3.11.86 \(\int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx\) [1086]

Optimal. Leaf size=118 \[ \frac {(A b-a B) (b d-a e)^2 (a+b x)^{11}}{11 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^{12}}{12 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^{13}}{13 b^4}+\frac {B e^2 (a+b x)^{14}}{14 b^4} \]

[Out]

1/11*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^11/b^4+1/12*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^12/b^4+1/13*e*(A*b*
e-3*B*a*e+2*B*b*d)*(b*x+a)^13/b^4+1/14*B*e^2*(b*x+a)^14/b^4

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Rubi [A]
time = 0.44, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {e (a+b x)^{13} (-3 a B e+A b e+2 b B d)}{13 b^4}+\frac {(a+b x)^{12} (b d-a e) (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^2}{11 b^4}+\frac {B e^2 (a+b x)^{14}}{14 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]

[Out]

((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^11)/(11*b^4) + ((b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^12)/(12
*b^4) + (e*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^13)/(13*b^4) + (B*e^2*(a + b*x)^14)/(14*b^4)

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*} \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^2 (a+b x)^{10}}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^{11}}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^{12}}{b^3}+\frac {B e^2 (a+b x)^{13}}{b^3}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^{11}}{11 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^{12}}{12 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^{13}}{13 b^4}+\frac {B e^2 (a+b x)^{14}}{14 b^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(118)=236\).
time = 0.28, size = 614, normalized size = 5.20 \begin {gather*} \frac {x \left (1001 a^{10} \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+2002 a^9 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+9009 a^8 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+3432 a^7 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3003 a^6 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+6006 a^5 b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+1001 a^4 b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )+364 a^3 b^7 x^7 \left (11 A \left (45 d^2+80 d e x+36 e^2 x^2\right )+8 B x \left (55 d^2+99 d e x+45 e^2 x^2\right )\right )+273 a^2 b^8 x^8 \left (4 A \left (55 d^2+99 d e x+45 e^2 x^2\right )+3 B x \left (66 d^2+120 d e x+55 e^2 x^2\right )\right )+14 a b^9 x^9 \left (13 A \left (66 d^2+120 d e x+55 e^2 x^2\right )+10 B x \left (78 d^2+143 d e x+66 e^2 x^2\right )\right )+b^{10} x^{10} \left (14 A \left (78 d^2+143 d e x+66 e^2 x^2\right )+11 B x \left (91 d^2+168 d e x+78 e^2 x^2\right )\right )\right )}{12012} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]

[Out]

(x*(1001*a^10*(4*A*(3*d^2 + 3*d*e*x + e^2*x^2) + B*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)) + 2002*a^9*b*x*(5*A*(6*d^2
+ 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*e*x + 6*e^2*x^2)) + 9009*a^8*b^2*x^2*(2*A*(10*d^2 + 15*d*e*x +
6*e^2*x^2) + B*x*(15*d^2 + 24*d*e*x + 10*e^2*x^2)) + 3432*a^7*b^3*x^3*(7*A*(15*d^2 + 24*d*e*x + 10*e^2*x^2) +
4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2)) + 3003*a^6*b^4*x^4*(8*A*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 5*B*x*(28*d
^2 + 48*d*e*x + 21*e^2*x^2)) + 6006*a^5*b^5*x^5*(3*A*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 2*B*x*(36*d^2 + 63*d*e
*x + 28*e^2*x^2)) + 1001*a^4*b^6*x^6*(10*A*(36*d^2 + 63*d*e*x + 28*e^2*x^2) + 7*B*x*(45*d^2 + 80*d*e*x + 36*e^
2*x^2)) + 364*a^3*b^7*x^7*(11*A*(45*d^2 + 80*d*e*x + 36*e^2*x^2) + 8*B*x*(55*d^2 + 99*d*e*x + 45*e^2*x^2)) + 2
73*a^2*b^8*x^8*(4*A*(55*d^2 + 99*d*e*x + 45*e^2*x^2) + 3*B*x*(66*d^2 + 120*d*e*x + 55*e^2*x^2)) + 14*a*b^9*x^9
*(13*A*(66*d^2 + 120*d*e*x + 55*e^2*x^2) + 10*B*x*(78*d^2 + 143*d*e*x + 66*e^2*x^2)) + b^10*x^10*(14*A*(78*d^2
+ 143*d*e*x + 66*e^2*x^2) + 11*B*x*(91*d^2 + 168*d*e*x + 78*e^2*x^2))))/12012

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(768\) vs. \(2(110)=220\).
time = 0.08, size = 769, normalized size = 6.52 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)*(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/14*b^10*B*e^2*x^14+1/13*((A*b^10+10*B*a*b^9)*e^2+2*b^10*B*d*e)*x^13+1/12*((10*A*a*b^9+45*B*a^2*b^8)*e^2+2*(A
*b^10+10*B*a*b^9)*d*e+b^10*B*d^2)*x^12+1/11*((45*A*a^2*b^8+120*B*a^3*b^7)*e^2+2*(10*A*a*b^9+45*B*a^2*b^8)*d*e+
(A*b^10+10*B*a*b^9)*d^2)*x^11+1/10*((120*A*a^3*b^7+210*B*a^4*b^6)*e^2+2*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e+(10*A
*a*b^9+45*B*a^2*b^8)*d^2)*x^10+1/9*((210*A*a^4*b^6+252*B*a^5*b^5)*e^2+2*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e+(45*
A*a^2*b^8+120*B*a^3*b^7)*d^2)*x^9+1/8*((252*A*a^5*b^5+210*B*a^6*b^4)*e^2+2*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e+(
120*A*a^3*b^7+210*B*a^4*b^6)*d^2)*x^8+1/7*((210*A*a^6*b^4+120*B*a^7*b^3)*e^2+2*(252*A*a^5*b^5+210*B*a^6*b^4)*d
*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^2)*x^7+1/6*((120*A*a^7*b^3+45*B*a^8*b^2)*e^2+2*(210*A*a^6*b^4+120*B*a^7*b^3
)*d*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^2)*x^6+1/5*((45*A*a^8*b^2+10*B*a^9*b)*e^2+2*(120*A*a^7*b^3+45*B*a^8*b^2)
*d*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^2)*x^5+1/4*((10*A*a^9*b+B*a^10)*e^2+2*(45*A*a^8*b^2+10*B*a^9*b)*d*e+(120*
A*a^7*b^3+45*B*a^8*b^2)*d^2)*x^4+1/3*(a^10*A*e^2+2*(10*A*a^9*b+B*a^10)*d*e+(45*A*a^8*b^2+10*B*a^9*b)*d^2)*x^3+
1/2*(2*a^10*A*d*e+(10*A*a^9*b+B*a^10)*d^2)*x^2+a^10*A*d^2*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (116) = 232\).
time = 0.31, size = 794, normalized size = 6.73 \begin {gather*} \frac {1}{14} \, B b^{10} x^{14} e^{2} + A a^{10} d^{2} x + \frac {1}{13} \, {\left (2 \, B b^{10} d e + 10 \, B a b^{9} e^{2} + A b^{10} e^{2}\right )} x^{13} + \frac {1}{12} \, {\left (B b^{10} d^{2} + 45 \, B a^{2} b^{8} e^{2} + 10 \, A a b^{9} e^{2} + 2 \, {\left (10 \, B a b^{9} e + A b^{10} e\right )} d\right )} x^{12} + \frac {1}{11} \, {\left (120 \, B a^{3} b^{7} e^{2} + 45 \, A a^{2} b^{8} e^{2} + {\left (10 \, B a b^{9} + A b^{10}\right )} d^{2} + 10 \, {\left (9 \, B a^{2} b^{8} e + 2 \, A a b^{9} e\right )} d\right )} x^{11} + \frac {1}{2} \, {\left (42 \, B a^{4} b^{6} e^{2} + 24 \, A a^{3} b^{7} e^{2} + {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d^{2} + 6 \, {\left (8 \, B a^{3} b^{7} e + 3 \, A a^{2} b^{8} e\right )} d\right )} x^{10} + \frac {1}{3} \, {\left (84 \, B a^{5} b^{5} e^{2} + 70 \, A a^{4} b^{6} e^{2} + 5 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d^{2} + 20 \, {\left (7 \, B a^{4} b^{6} e + 4 \, A a^{3} b^{7} e\right )} d\right )} x^{9} + \frac {3}{4} \, {\left (35 \, B a^{6} b^{4} e^{2} + 42 \, A a^{5} b^{5} e^{2} + 5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d^{2} + 14 \, {\left (6 \, B a^{5} b^{5} e + 5 \, A a^{4} b^{6} e\right )} d\right )} x^{8} + \frac {6}{7} \, {\left (20 \, B a^{7} b^{3} e^{2} + 35 \, A a^{6} b^{4} e^{2} + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d^{2} + 14 \, {\left (5 \, B a^{6} b^{4} e + 6 \, A a^{5} b^{5} e\right )} d\right )} x^{7} + \frac {1}{2} \, {\left (15 \, B a^{8} b^{2} e^{2} + 40 \, A a^{7} b^{3} e^{2} + 14 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d^{2} + 20 \, {\left (4 \, B a^{7} b^{3} e + 7 \, A a^{6} b^{4} e\right )} d\right )} x^{6} + {\left (2 \, B a^{9} b e^{2} + 9 \, A a^{8} b^{2} e^{2} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d^{2} + 6 \, {\left (3 \, B a^{8} b^{2} e + 8 \, A a^{7} b^{3} e\right )} d\right )} x^{5} + \frac {1}{4} \, {\left (B a^{10} e^{2} + 10 \, A a^{9} b e^{2} + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d^{2} + 10 \, {\left (2 \, B a^{9} b e + 9 \, A a^{8} b^{2} e\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (A a^{10} e^{2} + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d^{2} + 2 \, {\left (B a^{10} e + 10 \, A a^{9} b e\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{10} d e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="maxima")

[Out]

1/14*B*b^10*x^14*e^2 + A*a^10*d^2*x + 1/13*(2*B*b^10*d*e + 10*B*a*b^9*e^2 + A*b^10*e^2)*x^13 + 1/12*(B*b^10*d^
2 + 45*B*a^2*b^8*e^2 + 10*A*a*b^9*e^2 + 2*(10*B*a*b^9*e + A*b^10*e)*d)*x^12 + 1/11*(120*B*a^3*b^7*e^2 + 45*A*a
^2*b^8*e^2 + (10*B*a*b^9 + A*b^10)*d^2 + 10*(9*B*a^2*b^8*e + 2*A*a*b^9*e)*d)*x^11 + 1/2*(42*B*a^4*b^6*e^2 + 24
*A*a^3*b^7*e^2 + (9*B*a^2*b^8 + 2*A*a*b^9)*d^2 + 6*(8*B*a^3*b^7*e + 3*A*a^2*b^8*e)*d)*x^10 + 1/3*(84*B*a^5*b^5
*e^2 + 70*A*a^4*b^6*e^2 + 5*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2 + 20*(7*B*a^4*b^6*e + 4*A*a^3*b^7*e)*d)*x^9 + 3/4*
(35*B*a^6*b^4*e^2 + 42*A*a^5*b^5*e^2 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2 + 14*(6*B*a^5*b^5*e + 5*A*a^4*b^6*e)*
d)*x^8 + 6/7*(20*B*a^7*b^3*e^2 + 35*A*a^6*b^4*e^2 + 7*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2 + 14*(5*B*a^6*b^4*e + 6*
A*a^5*b^5*e)*d)*x^7 + 1/2*(15*B*a^8*b^2*e^2 + 40*A*a^7*b^3*e^2 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2 + 20*(4*B*
a^7*b^3*e + 7*A*a^6*b^4*e)*d)*x^6 + (2*B*a^9*b*e^2 + 9*A*a^8*b^2*e^2 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2 + 6*(
3*B*a^8*b^2*e + 8*A*a^7*b^3*e)*d)*x^5 + 1/4*(B*a^10*e^2 + 10*A*a^9*b*e^2 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2
+ 10*(2*B*a^9*b*e + 9*A*a^8*b^2*e)*d)*x^4 + 1/3*(A*a^10*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2 + 2*(B*a^10*e +
10*A*a^9*b*e)*d)*x^3 + 1/2*(2*A*a^10*d*e + (B*a^10 + 10*A*a^9*b)*d^2)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (116) = 232\).
time = 1.15, size = 782, normalized size = 6.63 \begin {gather*} \frac {1}{12} \, B b^{10} d^{2} x^{12} + A a^{10} d^{2} x + \frac {1}{11} \, {\left (10 \, B a b^{9} + A b^{10}\right )} d^{2} x^{11} + \frac {1}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d^{2} x^{10} + \frac {5}{3} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d^{2} x^{9} + \frac {15}{4} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d^{2} x^{8} + 6 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d^{2} x^{7} + 7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d^{2} x^{6} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d^{2} x^{5} + \frac {15}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d^{2} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} d^{2} x^{2} + \frac {1}{12012} \, {\left (858 \, B b^{10} x^{14} + 4004 \, A a^{10} x^{3} + 924 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + 5005 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + 16380 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 36036 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + 56056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + 63063 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + 51480 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + 30030 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} + 12012 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + 3003 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4}\right )} e^{2} + \frac {1}{858} \, {\left (132 \, B b^{10} d x^{13} + 858 \, A a^{10} d x^{2} + 143 \, {\left (10 \, B a b^{9} + A b^{10}\right )} d x^{12} + 780 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d x^{11} + 2574 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d x^{10} + 5720 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d x^{9} + 9009 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d x^{8} + 10296 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d x^{7} + 8580 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d x^{6} + 5148 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d x^{5} + 2145 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d x^{4} + 572 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} d x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="fricas")

[Out]

1/12*B*b^10*d^2*x^12 + A*a^10*d^2*x + 1/11*(10*B*a*b^9 + A*b^10)*d^2*x^11 + 1/2*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*
x^10 + 5/3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*x^9 + 15/4*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*x^8 + 6*(6*B*a^5*b^5 + 5
*A*a^4*b^6)*d^2*x^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*x^6 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*x^5 + 15/4*(3*
B*a^8*b^2 + 8*A*a^7*b^3)*d^2*x^4 + 5/3*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*x^3 + 1/2*(B*a^10 + 10*A*a^9*b)*d^2*x^2 +
1/12012*(858*B*b^10*x^14 + 4004*A*a^10*x^3 + 924*(10*B*a*b^9 + A*b^10)*x^13 + 5005*(9*B*a^2*b^8 + 2*A*a*b^9)*
x^12 + 16380*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^11 + 36036*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^10 + 56056*(6*B*a^5*b^5 +
5*A*a^4*b^6)*x^9 + 63063*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^8 + 51480*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^7 + 30030*(3*B*
a^8*b^2 + 8*A*a^7*b^3)*x^6 + 12012*(2*B*a^9*b + 9*A*a^8*b^2)*x^5 + 3003*(B*a^10 + 10*A*a^9*b)*x^4)*e^2 + 1/858
*(132*B*b^10*d*x^13 + 858*A*a^10*d*x^2 + 143*(10*B*a*b^9 + A*b^10)*d*x^12 + 780*(9*B*a^2*b^8 + 2*A*a*b^9)*d*x^
11 + 2574*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*x^10 + 5720*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*x^9 + 9009*(6*B*a^5*b^5 + 5*
A*a^4*b^6)*d*x^8 + 10296*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*x^7 + 8580*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*x^6 + 5148*(3*
B*a^8*b^2 + 8*A*a^7*b^3)*d*x^5 + 2145*(2*B*a^9*b + 9*A*a^8*b^2)*d*x^4 + 572*(B*a^10 + 10*A*a^9*b)*d*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (116) = 232\).
time = 0.08, size = 921, normalized size = 7.81 \begin {gather*} A a^{10} d^{2} x + \frac {B b^{10} e^{2} x^{14}}{14} + x^{13} \left (\frac {A b^{10} e^{2}}{13} + \frac {10 B a b^{9} e^{2}}{13} + \frac {2 B b^{10} d e}{13}\right ) + x^{12} \cdot \left (\frac {5 A a b^{9} e^{2}}{6} + \frac {A b^{10} d e}{6} + \frac {15 B a^{2} b^{8} e^{2}}{4} + \frac {5 B a b^{9} d e}{3} + \frac {B b^{10} d^{2}}{12}\right ) + x^{11} \cdot \left (\frac {45 A a^{2} b^{8} e^{2}}{11} + \frac {20 A a b^{9} d e}{11} + \frac {A b^{10} d^{2}}{11} + \frac {120 B a^{3} b^{7} e^{2}}{11} + \frac {90 B a^{2} b^{8} d e}{11} + \frac {10 B a b^{9} d^{2}}{11}\right ) + x^{10} \cdot \left (12 A a^{3} b^{7} e^{2} + 9 A a^{2} b^{8} d e + A a b^{9} d^{2} + 21 B a^{4} b^{6} e^{2} + 24 B a^{3} b^{7} d e + \frac {9 B a^{2} b^{8} d^{2}}{2}\right ) + x^{9} \cdot \left (\frac {70 A a^{4} b^{6} e^{2}}{3} + \frac {80 A a^{3} b^{7} d e}{3} + 5 A a^{2} b^{8} d^{2} + 28 B a^{5} b^{5} e^{2} + \frac {140 B a^{4} b^{6} d e}{3} + \frac {40 B a^{3} b^{7} d^{2}}{3}\right ) + x^{8} \cdot \left (\frac {63 A a^{5} b^{5} e^{2}}{2} + \frac {105 A a^{4} b^{6} d e}{2} + 15 A a^{3} b^{7} d^{2} + \frac {105 B a^{6} b^{4} e^{2}}{4} + 63 B a^{5} b^{5} d e + \frac {105 B a^{4} b^{6} d^{2}}{4}\right ) + x^{7} \cdot \left (30 A a^{6} b^{4} e^{2} + 72 A a^{5} b^{5} d e + 30 A a^{4} b^{6} d^{2} + \frac {120 B a^{7} b^{3} e^{2}}{7} + 60 B a^{6} b^{4} d e + 36 B a^{5} b^{5} d^{2}\right ) + x^{6} \cdot \left (20 A a^{7} b^{3} e^{2} + 70 A a^{6} b^{4} d e + 42 A a^{5} b^{5} d^{2} + \frac {15 B a^{8} b^{2} e^{2}}{2} + 40 B a^{7} b^{3} d e + 35 B a^{6} b^{4} d^{2}\right ) + x^{5} \cdot \left (9 A a^{8} b^{2} e^{2} + 48 A a^{7} b^{3} d e + 42 A a^{6} b^{4} d^{2} + 2 B a^{9} b e^{2} + 18 B a^{8} b^{2} d e + 24 B a^{7} b^{3} d^{2}\right ) + x^{4} \cdot \left (\frac {5 A a^{9} b e^{2}}{2} + \frac {45 A a^{8} b^{2} d e}{2} + 30 A a^{7} b^{3} d^{2} + \frac {B a^{10} e^{2}}{4} + 5 B a^{9} b d e + \frac {45 B a^{8} b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{10} e^{2}}{3} + \frac {20 A a^{9} b d e}{3} + 15 A a^{8} b^{2} d^{2} + \frac {2 B a^{10} d e}{3} + \frac {10 B a^{9} b d^{2}}{3}\right ) + x^{2} \left (A a^{10} d e + 5 A a^{9} b d^{2} + \frac {B a^{10} d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)*(e*x+d)**2,x)

[Out]

A*a**10*d**2*x + B*b**10*e**2*x**14/14 + x**13*(A*b**10*e**2/13 + 10*B*a*b**9*e**2/13 + 2*B*b**10*d*e/13) + x*
*12*(5*A*a*b**9*e**2/6 + A*b**10*d*e/6 + 15*B*a**2*b**8*e**2/4 + 5*B*a*b**9*d*e/3 + B*b**10*d**2/12) + x**11*(
45*A*a**2*b**8*e**2/11 + 20*A*a*b**9*d*e/11 + A*b**10*d**2/11 + 120*B*a**3*b**7*e**2/11 + 90*B*a**2*b**8*d*e/1
1 + 10*B*a*b**9*d**2/11) + x**10*(12*A*a**3*b**7*e**2 + 9*A*a**2*b**8*d*e + A*a*b**9*d**2 + 21*B*a**4*b**6*e**
2 + 24*B*a**3*b**7*d*e + 9*B*a**2*b**8*d**2/2) + x**9*(70*A*a**4*b**6*e**2/3 + 80*A*a**3*b**7*d*e/3 + 5*A*a**2
*b**8*d**2 + 28*B*a**5*b**5*e**2 + 140*B*a**4*b**6*d*e/3 + 40*B*a**3*b**7*d**2/3) + x**8*(63*A*a**5*b**5*e**2/
2 + 105*A*a**4*b**6*d*e/2 + 15*A*a**3*b**7*d**2 + 105*B*a**6*b**4*e**2/4 + 63*B*a**5*b**5*d*e + 105*B*a**4*b**
6*d**2/4) + x**7*(30*A*a**6*b**4*e**2 + 72*A*a**5*b**5*d*e + 30*A*a**4*b**6*d**2 + 120*B*a**7*b**3*e**2/7 + 60
*B*a**6*b**4*d*e + 36*B*a**5*b**5*d**2) + x**6*(20*A*a**7*b**3*e**2 + 70*A*a**6*b**4*d*e + 42*A*a**5*b**5*d**2
+ 15*B*a**8*b**2*e**2/2 + 40*B*a**7*b**3*d*e + 35*B*a**6*b**4*d**2) + x**5*(9*A*a**8*b**2*e**2 + 48*A*a**7*b*
*3*d*e + 42*A*a**6*b**4*d**2 + 2*B*a**9*b*e**2 + 18*B*a**8*b**2*d*e + 24*B*a**7*b**3*d**2) + x**4*(5*A*a**9*b*
e**2/2 + 45*A*a**8*b**2*d*e/2 + 30*A*a**7*b**3*d**2 + B*a**10*e**2/4 + 5*B*a**9*b*d*e + 45*B*a**8*b**2*d**2/4)
+ x**3*(A*a**10*e**2/3 + 20*A*a**9*b*d*e/3 + 15*A*a**8*b**2*d**2 + 2*B*a**10*d*e/3 + 10*B*a**9*b*d**2/3) + x*
*2*(A*a**10*d*e + 5*A*a**9*b*d**2 + B*a**10*d**2/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (116) = 232\).
time = 0.94, size = 904, normalized size = 7.66 \begin {gather*} \frac {1}{14} \, B b^{10} x^{14} e^{2} + \frac {2}{13} \, B b^{10} d x^{13} e + \frac {1}{12} \, B b^{10} d^{2} x^{12} + \frac {10}{13} \, B a b^{9} x^{13} e^{2} + \frac {1}{13} \, A b^{10} x^{13} e^{2} + \frac {5}{3} \, B a b^{9} d x^{12} e + \frac {1}{6} \, A b^{10} d x^{12} e + \frac {10}{11} \, B a b^{9} d^{2} x^{11} + \frac {1}{11} \, A b^{10} d^{2} x^{11} + \frac {15}{4} \, B a^{2} b^{8} x^{12} e^{2} + \frac {5}{6} \, A a b^{9} x^{12} e^{2} + \frac {90}{11} \, B a^{2} b^{8} d x^{11} e + \frac {20}{11} \, A a b^{9} d x^{11} e + \frac {9}{2} \, B a^{2} b^{8} d^{2} x^{10} + A a b^{9} d^{2} x^{10} + \frac {120}{11} \, B a^{3} b^{7} x^{11} e^{2} + \frac {45}{11} \, A a^{2} b^{8} x^{11} e^{2} + 24 \, B a^{3} b^{7} d x^{10} e + 9 \, A a^{2} b^{8} d x^{10} e + \frac {40}{3} \, B a^{3} b^{7} d^{2} x^{9} + 5 \, A a^{2} b^{8} d^{2} x^{9} + 21 \, B a^{4} b^{6} x^{10} e^{2} + 12 \, A a^{3} b^{7} x^{10} e^{2} + \frac {140}{3} \, B a^{4} b^{6} d x^{9} e + \frac {80}{3} \, A a^{3} b^{7} d x^{9} e + \frac {105}{4} \, B a^{4} b^{6} d^{2} x^{8} + 15 \, A a^{3} b^{7} d^{2} x^{8} + 28 \, B a^{5} b^{5} x^{9} e^{2} + \frac {70}{3} \, A a^{4} b^{6} x^{9} e^{2} + 63 \, B a^{5} b^{5} d x^{8} e + \frac {105}{2} \, A a^{4} b^{6} d x^{8} e + 36 \, B a^{5} b^{5} d^{2} x^{7} + 30 \, A a^{4} b^{6} d^{2} x^{7} + \frac {105}{4} \, B a^{6} b^{4} x^{8} e^{2} + \frac {63}{2} \, A a^{5} b^{5} x^{8} e^{2} + 60 \, B a^{6} b^{4} d x^{7} e + 72 \, A a^{5} b^{5} d x^{7} e + 35 \, B a^{6} b^{4} d^{2} x^{6} + 42 \, A a^{5} b^{5} d^{2} x^{6} + \frac {120}{7} \, B a^{7} b^{3} x^{7} e^{2} + 30 \, A a^{6} b^{4} x^{7} e^{2} + 40 \, B a^{7} b^{3} d x^{6} e + 70 \, A a^{6} b^{4} d x^{6} e + 24 \, B a^{7} b^{3} d^{2} x^{5} + 42 \, A a^{6} b^{4} d^{2} x^{5} + \frac {15}{2} \, B a^{8} b^{2} x^{6} e^{2} + 20 \, A a^{7} b^{3} x^{6} e^{2} + 18 \, B a^{8} b^{2} d x^{5} e + 48 \, A a^{7} b^{3} d x^{5} e + \frac {45}{4} \, B a^{8} b^{2} d^{2} x^{4} + 30 \, A a^{7} b^{3} d^{2} x^{4} + 2 \, B a^{9} b x^{5} e^{2} + 9 \, A a^{8} b^{2} x^{5} e^{2} + 5 \, B a^{9} b d x^{4} e + \frac {45}{2} \, A a^{8} b^{2} d x^{4} e + \frac {10}{3} \, B a^{9} b d^{2} x^{3} + 15 \, A a^{8} b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{10} x^{4} e^{2} + \frac {5}{2} \, A a^{9} b x^{4} e^{2} + \frac {2}{3} \, B a^{10} d x^{3} e + \frac {20}{3} \, A a^{9} b d x^{3} e + \frac {1}{2} \, B a^{10} d^{2} x^{2} + 5 \, A a^{9} b d^{2} x^{2} + \frac {1}{3} \, A a^{10} x^{3} e^{2} + A a^{10} d x^{2} e + A a^{10} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="giac")

[Out]

1/14*B*b^10*x^14*e^2 + 2/13*B*b^10*d*x^13*e + 1/12*B*b^10*d^2*x^12 + 10/13*B*a*b^9*x^13*e^2 + 1/13*A*b^10*x^13
*e^2 + 5/3*B*a*b^9*d*x^12*e + 1/6*A*b^10*d*x^12*e + 10/11*B*a*b^9*d^2*x^11 + 1/11*A*b^10*d^2*x^11 + 15/4*B*a^2
*b^8*x^12*e^2 + 5/6*A*a*b^9*x^12*e^2 + 90/11*B*a^2*b^8*d*x^11*e + 20/11*A*a*b^9*d*x^11*e + 9/2*B*a^2*b^8*d^2*x
^10 + A*a*b^9*d^2*x^10 + 120/11*B*a^3*b^7*x^11*e^2 + 45/11*A*a^2*b^8*x^11*e^2 + 24*B*a^3*b^7*d*x^10*e + 9*A*a^
2*b^8*d*x^10*e + 40/3*B*a^3*b^7*d^2*x^9 + 5*A*a^2*b^8*d^2*x^9 + 21*B*a^4*b^6*x^10*e^2 + 12*A*a^3*b^7*x^10*e^2
+ 140/3*B*a^4*b^6*d*x^9*e + 80/3*A*a^3*b^7*d*x^9*e + 105/4*B*a^4*b^6*d^2*x^8 + 15*A*a^3*b^7*d^2*x^8 + 28*B*a^5
*b^5*x^9*e^2 + 70/3*A*a^4*b^6*x^9*e^2 + 63*B*a^5*b^5*d*x^8*e + 105/2*A*a^4*b^6*d*x^8*e + 36*B*a^5*b^5*d^2*x^7
+ 30*A*a^4*b^6*d^2*x^7 + 105/4*B*a^6*b^4*x^8*e^2 + 63/2*A*a^5*b^5*x^8*e^2 + 60*B*a^6*b^4*d*x^7*e + 72*A*a^5*b^
5*d*x^7*e + 35*B*a^6*b^4*d^2*x^6 + 42*A*a^5*b^5*d^2*x^6 + 120/7*B*a^7*b^3*x^7*e^2 + 30*A*a^6*b^4*x^7*e^2 + 40*
B*a^7*b^3*d*x^6*e + 70*A*a^6*b^4*d*x^6*e + 24*B*a^7*b^3*d^2*x^5 + 42*A*a^6*b^4*d^2*x^5 + 15/2*B*a^8*b^2*x^6*e^
2 + 20*A*a^7*b^3*x^6*e^2 + 18*B*a^8*b^2*d*x^5*e + 48*A*a^7*b^3*d*x^5*e + 45/4*B*a^8*b^2*d^2*x^4 + 30*A*a^7*b^3
*d^2*x^4 + 2*B*a^9*b*x^5*e^2 + 9*A*a^8*b^2*x^5*e^2 + 5*B*a^9*b*d*x^4*e + 45/2*A*a^8*b^2*d*x^4*e + 10/3*B*a^9*b
*d^2*x^3 + 15*A*a^8*b^2*d^2*x^3 + 1/4*B*a^10*x^4*e^2 + 5/2*A*a^9*b*x^4*e^2 + 2/3*B*a^10*d*x^3*e + 20/3*A*a^9*b
*d*x^3*e + 1/2*B*a^10*d^2*x^2 + 5*A*a^9*b*d^2*x^2 + 1/3*A*a^10*x^3*e^2 + A*a^10*d*x^2*e + A*a^10*d^2*x

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Mupad [B]
time = 1.37, size = 757, normalized size = 6.42 \begin {gather*} x^6\,\left (\frac {15\,B\,a^8\,b^2\,e^2}{2}+40\,B\,a^7\,b^3\,d\,e+20\,A\,a^7\,b^3\,e^2+35\,B\,a^6\,b^4\,d^2+70\,A\,a^6\,b^4\,d\,e+42\,A\,a^5\,b^5\,d^2\right )+x^7\,\left (\frac {120\,B\,a^7\,b^3\,e^2}{7}+60\,B\,a^6\,b^4\,d\,e+30\,A\,a^6\,b^4\,e^2+36\,B\,a^5\,b^5\,d^2+72\,A\,a^5\,b^5\,d\,e+30\,A\,a^4\,b^6\,d^2\right )+x^9\,\left (28\,B\,a^5\,b^5\,e^2+\frac {140\,B\,a^4\,b^6\,d\,e}{3}+\frac {70\,A\,a^4\,b^6\,e^2}{3}+\frac {40\,B\,a^3\,b^7\,d^2}{3}+\frac {80\,A\,a^3\,b^7\,d\,e}{3}+5\,A\,a^2\,b^8\,d^2\right )+x^8\,\left (\frac {105\,B\,a^6\,b^4\,e^2}{4}+63\,B\,a^5\,b^5\,d\,e+\frac {63\,A\,a^5\,b^5\,e^2}{2}+\frac {105\,B\,a^4\,b^6\,d^2}{4}+\frac {105\,A\,a^4\,b^6\,d\,e}{2}+15\,A\,a^3\,b^7\,d^2\right )+x^4\,\left (\frac {B\,a^{10}\,e^2}{4}+5\,B\,a^9\,b\,d\,e+\frac {5\,A\,a^9\,b\,e^2}{2}+\frac {45\,B\,a^8\,b^2\,d^2}{4}+\frac {45\,A\,a^8\,b^2\,d\,e}{2}+30\,A\,a^7\,b^3\,d^2\right )+x^{11}\,\left (\frac {120\,B\,a^3\,b^7\,e^2}{11}+\frac {90\,B\,a^2\,b^8\,d\,e}{11}+\frac {45\,A\,a^2\,b^8\,e^2}{11}+\frac {10\,B\,a\,b^9\,d^2}{11}+\frac {20\,A\,a\,b^9\,d\,e}{11}+\frac {A\,b^{10}\,d^2}{11}\right )+x^{10}\,\left (21\,B\,a^4\,b^6\,e^2+24\,B\,a^3\,b^7\,d\,e+12\,A\,a^3\,b^7\,e^2+\frac {9\,B\,a^2\,b^8\,d^2}{2}+9\,A\,a^2\,b^8\,d\,e+A\,a\,b^9\,d^2\right )+x^5\,\left (2\,B\,a^9\,b\,e^2+18\,B\,a^8\,b^2\,d\,e+9\,A\,a^8\,b^2\,e^2+24\,B\,a^7\,b^3\,d^2+48\,A\,a^7\,b^3\,d\,e+42\,A\,a^6\,b^4\,d^2\right )+x^3\,\left (\frac {2\,B\,a^{10}\,d\,e}{3}+\frac {A\,a^{10}\,e^2}{3}+\frac {10\,B\,a^9\,b\,d^2}{3}+\frac {20\,A\,a^9\,b\,d\,e}{3}+15\,A\,a^8\,b^2\,d^2\right )+x^{12}\,\left (\frac {15\,B\,a^2\,b^8\,e^2}{4}+\frac {5\,B\,a\,b^9\,d\,e}{3}+\frac {5\,A\,a\,b^9\,e^2}{6}+\frac {B\,b^{10}\,d^2}{12}+\frac {A\,b^{10}\,d\,e}{6}\right )+A\,a^{10}\,d^2\,x+\frac {a^9\,d\,x^2\,\left (2\,A\,a\,e+10\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^9\,e\,x^{13}\,\left (A\,b\,e+10\,B\,a\,e+2\,B\,b\,d\right )}{13}+\frac {B\,b^{10}\,e^2\,x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^10*(d + e*x)^2,x)

[Out]

x^6*(42*A*a^5*b^5*d^2 + 20*A*a^7*b^3*e^2 + 35*B*a^6*b^4*d^2 + (15*B*a^8*b^2*e^2)/2 + 70*A*a^6*b^4*d*e + 40*B*a
^7*b^3*d*e) + x^7*(30*A*a^4*b^6*d^2 + 30*A*a^6*b^4*e^2 + 36*B*a^5*b^5*d^2 + (120*B*a^7*b^3*e^2)/7 + 72*A*a^5*b
^5*d*e + 60*B*a^6*b^4*d*e) + x^9*(5*A*a^2*b^8*d^2 + (70*A*a^4*b^6*e^2)/3 + (40*B*a^3*b^7*d^2)/3 + 28*B*a^5*b^5
*e^2 + (80*A*a^3*b^7*d*e)/3 + (140*B*a^4*b^6*d*e)/3) + x^8*(15*A*a^3*b^7*d^2 + (63*A*a^5*b^5*e^2)/2 + (105*B*a
^4*b^6*d^2)/4 + (105*B*a^6*b^4*e^2)/4 + (105*A*a^4*b^6*d*e)/2 + 63*B*a^5*b^5*d*e) + x^4*((B*a^10*e^2)/4 + (5*A
*a^9*b*e^2)/2 + 30*A*a^7*b^3*d^2 + (45*B*a^8*b^2*d^2)/4 + 5*B*a^9*b*d*e + (45*A*a^8*b^2*d*e)/2) + x^11*((A*b^1
0*d^2)/11 + (10*B*a*b^9*d^2)/11 + (45*A*a^2*b^8*e^2)/11 + (120*B*a^3*b^7*e^2)/11 + (20*A*a*b^9*d*e)/11 + (90*B
*a^2*b^8*d*e)/11) + x^10*(A*a*b^9*d^2 + 12*A*a^3*b^7*e^2 + (9*B*a^2*b^8*d^2)/2 + 21*B*a^4*b^6*e^2 + 9*A*a^2*b^
8*d*e + 24*B*a^3*b^7*d*e) + x^5*(2*B*a^9*b*e^2 + 42*A*a^6*b^4*d^2 + 9*A*a^8*b^2*e^2 + 24*B*a^7*b^3*d^2 + 48*A*
a^7*b^3*d*e + 18*B*a^8*b^2*d*e) + x^3*((A*a^10*e^2)/3 + (2*B*a^10*d*e)/3 + (10*B*a^9*b*d^2)/3 + 15*A*a^8*b^2*d
^2 + (20*A*a^9*b*d*e)/3) + x^12*((B*b^10*d^2)/12 + (A*b^10*d*e)/6 + (5*A*a*b^9*e^2)/6 + (15*B*a^2*b^8*e^2)/4 +
(5*B*a*b^9*d*e)/3) + A*a^10*d^2*x + (a^9*d*x^2*(2*A*a*e + 10*A*b*d + B*a*d))/2 + (b^9*e*x^13*(A*b*e + 10*B*a*
e + 2*B*b*d))/13 + (B*b^10*e^2*x^14)/14

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