivative size = 2198, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="maxima")

[Out]

1/19*B*b^10*e^7*x^19 + A*a^10*d^7*x + 1/18*(7*B*b^10*d*e^6 + (10*B*a*b^9 + A*b^10)*e^7)*x^18 + 1/17*(21*B*b^10
*d^2*e^5 + 7*(10*B*a*b^9 + A*b^10)*d*e^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^17 + 1/16*(35*B*b^10*d^3*e^4 + 2
1*(10*B*a*b^9 + A*b^10)*d^2*e^5 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^6 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^1
6 + 1/3*(7*B*b^10*d^4*e^3 + 7*(10*B*a*b^9 + A*b^10)*d^3*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 + 21*(8*B*a
^3*b^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^7)*x^15 + 1/2*(3*B*b^10*d^5*e^2 + 5*(10*B*a*b^9
+ A*b^10)*d^4*e^3 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^4 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 30*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^7)*x^14 + 7/13*(B*b^10*d^6*e + 3*(10*B*a*b^9 + A*
b^10)*d^5*e^2 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^3 + 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 90*(7*B*a^4*b^
6 + 4*A*a^3*b^7)*d^2*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^13 + 1/
12*(B*b^10*d^7 + 7*(10*B*a*b^9 + A*b^10)*d^6*e + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 + 525*(8*B*a^3*b^7 + 3*
A*a^2*b^8)*d^4*e^3 + 1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^5 + 294*
(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^6 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^7)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^7
+ 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^2 + 1050*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d^4*e^3 + 1470*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^4 + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^5 + 210*(4*B*a
^7*b^3 + 7*A*a^6*b^4)*d*e^6 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^7)*x^11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^7 +
21*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e + 126*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^2 + 294*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*d^4*e^3 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^4 + 126*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^5 + 21*(3*B*a^8*b
^2 + 8*A*a^7*b^3)*d*e^6 + (2*B*a^9*b + 9*A*a^8*b^2)*e^7)*x^10 + 1/9*(15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7 + 210*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e + 882*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^2 + 1470*(5*B*a^6*b^4 + 6*A*a^5*b^5)
*d^4*e^3 + 1050*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^4 + 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^5 + 35*(2*B*a^9*b
+ 9*A*a^8*b^2)*d*e^6 + (B*a^10 + 10*A*a^9*b)*e^7)*x^9 + 1/8*(A*a^10*e^7 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7 +
294*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e + 882*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^2 + 1050*(4*B*a^7*b^3 + 7*A*a^6
*b^4)*d^4*e^3 + 525*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^4 + 105*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^5 + 7*(B*a^10 +
10*A*a^9*b)*d*e^6)*x^8 + (A*a^10*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^7 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^
6*e + 90*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^5*e^2 + 75*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^4*e^3 + 25*(2*B*a^9*b + 9*A*a^
8*b^2)*d^3*e^4 + 3*(B*a^10 + 10*A*a^9*b)*d^2*e^5)*x^7 + 7/6*(3*A*a^10*d^2*e^5 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*
d^7 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^6*e + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^5*e^2 + 25*(2*B*a^9*b + 9*A*a^8*
b^2)*d^4*e^3 + 5*(B*a^10 + 10*A*a^9*b)*d^3*e^4)*x^6 + (7*A*a^10*d^3*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^7 +
21*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^6*e + 21*(2*B*a^9*b + 9*A*a^8*b^2)*d^5*e^2 + 7*(B*a^10 + 10*A*a^9*b)*d^4*e^3)
*x^5 + 1/4*(35*A*a^10*d^4*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^7 + 35*(2*B*a^9*b + 9*A*a^8*b^2)*d^6*e + 21*(
B*a^10 + 10*A*a^9*b)*d^5*e^2)*x^4 + 1/3*(21*A*a^10*d^5*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^7 + 7*(B*a^10 + 10*
A*a^9*b)*d^6*e)*x^3 + 1/2*(7*A*a^10*d^6*e + (B*a^10 + 10*A*a^9*b)*d^7)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2783 vs. \(2 (311) = 622\).

Time = 0.31 (sec) , antiderivative size = 2783, normalized size of antiderivative = 8.46 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="giac")

[Out]

1/19*B*b^10*e^7*x^19 + 7/18*B*b^10*d*e^6*x^18 + 5/9*B*a*b^9*e^7*x^18 + 1/18*A*b^10*e^7*x^18 + 21/17*B*b^10*d^2
*e^5*x^17 + 70/17*B*a*b^9*d*e^6*x^17 + 7/17*A*b^10*d*e^6*x^17 + 45/17*B*a^2*b^8*e^7*x^17 + 10/17*A*a*b^9*e^7*x
^17 + 35/16*B*b^10*d^3*e^4*x^16 + 105/8*B*a*b^9*d^2*e^5*x^16 + 21/16*A*b^10*d^2*e^5*x^16 + 315/16*B*a^2*b^8*d*
e^6*x^16 + 35/8*A*a*b^9*d*e^6*x^16 + 15/2*B*a^3*b^7*e^7*x^16 + 45/16*A*a^2*b^8*e^7*x^16 + 7/3*B*b^10*d^4*e^3*x
^15 + 70/3*B*a*b^9*d^3*e^4*x^15 + 7/3*A*b^10*d^3*e^4*x^15 + 63*B*a^2*b^8*d^2*e^5*x^15 + 14*A*a*b^9*d^2*e^5*x^1
5 + 56*B*a^3*b^7*d*e^6*x^15 + 21*A*a^2*b^8*d*e^6*x^15 + 14*B*a^4*b^6*e^7*x^15 + 8*A*a^3*b^7*e^7*x^15 + 3/2*B*b
^10*d^5*e^2*x^14 + 25*B*a*b^9*d^4*e^3*x^14 + 5/2*A*b^10*d^4*e^3*x^14 + 225/2*B*a^2*b^8*d^3*e^4*x^14 + 25*A*a*b
^9*d^3*e^4*x^14 + 180*B*a^3*b^7*d^2*e^5*x^14 + 135/2*A*a^2*b^8*d^2*e^5*x^14 + 105*B*a^4*b^6*d*e^6*x^14 + 60*A*
a^3*b^7*d*e^6*x^14 + 18*B*a^5*b^5*e^7*x^14 + 15*A*a^4*b^6*e^7*x^14 + 7/13*B*b^10*d^6*e*x^13 + 210/13*B*a*b^9*d
^5*e^2*x^13 + 21/13*A*b^10*d^5*e^2*x^13 + 1575/13*B*a^2*b^8*d^4*e^3*x^13 + 350/13*A*a*b^9*d^4*e^3*x^13 + 4200/
13*B*a^3*b^7*d^3*e^4*x^13 + 1575/13*A*a^2*b^8*d^3*e^4*x^13 + 4410/13*B*a^4*b^6*d^2*e^5*x^13 + 2520/13*A*a^3*b^
7*d^2*e^5*x^13 + 1764/13*B*a^5*b^5*d*e^6*x^13 + 1470/13*A*a^4*b^6*d*e^6*x^13 + 210/13*B*a^6*b^4*e^7*x^13 + 252
/13*A*a^5*b^5*e^7*x^13 + 1/12*B*b^10*d^7*x^12 + 35/6*B*a*b^9*d^6*e*x^12 + 7/12*A*b^10*d^6*e*x^12 + 315/4*B*a^2
*b^8*d^5*e^2*x^12 + 35/2*A*a*b^9*d^5*e^2*x^12 + 350*B*a^3*b^7*d^4*e^3*x^12 + 525/4*A*a^2*b^8*d^4*e^3*x^12 + 12
25/2*B*a^4*b^6*d^3*e^4*x^12 + 350*A*a^3*b^7*d^3*e^4*x^12 + 441*B*a^5*b^5*d^2*e^5*x^12 + 735/2*A*a^4*b^6*d^2*e^
5*x^12 + 245/2*B*a^6*b^4*d*e^6*x^12 + 147*A*a^5*b^5*d*e^6*x^12 + 10*B*a^7*b^3*e^7*x^12 + 35/2*A*a^6*b^4*e^7*x^
12 + 10/11*B*a*b^9*d^7*x^11 + 1/11*A*b^10*d^7*x^11 + 315/11*B*a^2*b^8*d^6*e*x^11 + 70/11*A*a*b^9*d^6*e*x^11 +
2520/11*B*a^3*b^7*d^5*e^2*x^11 + 945/11*A*a^2*b^8*d^5*e^2*x^11 + 7350/11*B*a^4*b^6*d^4*e^3*x^11 + 4200/11*A*a^
3*b^7*d^4*e^3*x^11 + 8820/11*B*a^5*b^5*d^3*e^4*x^11 + 7350/11*A*a^4*b^6*d^3*e^4*x^11 + 4410/11*B*a^6*b^4*d^2*e
^5*x^11 + 5292/11*A*a^5*b^5*d^2*e^5*x^11 + 840/11*B*a^7*b^3*d*e^6*x^11 + 1470/11*A*a^6*b^4*d*e^6*x^11 + 45/11*
B*a^8*b^2*e^7*x^11 + 120/11*A*a^7*b^3*e^7*x^11 + 9/2*B*a^2*b^8*d^7*x^10 + A*a*b^9*d^7*x^10 + 84*B*a^3*b^7*d^6*
e*x^10 + 63/2*A*a^2*b^8*d^6*e*x^10 + 441*B*a^4*b^6*d^5*e^2*x^10 + 252*A*a^3*b^7*d^5*e^2*x^10 + 882*B*a^5*b^5*d
^4*e^3*x^10 + 735*A*a^4*b^6*d^4*e^3*x^10 + 735*B*a^6*b^4*d^3*e^4*x^10 + 882*A*a^5*b^5*d^3*e^4*x^10 + 252*B*a^7
*b^3*d^2*e^5*x^10 + 441*A*a^6*b^4*d^2*e^5*x^10 + 63/2*B*a^8*b^2*d*e^6*x^10 + 84*A*a^7*b^3*d*e^6*x^10 + B*a^9*b
*e^7*x^10 + 9/2*A*a^8*b^2*e^7*x^10 + 40/3*B*a^3*b^7*d^7*x^9 + 5*A*a^2*b^8*d^7*x^9 + 490/3*B*a^4*b^6*d^6*e*x^9
+ 280/3*A*a^3*b^7*d^6*e*x^9 + 588*B*a^5*b^5*d^5*e^2*x^9 + 490*A*a^4*b^6*d^5*e^2*x^9 + 2450/3*B*a^6*b^4*d^4*e^3
*x^9 + 980*A*a^5*b^5*d^4*e^3*x^9 + 1400/3*B*a^7*b^3*d^3*e^4*x^9 + 2450/3*A*a^6*b^4*d^3*e^4*x^9 + 105*B*a^8*b^2
*d^2*e^5*x^9 + 280*A*a^7*b^3*d^2*e^5*x^9 + 70/9*B*a^9*b*d*e^6*x^9 + 35*A*a^8*b^2*d*e^6*x^9 + 1/9*B*a^10*e^7*x^
9 + 10/9*A*a^9*b*e^7*x^9 + 105/4*B*a^4*b^6*d^7*x^8 + 15*A*a^3*b^7*d^7*x^8 + 441/2*B*a^5*b^5*d^6*e*x^8 + 735/4*
A*a^4*b^6*d^6*e*x^8 + 2205/4*B*a^6*b^4*d^5*e^2*x^8 + 1323/2*A*a^5*b^5*d^5*e^2*x^8 + 525*B*a^7*b^3*d^4*e^3*x^8
+ 3675/4*A*a^6*b^4*d^4*e^3*x^8 + 1575/8*B*a^8*b^2*d^3*e^4*x^8 + 525*A*a^7*b^3*d^3*e^4*x^8 + 105/4*B*a^9*b*d^2*
e^5*x^8 + 945/8*A*a^8*b^2*d^2*e^5*x^8 + 7/8*B*a^10*d*e^6*x^8 + 35/4*A*a^9*b*d*e^6*x^8 + 1/8*A*a^10*e^7*x^8 + 3
6*B*a^5*b^5*d^7*x^7 + 30*A*a^4*b^6*d^7*x^7 + 210*B*a^6*b^4*d^6*e*x^7 + 252*A*a^5*b^5*d^6*e*x^7 + 360*B*a^7*b^3
*d^5*e^2*x^7 + 630*A*a^6*b^4*d^5*e^2*x^7 + 225*B*a^8*b^2*d^4*e^3*x^7 + 600*A*a^7*b^3*d^4*e^3*x^7 + 50*B*a^9*b*
d^3*e^4*x^7 + 225*A*a^8*b^2*d^3*e^4*x^7 + 3*B*a^10*d^2*e^5*x^7 + 30*A*a^9*b*d^2*e^5*x^7 + A*a^10*d*e^6*x^7 + 3
5*B*a^6*b^4*d^7*x^6 + 42*A*a^5*b^5*d^7*x^6 + 140*B*a^7*b^3*d^6*e*x^6 + 245*A*a^6*b^4*d^6*e*x^6 + 315/2*B*a^8*b
^2*d^5*e^2*x^6 + 420*A*a^7*b^3*d^5*e^2*x^6 + 175/3*B*a^9*b*d^4*e^3*x^6 + 525/2*A*a^8*b^2*d^4*e^3*x^6 + 35/6*B*
a^10*d^3*e^4*x^6 + 175/3*A*a^9*b*d^3*e^4*x^6 + 7/2*A*a^10*d^2*e^5*x^6 + 24*B*a^7*b^3*d^7*x^5 + 42*A*a^6*b^4*d^
7*x^5 + 63*B*a^8*b^2*d^6*e*x^5 + 168*A*a^7*b^3*d^6*e*x^5 + 42*B*a^9*b*d^5*e^2*x^5 + 189*A*a^8*b^2*d^5*e^2*x^5
+ 7*B*a^10*d^4*e^3*x^5 + 70*A*a^9*b*d^4*e^3*x^5 + 7*A*a^10*d^3*e^4*x^5 + 45/4*B*a^8*b^2*d^7*x^4 + 30*A*a^7*b^3
*d^7*x^4 + 35/2*B*a^9*b*d^6*e*x^4 + 315/4*A*a^8*b^2*d^6*e*x^4 + 21/4*B*a^10*d^5*e^2*x^4 + 105/2*A*a^9*b*d^5*e^
2*x^4 + 35/4*A*a^10*d^4*e^3*x^4 + 10/3*B*a^9*b*d^7*x^3 + 15*A*a^8*b^2*d^7*x^3 + 7/3*B*a^10*d^6*e*x^3 + 70/3*A*
a^9*b*d^6*e*x^3 + 7*A*a^10*d^5*e^2*x^3 + 1/2*B*a^10*d^7*x^2 + 5*A*a^9*b*d^7*x^2 + 7/2*A*a^10*d^6*e*x^2 + A*a^1
0*d^7*x

Mupad [B] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 2316, normalized size of antiderivative = 7.04 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \]

[In]

int((A + B*x)*(a + b*x)^10*(d + e*x)^7,x)

[Out]

x^5*(42*A*a^6*b^4*d^7 + 24*B*a^7*b^3*d^7 + 7*A*a^10*d^3*e^4 + 7*B*a^10*d^4*e^3 + 168*A*a^7*b^3*d^6*e + 70*A*a^
9*b*d^4*e^3 + 63*B*a^8*b^2*d^6*e + 42*B*a^9*b*d^5*e^2 + 189*A*a^8*b^2*d^5*e^2) + x^15*(8*A*a^3*b^7*e^7 + 14*B*
a^4*b^6*e^7 + (7*A*b^10*d^3*e^4)/3 + (7*B*b^10*d^4*e^3)/3 + 14*A*a*b^9*d^2*e^5 + 21*A*a^2*b^8*d*e^6 + (70*B*a*
b^9*d^3*e^4)/3 + 56*B*a^3*b^7*d*e^6 + 63*B*a^2*b^8*d^2*e^5) + x^9*((B*a^10*e^7)/9 + (10*A*a^9*b*e^7)/9 + 5*A*a
^2*b^8*d^7 + (40*B*a^3*b^7*d^7)/3 + (280*A*a^3*b^7*d^6*e)/3 + 35*A*a^8*b^2*d*e^6 + (490*B*a^4*b^6*d^6*e)/3 + 4
90*A*a^4*b^6*d^5*e^2 + 980*A*a^5*b^5*d^4*e^3 + (2450*A*a^6*b^4*d^3*e^4)/3 + 280*A*a^7*b^3*d^2*e^5 + 588*B*a^5*
b^5*d^5*e^2 + (2450*B*a^6*b^4*d^4*e^3)/3 + (1400*B*a^7*b^3*d^3*e^4)/3 + 105*B*a^8*b^2*d^2*e^5 + (70*B*a^9*b*d*
e^6)/9) + x^11*((A*b^10*d^7)/11 + (10*B*a*b^9*d^7)/11 + (120*A*a^7*b^3*e^7)/11 + (45*B*a^8*b^2*e^7)/11 + (1470
*A*a^6*b^4*d*e^6)/11 + (315*B*a^2*b^8*d^6*e)/11 + (840*B*a^7*b^3*d*e^6)/11 + (945*A*a^2*b^8*d^5*e^2)/11 + (420
0*A*a^3*b^7*d^4*e^3)/11 + (7350*A*a^4*b^6*d^3*e^4)/11 + (5292*A*a^5*b^5*d^2*e^5)/11 + (2520*B*a^3*b^7*d^5*e^2)
/11 + (7350*B*a^4*b^6*d^4*e^3)/11 + (8820*B*a^5*b^5*d^3*e^4)/11 + (4410*B*a^6*b^4*d^2*e^5)/11 + (70*A*a*b^9*d^
6*e)/11) + x^7*(A*a^10*d*e^6 + 30*A*a^4*b^6*d^7 + 36*B*a^5*b^5*d^7 + 3*B*a^10*d^2*e^5 + 252*A*a^5*b^5*d^6*e +
30*A*a^9*b*d^2*e^5 + 210*B*a^6*b^4*d^6*e + 50*B*a^9*b*d^3*e^4 + 630*A*a^6*b^4*d^5*e^2 + 600*A*a^7*b^3*d^4*e^3
+ 225*A*a^8*b^2*d^3*e^4 + 360*B*a^7*b^3*d^5*e^2 + 225*B*a^8*b^2*d^4*e^3) + x^13*((7*B*b^10*d^6*e)/13 + (252*A*
a^5*b^5*e^7)/13 + (210*B*a^6*b^4*e^7)/13 + (21*A*b^10*d^5*e^2)/13 + (350*A*a*b^9*d^4*e^3)/13 + (1470*A*a^4*b^6
*d*e^6)/13 + (210*B*a*b^9*d^5*e^2)/13 + (1764*B*a^5*b^5*d*e^6)/13 + (1575*A*a^2*b^8*d^3*e^4)/13 + (2520*A*a^3*
b^7*d^2*e^5)/13 + (1575*B*a^2*b^8*d^4*e^3)/13 + (4200*B*a^3*b^7*d^3*e^4)/13 + (4410*B*a^4*b^6*d^2*e^5)/13) + x
^6*(42*A*a^5*b^5*d^7 + 35*B*a^6*b^4*d^7 + (7*A*a^10*d^2*e^5)/2 + (35*B*a^10*d^3*e^4)/6 + 245*A*a^6*b^4*d^6*e +
(175*A*a^9*b*d^3*e^4)/3 + 140*B*a^7*b^3*d^6*e + (175*B*a^9*b*d^4*e^3)/3 + 420*A*a^7*b^3*d^5*e^2 + (525*A*a^8*
b^2*d^4*e^3)/2 + (315*B*a^8*b^2*d^5*e^2)/2) + x^14*(15*A*a^4*b^6*e^7 + 18*B*a^5*b^5*e^7 + (5*A*b^10*d^4*e^3)/2
+ (3*B*b^10*d^5*e^2)/2 + 25*A*a*b^9*d^3*e^4 + 60*A*a^3*b^7*d*e^6 + 25*B*a*b^9*d^4*e^3 + 105*B*a^4*b^6*d*e^6 +
(135*A*a^2*b^8*d^2*e^5)/2 + (225*B*a^2*b^8*d^3*e^4)/2 + 180*B*a^3*b^7*d^2*e^5) + x^8*((A*a^10*e^7)/8 + (7*B*a
^10*d*e^6)/8 + 15*A*a^3*b^7*d^7 + (105*B*a^4*b^6*d^7)/4 + (735*A*a^4*b^6*d^6*e)/4 + (441*B*a^5*b^5*d^6*e)/2 +
(105*B*a^9*b*d^2*e^5)/4 + (1323*A*a^5*b^5*d^5*e^2)/2 + (3675*A*a^6*b^4*d^4*e^3)/4 + 525*A*a^7*b^3*d^3*e^4 + (9
45*A*a^8*b^2*d^2*e^5)/8 + (2205*B*a^6*b^4*d^5*e^2)/4 + 525*B*a^7*b^3*d^4*e^3 + (1575*B*a^8*b^2*d^3*e^4)/8 + (3
5*A*a^9*b*d*e^6)/4) + x^12*((B*b^10*d^7)/12 + (7*A*b^10*d^6*e)/12 + (35*A*a^6*b^4*e^7)/2 + 10*B*a^7*b^3*e^7 +
(35*A*a*b^9*d^5*e^2)/2 + 147*A*a^5*b^5*d*e^6 + (245*B*a^6*b^4*d*e^6)/2 + (525*A*a^2*b^8*d^4*e^3)/4 + 350*A*a^3
*b^7*d^3*e^4 + (735*A*a^4*b^6*d^2*e^5)/2 + (315*B*a^2*b^8*d^5*e^2)/4 + 350*B*a^3*b^7*d^4*e^3 + (1225*B*a^4*b^6
*d^3*e^4)/2 + 441*B*a^5*b^5*d^2*e^5 + (35*B*a*b^9*d^6*e)/6) + x^4*(30*A*a^7*b^3*d^7 + (45*B*a^8*b^2*d^7)/4 + (
35*A*a^10*d^4*e^3)/4 + (21*B*a^10*d^5*e^2)/4 + (315*A*a^8*b^2*d^6*e)/4 + (105*A*a^9*b*d^5*e^2)/2 + (35*B*a^9*b
*d^6*e)/2) + x^16*((45*A*a^2*b^8*e^7)/16 + (15*B*a^3*b^7*e^7)/2 + (21*A*b^10*d^2*e^5)/16 + (35*B*b^10*d^3*e^4)
/16 + (105*B*a*b^9*d^2*e^5)/8 + (315*B*a^2*b^8*d*e^6)/16 + (35*A*a*b^9*d*e^6)/8) + x^10*(A*a*b^9*d^7 + B*a^9*b
*e^7 + (9*A*a^8*b^2*e^7)/2 + (9*B*a^2*b^8*d^7)/2 + (63*A*a^2*b^8*d^6*e)/2 + 84*A*a^7*b^3*d*e^6 + 84*B*a^3*b^7*
d^6*e + (63*B*a^8*b^2*d*e^6)/2 + 252*A*a^3*b^7*d^5*e^2 + 735*A*a^4*b^6*d^4*e^3 + 882*A*a^5*b^5*d^3*e^4 + 441*A
*a^6*b^4*d^2*e^5 + 441*B*a^4*b^6*d^5*e^2 + 882*B*a^5*b^5*d^4*e^3 + 735*B*a^6*b^4*d^3*e^4 + 252*B*a^7*b^3*d^2*e
^5) + (a^9*d^6*x^2*(7*A*a*e + 10*A*b*d + B*a*d))/2 + (b^9*e^6*x^18*(A*b*e + 10*B*a*e + 7*B*b*d))/18 + A*a^10*d
^7*x + (a^8*d^5*x^3*(21*A*a^2*e^2 + 45*A*b^2*d^2 + 10*B*a*b*d^2 + 7*B*a^2*d*e + 70*A*a*b*d*e))/3 + (b^8*e^5*x^
17*(45*B*a^2*e^2 + 21*B*b^2*d^2 + 10*A*a*b*e^2 + 7*A*b^2*d*e + 70*B*a*b*d*e))/17 + (B*b^10*e^7*x^19)/19