3.11.51 \(\int (a+b x)^6 (A+B x) (d+e x)^8 \, dx\) [1051]

Optimal. Leaf size=292 \[ -\frac {(b d-a e)^6 (B d-A e) (d+e x)^9}{9 e^8}+\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{10}}{10 e^8}-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{11}}{11 e^8}+\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{12}}{12 e^8}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{13}}{13 e^8}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{14}}{14 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^{15}}{15 e^8}+\frac {b^6 B (d+e x)^{16}}{16 e^8} \]

[Out]

-1/9*(-a*e+b*d)^6*(-A*e+B*d)*(e*x+d)^9/e^8+1/10*(-a*e+b*d)^5*(-6*A*b*e-B*a*e+7*B*b*d)*(e*x+d)^10/e^8-3/11*b*(-
a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)*(e*x+d)^11/e^8+5/12*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(e*x+d)^
12/e^8-5/13*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d)^13/e^8+3/14*b^4*(-a*e+b*d)*(-2*A*b*e-5*B*a*e+7
*B*b*d)*(e*x+d)^14/e^8-1/15*b^5*(-A*b*e-6*B*a*e+7*B*b*d)*(e*x+d)^15/e^8+1/16*b^6*B*(e*x+d)^16/e^8

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Rubi [A]
time = 0.90, antiderivative size = 292, 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 {b^5 (d+e x)^{15} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac {3 b^4 (d+e x)^{14} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{14 e^8}-\frac {5 b^3 (d+e x)^{13} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac {5 b^2 (d+e x)^{12} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{12 e^8}-\frac {3 b (d+e x)^{11} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac {(d+e x)^{10} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{10 e^8}-\frac {(d+e x)^9 (b d-a e)^6 (B d-A e)}{9 e^8}+\frac {b^6 B (d+e x)^{16}}{16 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]

[Out]

-1/9*((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^9)/e^8 + ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^10)/(1
0*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^11)/(11*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B
*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^12)/(12*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^
13)/(13*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^14)/(14*e^8) - (b^5*(7*b*B*d - A*b*e
- 6*a*B*e)*(d + e*x)^15)/(15*e^8) + (b^6*B*(d + e*x)^16)/(16*e^8)

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1385\) vs. \(2(292)=584\).
time = 0.34, size = 1385, normalized size = 4.74 \begin {gather*} a^6 A d^8 x+\frac {1}{2} a^5 d^7 (6 A b d+a B d+8 a A e) x^2+\frac {1}{3} a^4 d^6 \left (2 a B d (3 b d+4 a e)+A \left (15 b^2 d^2+48 a b d e+28 a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 d^5 \left (a B d \left (15 b^2 d^2+48 a b d e+28 a^2 e^2\right )+4 A \left (5 b^3 d^3+30 a b^2 d^2 e+42 a^2 b d e^2+14 a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 d^4 \left (4 a B d \left (5 b^3 d^3+30 a b^2 d^2 e+42 a^2 b d e^2+14 a^3 e^3\right )+A \left (15 b^4 d^4+160 a b^3 d^3 e+420 a^2 b^2 d^2 e^2+336 a^3 b d e^3+70 a^4 e^4\right )\right ) x^5+\frac {1}{6} a d^3 \left (a B d \left (15 b^4 d^4+160 a b^3 d^3 e+420 a^2 b^2 d^2 e^2+336 a^3 b d e^3+70 a^4 e^4\right )+2 A \left (3 b^5 d^5+60 a b^4 d^4 e+280 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+210 a^4 b d e^4+28 a^5 e^5\right )\right ) x^6+\frac {1}{7} d^2 \left (2 a B d \left (3 b^5 d^5+60 a b^4 d^4 e+280 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+210 a^4 b d e^4+28 a^5 e^5\right )+A \left (b^6 d^6+48 a b^5 d^5 e+420 a^2 b^4 d^4 e^2+1120 a^3 b^3 d^3 e^3+1050 a^4 b^2 d^2 e^4+336 a^5 b d e^5+28 a^6 e^6\right )\right ) x^7+\frac {1}{8} d \left (168 a^5 b d e^5 (2 B d+A e)+420 a^2 b^4 d^4 e^2 (B d+2 A e)+4 a^6 e^6 (7 B d+2 A e)+210 a^4 b^2 d^2 e^4 (5 B d+4 A e)+280 a^3 b^3 d^3 e^3 (4 B d+5 A e)+24 a b^5 d^5 e (2 B d+7 A e)+b^6 d^6 (B d+8 A e)\right ) x^8+\frac {1}{9} e \left (420 a^4 b^2 d^2 e^4 (2 B d+A e)+a^6 e^6 (8 B d+A e)+168 a b^5 d^5 e (B d+2 A e)+24 a^5 b d e^5 (7 B d+2 A e)+280 a^3 b^3 d^3 e^3 (5 B d+4 A e)+210 a^2 b^4 d^4 e^2 (4 B d+5 A e)+4 b^6 d^6 (2 B d+7 A e)\right ) x^9+\frac {1}{10} e^2 \left (a^6 B e^6+560 a^3 b^3 d^2 e^3 (2 B d+A e)+6 a^5 b e^5 (8 B d+A e)+28 b^6 d^5 (B d+2 A e)+60 a^4 b^2 d e^4 (7 B d+2 A e)+210 a^2 b^4 d^3 e^2 (5 B d+4 A e)+84 a b^5 d^4 e (4 B d+5 A e)\right ) x^{10}+\frac {1}{11} b e^3 \left (6 a^5 B e^5+420 a^2 b^3 d^2 e^2 (2 B d+A e)+15 a^4 b e^4 (8 B d+A e)+80 a^3 b^2 d e^3 (7 B d+2 A e)+84 a b^4 d^3 e (5 B d+4 A e)+14 b^5 d^4 (4 B d+5 A e)\right ) x^{11}+\frac {1}{12} b^2 e^4 \left (15 a^4 B e^4+168 a b^3 d^2 e (2 B d+A e)+20 a^3 b e^3 (8 B d+A e)+60 a^2 b^2 d e^2 (7 B d+2 A e)+14 b^4 d^3 (5 B d+4 A e)\right ) x^{12}+\frac {1}{13} b^3 e^5 \left (20 a^3 B e^3+28 b^3 d^2 (2 B d+A e)+15 a^2 b e^2 (8 B d+A e)+24 a b^2 d e (7 B d+2 A e)\right ) x^{13}+\frac {1}{14} b^4 e^6 \left (15 a^2 B e^2+6 a b e (8 B d+A e)+4 b^2 d (7 B d+2 A e)\right ) x^{14}+\frac {1}{15} b^5 e^7 (8 b B d+A b e+6 a B e) x^{15}+\frac {1}{16} b^6 B e^8 x^{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]

[Out]

a^6*A*d^8*x + (a^5*d^7*(6*A*b*d + a*B*d + 8*a*A*e)*x^2)/2 + (a^4*d^6*(2*a*B*d*(3*b*d + 4*a*e) + A*(15*b^2*d^2
+ 48*a*b*d*e + 28*a^2*e^2))*x^3)/3 + (a^3*d^5*(a*B*d*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2) + 4*A*(5*b^3*d^3 +
30*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 14*a^3*e^3))*x^4)/4 + (a^2*d^4*(4*a*B*d*(5*b^3*d^3 + 30*a*b^2*d^2*e + 42*a^
2*b*d*e^2 + 14*a^3*e^3) + A*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4
))*x^5)/5 + (a*d^3*(a*B*d*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4)
+ 2*A*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5))
*x^6)/6 + (d^2*(2*a*B*d*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*
e^4 + 28*a^5*e^5) + A*(b^6*d^6 + 48*a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^
2*e^4 + 336*a^5*b*d*e^5 + 28*a^6*e^6))*x^7)/7 + (d*(168*a^5*b*d*e^5*(2*B*d + A*e) + 420*a^2*b^4*d^4*e^2*(B*d +
2*A*e) + 4*a^6*e^6*(7*B*d + 2*A*e) + 210*a^4*b^2*d^2*e^4*(5*B*d + 4*A*e) + 280*a^3*b^3*d^3*e^3*(4*B*d + 5*A*e
) + 24*a*b^5*d^5*e*(2*B*d + 7*A*e) + b^6*d^6*(B*d + 8*A*e))*x^8)/8 + (e*(420*a^4*b^2*d^2*e^4*(2*B*d + A*e) + a
^6*e^6*(8*B*d + A*e) + 168*a*b^5*d^5*e*(B*d + 2*A*e) + 24*a^5*b*d*e^5*(7*B*d + 2*A*e) + 280*a^3*b^3*d^3*e^3*(5
*B*d + 4*A*e) + 210*a^2*b^4*d^4*e^2*(4*B*d + 5*A*e) + 4*b^6*d^6*(2*B*d + 7*A*e))*x^9)/9 + (e^2*(a^6*B*e^6 + 56
0*a^3*b^3*d^2*e^3*(2*B*d + A*e) + 6*a^5*b*e^5*(8*B*d + A*e) + 28*b^6*d^5*(B*d + 2*A*e) + 60*a^4*b^2*d*e^4*(7*B
*d + 2*A*e) + 210*a^2*b^4*d^3*e^2*(5*B*d + 4*A*e) + 84*a*b^5*d^4*e*(4*B*d + 5*A*e))*x^10)/10 + (b*e^3*(6*a^5*B
*e^5 + 420*a^2*b^3*d^2*e^2*(2*B*d + A*e) + 15*a^4*b*e^4*(8*B*d + A*e) + 80*a^3*b^2*d*e^3*(7*B*d + 2*A*e) + 84*
a*b^4*d^3*e*(5*B*d + 4*A*e) + 14*b^5*d^4*(4*B*d + 5*A*e))*x^11)/11 + (b^2*e^4*(15*a^4*B*e^4 + 168*a*b^3*d^2*e*
(2*B*d + A*e) + 20*a^3*b*e^3*(8*B*d + A*e) + 60*a^2*b^2*d*e^2*(7*B*d + 2*A*e) + 14*b^4*d^3*(5*B*d + 4*A*e))*x^
12)/12 + (b^3*e^5*(20*a^3*B*e^3 + 28*b^3*d^2*(2*B*d + A*e) + 15*a^2*b*e^2*(8*B*d + A*e) + 24*a*b^2*d*e*(7*B*d
+ 2*A*e))*x^13)/13 + (b^4*e^6*(15*a^2*B*e^2 + 6*a*b*e*(8*B*d + A*e) + 4*b^2*d*(7*B*d + 2*A*e))*x^14)/14 + (b^5
*e^7*(8*b*B*d + A*b*e + 6*a*B*e)*x^15)/15 + (b^6*B*e^8*x^16)/16

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1524\) vs. \(2(276)=552\).
time = 0.07, size = 1525, normalized size = 5.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1525\)
norman	\(\text {Expression too large to display}\)	\(1642\)
gosper	\(\text {Expression too large to display}\)	\(1944\)
risch	\(\text {Expression too large to display}\)	\(1944\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

1/16*b^6*B*e^8*x^16+1/15*((A*b^6+6*B*a*b^5)*e^8+8*b^6*B*d*e^7)*x^15+1/14*((6*A*a*b^5+15*B*a^2*b^4)*e^8+8*(A*b^
6+6*B*a*b^5)*d*e^7+28*b^6*B*d^2*e^6)*x^14+1/13*((15*A*a^2*b^4+20*B*a^3*b^3)*e^8+8*(6*A*a*b^5+15*B*a^2*b^4)*d*e
^7+28*(A*b^6+6*B*a*b^5)*d^2*e^6+56*b^6*B*d^3*e^5)*x^13+1/12*((20*A*a^3*b^3+15*B*a^4*b^2)*e^8+8*(15*A*a^2*b^4+2
0*B*a^3*b^3)*d*e^7+28*(6*A*a*b^5+15*B*a^2*b^4)*d^2*e^6+56*(A*b^6+6*B*a*b^5)*d^3*e^5+70*b^6*B*d^4*e^4)*x^12+1/1
1*((15*A*a^4*b^2+6*B*a^5*b)*e^8+8*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^7+28*(15*A*a^2*b^4+20*B*a^3*b^3)*d^2*e^6+56*
(6*A*a*b^5+15*B*a^2*b^4)*d^3*e^5+70*(A*b^6+6*B*a*b^5)*d^4*e^4+56*b^6*B*d^5*e^3)*x^11+1/10*((6*A*a^5*b+B*a^6)*e
^8+8*(15*A*a^4*b^2+6*B*a^5*b)*d*e^7+28*(20*A*a^3*b^3+15*B*a^4*b^2)*d^2*e^6+56*(15*A*a^2*b^4+20*B*a^3*b^3)*d^3*
e^5+70*(6*A*a*b^5+15*B*a^2*b^4)*d^4*e^4+56*(A*b^6+6*B*a*b^5)*d^5*e^3+28*b^6*B*d^6*e^2)*x^10+1/9*(a^6*A*e^8+8*(
6*A*a^5*b+B*a^6)*d*e^7+28*(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^6+56*(20*A*a^3*b^3+15*B*a^4*b^2)*d^3*e^5+70*(15*A*a^2
*b^4+20*B*a^3*b^3)*d^4*e^4+56*(6*A*a*b^5+15*B*a^2*b^4)*d^5*e^3+28*(A*b^6+6*B*a*b^5)*d^6*e^2+8*b^6*B*d^7*e)*x^9
+1/8*(8*a^6*A*d*e^7+28*(6*A*a^5*b+B*a^6)*d^2*e^6+56*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^5+70*(20*A*a^3*b^3+15*B*a^4
*b^2)*d^4*e^4+56*(15*A*a^2*b^4+20*B*a^3*b^3)*d^5*e^3+28*(6*A*a*b^5+15*B*a^2*b^4)*d^6*e^2+8*(A*b^6+6*B*a*b^5)*d
^7*e+b^6*B*d^8)*x^8+1/7*(28*a^6*A*d^2*e^6+56*(6*A*a^5*b+B*a^6)*d^3*e^5+70*(15*A*a^4*b^2+6*B*a^5*b)*d^4*e^4+56*
(20*A*a^3*b^3+15*B*a^4*b^2)*d^5*e^3+28*(15*A*a^2*b^4+20*B*a^3*b^3)*d^6*e^2+8*(6*A*a*b^5+15*B*a^2*b^4)*d^7*e+(A
*b^6+6*B*a*b^5)*d^8)*x^7+1/6*(56*a^6*A*d^3*e^5+70*(6*A*a^5*b+B*a^6)*d^4*e^4+56*(15*A*a^4*b^2+6*B*a^5*b)*d^5*e^
3+28*(20*A*a^3*b^3+15*B*a^4*b^2)*d^6*e^2+8*(15*A*a^2*b^4+20*B*a^3*b^3)*d^7*e+(6*A*a*b^5+15*B*a^2*b^4)*d^8)*x^6
+1/5*(70*a^6*A*d^4*e^4+56*(6*A*a^5*b+B*a^6)*d^5*e^3+28*(15*A*a^4*b^2+6*B*a^5*b)*d^6*e^2+8*(20*A*a^3*b^3+15*B*a
^4*b^2)*d^7*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^8)*x^5+1/4*(56*a^6*A*d^5*e^3+28*(6*A*a^5*b+B*a^6)*d^6*e^2+8*(15*A*
a^4*b^2+6*B*a^5*b)*d^7*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d^8)*x^4+1/3*(28*a^6*A*d^6*e^2+8*(6*A*a^5*b+B*a^6)*d^7*e+
(15*A*a^4*b^2+6*B*a^5*b)*d^8)*x^3+1/2*(8*a^6*A*d^7*e+(6*A*a^5*b+B*a^6)*d^8)*x^2+a^6*A*d^8*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1564 vs. \(2 (295) = 590\).
time = 0.30, size = 1564, normalized size = 5.36 \begin {gather*} \frac {1}{16} \, B b^{6} x^{16} e^{8} + A a^{6} d^{8} x + \frac {1}{15} \, {\left (8 \, B b^{6} d e^{7} + 6 \, B a b^{5} e^{8} + A b^{6} e^{8}\right )} x^{15} + \frac {1}{14} \, {\left (28 \, B b^{6} d^{2} e^{6} + 15 \, B a^{2} b^{4} e^{8} + 6 \, A a b^{5} e^{8} + 8 \, {\left (6 \, B a b^{5} e^{7} + A b^{6} e^{7}\right )} d\right )} x^{14} + \frac {1}{13} \, {\left (56 \, B b^{6} d^{3} e^{5} + 20 \, B a^{3} b^{3} e^{8} + 15 \, A a^{2} b^{4} e^{8} + 28 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d^{2} + 24 \, {\left (5 \, B a^{2} b^{4} e^{7} + 2 \, A a b^{5} e^{7}\right )} d\right )} x^{13} + \frac {1}{12} \, {\left (70 \, B b^{6} d^{4} e^{4} + 15 \, B a^{4} b^{2} e^{8} + 20 \, A a^{3} b^{3} e^{8} + 56 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{3} + 84 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d^{2} + 40 \, {\left (4 \, B a^{3} b^{3} e^{7} + 3 \, A a^{2} b^{4} e^{7}\right )} d\right )} x^{12} + \frac {1}{11} \, {\left (56 \, B b^{6} d^{5} e^{3} + 6 \, B a^{5} b e^{8} + 15 \, A a^{4} b^{2} e^{8} + 70 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{4} + 168 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{3} + 140 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d^{2} + 40 \, {\left (3 \, B a^{4} b^{2} e^{7} + 4 \, A a^{3} b^{3} e^{7}\right )} d\right )} x^{11} + \frac {1}{10} \, {\left (28 \, B b^{6} d^{6} e^{2} + B a^{6} e^{8} + 6 \, A a^{5} b e^{8} + 56 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{5} + 210 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{4} + 280 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{3} + 140 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d^{2} + 24 \, {\left (2 \, B a^{5} b e^{7} + 5 \, A a^{4} b^{2} e^{7}\right )} d\right )} x^{10} + \frac {1}{9} \, {\left (8 \, B b^{6} d^{7} e + A a^{6} e^{8} + 28 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{6} + 168 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{5} + 350 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{4} + 280 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{3} + 84 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d^{2} + 8 \, {\left (B a^{6} e^{7} + 6 \, A a^{5} b e^{7}\right )} d\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{8} + 8 \, A a^{6} d e^{7} + 8 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{7} + 84 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{6} + 280 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{5} + 350 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{4} + 168 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{3} + 28 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left (28 \, A a^{6} d^{2} e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{8} + 24 \, {\left (5 \, B a^{2} b^{4} e + 2 \, A a b^{5} e\right )} d^{7} + 140 \, {\left (4 \, B a^{3} b^{3} e^{2} + 3 \, A a^{2} b^{4} e^{2}\right )} d^{6} + 280 \, {\left (3 \, B a^{4} b^{2} e^{3} + 4 \, A a^{3} b^{3} e^{3}\right )} d^{5} + 210 \, {\left (2 \, B a^{5} b e^{4} + 5 \, A a^{4} b^{2} e^{4}\right )} d^{4} + 56 \, {\left (B a^{6} e^{5} + 6 \, A a^{5} b e^{5}\right )} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (56 \, A a^{6} d^{3} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{8} + 40 \, {\left (4 \, B a^{3} b^{3} e + 3 \, A a^{2} b^{4} e\right )} d^{7} + 140 \, {\left (3 \, B a^{4} b^{2} e^{2} + 4 \, A a^{3} b^{3} e^{2}\right )} d^{6} + 168 \, {\left (2 \, B a^{5} b e^{3} + 5 \, A a^{4} b^{2} e^{3}\right )} d^{5} + 70 \, {\left (B a^{6} e^{4} + 6 \, A a^{5} b e^{4}\right )} d^{4}\right )} x^{6} + \frac {1}{5} \, {\left (70 \, A a^{6} d^{4} e^{4} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{8} + 40 \, {\left (3 \, B a^{4} b^{2} e + 4 \, A a^{3} b^{3} e\right )} d^{7} + 84 \, {\left (2 \, B a^{5} b e^{2} + 5 \, A a^{4} b^{2} e^{2}\right )} d^{6} + 56 \, {\left (B a^{6} e^{3} + 6 \, A a^{5} b e^{3}\right )} d^{5}\right )} x^{5} + \frac {1}{4} \, {\left (56 \, A a^{6} d^{5} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{8} + 24 \, {\left (2 \, B a^{5} b e + 5 \, A a^{4} b^{2} e\right )} d^{7} + 28 \, {\left (B a^{6} e^{2} + 6 \, A a^{5} b e^{2}\right )} d^{6}\right )} x^{4} + \frac {1}{3} \, {\left (28 \, A a^{6} d^{6} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{8} + 8 \, {\left (B a^{6} e + 6 \, A a^{5} b e\right )} d^{7}\right )} x^{3} + \frac {1}{2} \, {\left (8 \, A a^{6} d^{7} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{8}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="maxima")

[Out]

1/16*B*b^6*x^16*e^8 + A*a^6*d^8*x + 1/15*(8*B*b^6*d*e^7 + 6*B*a*b^5*e^8 + A*b^6*e^8)*x^15 + 1/14*(28*B*b^6*d^2
*e^6 + 15*B*a^2*b^4*e^8 + 6*A*a*b^5*e^8 + 8*(6*B*a*b^5*e^7 + A*b^6*e^7)*d)*x^14 + 1/13*(56*B*b^6*d^3*e^5 + 20*
B*a^3*b^3*e^8 + 15*A*a^2*b^4*e^8 + 28*(6*B*a*b^5*e^6 + A*b^6*e^6)*d^2 + 24*(5*B*a^2*b^4*e^7 + 2*A*a*b^5*e^7)*d
)*x^13 + 1/12*(70*B*b^6*d^4*e^4 + 15*B*a^4*b^2*e^8 + 20*A*a^3*b^3*e^8 + 56*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^3 + 8
4*(5*B*a^2*b^4*e^6 + 2*A*a*b^5*e^6)*d^2 + 40*(4*B*a^3*b^3*e^7 + 3*A*a^2*b^4*e^7)*d)*x^12 + 1/11*(56*B*b^6*d^5*
e^3 + 6*B*a^5*b*e^8 + 15*A*a^4*b^2*e^8 + 70*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^4 + 168*(5*B*a^2*b^4*e^5 + 2*A*a*b^5
*e^5)*d^3 + 140*(4*B*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d^2 + 40*(3*B*a^4*b^2*e^7 + 4*A*a^3*b^3*e^7)*d)*x^11 + 1/1
0*(28*B*b^6*d^6*e^2 + B*a^6*e^8 + 6*A*a^5*b*e^8 + 56*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^5 + 210*(5*B*a^2*b^4*e^4 +
2*A*a*b^5*e^4)*d^4 + 280*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^3 + 140*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)*d^2
+ 24*(2*B*a^5*b*e^7 + 5*A*a^4*b^2*e^7)*d)*x^10 + 1/9*(8*B*b^6*d^7*e + A*a^6*e^8 + 28*(6*B*a*b^5*e^2 + A*b^6*e
^2)*d^6 + 168*(5*B*a^2*b^4*e^3 + 2*A*a*b^5*e^3)*d^5 + 350*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^4 + 280*(3*B*a
^4*b^2*e^5 + 4*A*a^3*b^3*e^5)*d^3 + 84*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d^2 + 8*(B*a^6*e^7 + 6*A*a^5*b*e^7)*d
)*x^9 + 1/8*(B*b^6*d^8 + 8*A*a^6*d*e^7 + 8*(6*B*a*b^5*e + A*b^6*e)*d^7 + 84*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*
d^6 + 280*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d^5 + 350*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d^4 + 168*(2*B*a^5
*b*e^5 + 5*A*a^4*b^2*e^5)*d^3 + 28*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d^2)*x^8 + 1/7*(28*A*a^6*d^2*e^6 + (6*B*a*b^5 +
A*b^6)*d^8 + 24*(5*B*a^2*b^4*e + 2*A*a*b^5*e)*d^7 + 140*(4*B*a^3*b^3*e^2 + 3*A*a^2*b^4*e^2)*d^6 + 280*(3*B*a^
4*b^2*e^3 + 4*A*a^3*b^3*e^3)*d^5 + 210*(2*B*a^5*b*e^4 + 5*A*a^4*b^2*e^4)*d^4 + 56*(B*a^6*e^5 + 6*A*a^5*b*e^5)*
d^3)*x^7 + 1/6*(56*A*a^6*d^3*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8 + 40*(4*B*a^3*b^3*e + 3*A*a^2*b^4*e)*d^7 +
140*(3*B*a^4*b^2*e^2 + 4*A*a^3*b^3*e^2)*d^6 + 168*(2*B*a^5*b*e^3 + 5*A*a^4*b^2*e^3)*d^5 + 70*(B*a^6*e^4 + 6*A*
a^5*b*e^4)*d^4)*x^6 + 1/5*(70*A*a^6*d^4*e^4 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^8 + 40*(3*B*a^4*b^2*e + 4*A*a^3*
b^3*e)*d^7 + 84*(2*B*a^5*b*e^2 + 5*A*a^4*b^2*e^2)*d^6 + 56*(B*a^6*e^3 + 6*A*a^5*b*e^3)*d^5)*x^5 + 1/4*(56*A*a^
6*d^5*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^8 + 24*(2*B*a^5*b*e + 5*A*a^4*b^2*e)*d^7 + 28*(B*a^6*e^2 + 6*A*a^5
*b*e^2)*d^6)*x^4 + 1/3*(28*A*a^6*d^6*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^8 + 8*(B*a^6*e + 6*A*a^5*b*e)*d^7)*x^
3 + 1/2*(8*A*a^6*d^7*e + (B*a^6 + 6*A*a^5*b)*d^8)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1526 vs. \(2 (295) = 590\).
time = 1.06, size = 1526, normalized size = 5.23 \begin {gather*} \frac {1}{8} \, B b^{6} d^{8} x^{8} + A a^{6} d^{8} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{8} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{8} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{8} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{8} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{8} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{8} x^{2} + \frac {1}{720720} \, {\left (45045 \, B b^{6} x^{16} + 80080 \, A a^{6} x^{9} + 48048 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{15} + 154440 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{14} + 277200 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{13} + 300300 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{12} + 196560 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{11} + 72072 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{10}\right )} e^{8} + \frac {1}{45045} \, {\left (24024 \, B b^{6} d x^{15} + 45045 \, A a^{6} d x^{8} + 25740 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{14} + 83160 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{13} + 150150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{12} + 163800 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{11} + 108108 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x^{10} + 40040 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d x^{9}\right )} e^{7} + \frac {1}{858} \, {\left (1716 \, B b^{6} d^{2} x^{14} + 3432 \, A a^{6} d^{2} x^{7} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{13} + 6006 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{12} + 10920 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{11} + 12012 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x^{10} + 8008 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} x^{9} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} x^{8}\right )} e^{6} + \frac {1}{1287} \, {\left (5544 \, B b^{6} d^{3} x^{13} + 12012 \, A a^{6} d^{3} x^{6} + 6006 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{12} + 19656 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{11} + 36036 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x^{10} + 40040 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} x^{9} + 27027 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} x^{8} + 10296 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} x^{7}\right )} e^{5} + \frac {1}{396} \, {\left (2310 \, B b^{6} d^{4} x^{12} + 5544 \, A a^{6} d^{4} x^{5} + 2520 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{11} + 8316 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x^{10} + 15400 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} x^{9} + 17325 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} x^{8} + 11880 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{4} x^{7} + 4620 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{4} x^{6}\right )} e^{4} + \frac {1}{165} \, {\left (840 \, B b^{6} d^{5} x^{11} + 2310 \, A a^{6} d^{5} x^{4} + 924 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x^{10} + 3080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} x^{9} + 5775 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} x^{8} + 6600 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{5} x^{7} + 4620 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{5} x^{6} + 1848 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{5} x^{5}\right )} e^{3} + \frac {1}{90} \, {\left (252 \, B b^{6} d^{6} x^{10} + 840 \, A a^{6} d^{6} x^{3} + 280 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} x^{9} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{6} x^{8} + 1800 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{6} x^{7} + 2100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{6} x^{6} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{6} x^{5} + 630 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{6} x^{4}\right )} e^{2} + \frac {1}{63} \, {\left (56 \, B b^{6} d^{7} x^{9} + 252 \, A a^{6} d^{7} x^{2} + 63 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{7} x^{8} + 216 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{7} x^{7} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{7} x^{6} + 504 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{7} x^{5} + 378 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{7} x^{4} + 168 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{7} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="fricas")

[Out]

1/8*B*b^6*d^8*x^8 + A*a^6*d^8*x + 1/7*(6*B*a*b^5 + A*b^6)*d^8*x^7 + 1/2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8*x^6 + (4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^8*x^5 + 5/4*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^8*x^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^8*x^
3 + 1/2*(B*a^6 + 6*A*a^5*b)*d^8*x^2 + 1/720720*(45045*B*b^6*x^16 + 80080*A*a^6*x^9 + 48048*(6*B*a*b^5 + A*b^6)
*x^15 + 154440*(5*B*a^2*b^4 + 2*A*a*b^5)*x^14 + 277200*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^13 + 300300*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*x^12 + 196560*(2*B*a^5*b + 5*A*a^4*b^2)*x^11 + 72072*(B*a^6 + 6*A*a^5*b)*x^10)*e^8 + 1/45045*(2
4024*B*b^6*d*x^15 + 45045*A*a^6*d*x^8 + 25740*(6*B*a*b^5 + A*b^6)*d*x^14 + 83160*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x
^13 + 150150*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^12 + 163800*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^11 + 108108*(2*B*a^5*
b + 5*A*a^4*b^2)*d*x^10 + 40040*(B*a^6 + 6*A*a^5*b)*d*x^9)*e^7 + 1/858*(1716*B*b^6*d^2*x^14 + 3432*A*a^6*d^2*x
^7 + 1848*(6*B*a*b^5 + A*b^6)*d^2*x^13 + 6006*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^12 + 10920*(4*B*a^3*b^3 + 3*A*a^
2*b^4)*d^2*x^11 + 12012*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x^10 + 8008*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*x^9 + 3003*(
B*a^6 + 6*A*a^5*b)*d^2*x^8)*e^6 + 1/1287*(5544*B*b^6*d^3*x^13 + 12012*A*a^6*d^3*x^6 + 6006*(6*B*a*b^5 + A*b^6)
*d^3*x^12 + 19656*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^11 + 36036*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x^10 + 40040*(3*B
*a^4*b^2 + 4*A*a^3*b^3)*d^3*x^9 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*x^8 + 10296*(B*a^6 + 6*A*a^5*b)*d^3*x^7)
*e^5 + 1/396*(2310*B*b^6*d^4*x^12 + 5544*A*a^6*d^4*x^5 + 2520*(6*B*a*b^5 + A*b^6)*d^4*x^11 + 8316*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^4*x^10 + 15400*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*x^9 + 17325*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*x^8
+ 11880*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*x^7 + 4620*(B*a^6 + 6*A*a^5*b)*d^4*x^6)*e^4 + 1/165*(840*B*b^6*d^5*x^11
+ 2310*A*a^6*d^5*x^4 + 924*(6*B*a*b^5 + A*b^6)*d^5*x^10 + 3080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*x^9 + 5775*(4*B*
a^3*b^3 + 3*A*a^2*b^4)*d^5*x^8 + 6600*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*x^7 + 4620*(2*B*a^5*b + 5*A*a^4*b^2)*d^5
*x^6 + 1848*(B*a^6 + 6*A*a^5*b)*d^5*x^5)*e^3 + 1/90*(252*B*b^6*d^6*x^10 + 840*A*a^6*d^6*x^3 + 280*(6*B*a*b^5 +
A*b^6)*d^6*x^9 + 945*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*x^8 + 1800*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^6*x^7 + 2100*(3*B
*a^4*b^2 + 4*A*a^3*b^3)*d^6*x^6 + 1512*(2*B*a^5*b + 5*A*a^4*b^2)*d^6*x^5 + 630*(B*a^6 + 6*A*a^5*b)*d^6*x^4)*e^
2 + 1/63*(56*B*b^6*d^7*x^9 + 252*A*a^6*d^7*x^2 + 63*(6*B*a*b^5 + A*b^6)*d^7*x^8 + 216*(5*B*a^2*b^4 + 2*A*a*b^5
)*d^7*x^7 + 420*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*x^6 + 504*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^7*x^5 + 378*(2*B*a^5*b
+ 5*A*a^4*b^2)*d^7*x^4 + 168*(B*a^6 + 6*A*a^5*b)*d^7*x^3)*e

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1969 vs. \(2 (296) = 592\).
time = 0.14, size = 1969, normalized size = 6.74 \begin {gather*} A a^{6} d^{8} x + \frac {B b^{6} e^{8} x^{16}}{16} + x^{15} \left (\frac {A b^{6} e^{8}}{15} + \frac {2 B a b^{5} e^{8}}{5} + \frac {8 B b^{6} d e^{7}}{15}\right ) + x^{14} \cdot \left (\frac {3 A a b^{5} e^{8}}{7} + \frac {4 A b^{6} d e^{7}}{7} + \frac {15 B a^{2} b^{4} e^{8}}{14} + \frac {24 B a b^{5} d e^{7}}{7} + 2 B b^{6} d^{2} e^{6}\right ) + x^{13} \cdot \left (\frac {15 A a^{2} b^{4} e^{8}}{13} + \frac {48 A a b^{5} d e^{7}}{13} + \frac {28 A b^{6} d^{2} e^{6}}{13} + \frac {20 B a^{3} b^{3} e^{8}}{13} + \frac {120 B a^{2} b^{4} d e^{7}}{13} + \frac {168 B a b^{5} d^{2} e^{6}}{13} + \frac {56 B b^{6} d^{3} e^{5}}{13}\right ) + x^{12} \cdot \left (\frac {5 A a^{3} b^{3} e^{8}}{3} + 10 A a^{2} b^{4} d e^{7} + 14 A a b^{5} d^{2} e^{6} + \frac {14 A b^{6} d^{3} e^{5}}{3} + \frac {5 B a^{4} b^{2} e^{8}}{4} + \frac {40 B a^{3} b^{3} d e^{7}}{3} + 35 B a^{2} b^{4} d^{2} e^{6} + 28 B a b^{5} d^{3} e^{5} + \frac {35 B b^{6} d^{4} e^{4}}{6}\right ) + x^{11} \cdot \left (\frac {15 A a^{4} b^{2} e^{8}}{11} + \frac {160 A a^{3} b^{3} d e^{7}}{11} + \frac {420 A a^{2} b^{4} d^{2} e^{6}}{11} + \frac {336 A a b^{5} d^{3} e^{5}}{11} + \frac {70 A b^{6} d^{4} e^{4}}{11} + \frac {6 B a^{5} b e^{8}}{11} + \frac {120 B a^{4} b^{2} d e^{7}}{11} + \frac {560 B a^{3} b^{3} d^{2} e^{6}}{11} + \frac {840 B a^{2} b^{4} d^{3} e^{5}}{11} + \frac {420 B a b^{5} d^{4} e^{4}}{11} + \frac {56 B b^{6} d^{5} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a^{5} b e^{8}}{5} + 12 A a^{4} b^{2} d e^{7} + 56 A a^{3} b^{3} d^{2} e^{6} + 84 A a^{2} b^{4} d^{3} e^{5} + 42 A a b^{5} d^{4} e^{4} + \frac {28 A b^{6} d^{5} e^{3}}{5} + \frac {B a^{6} e^{8}}{10} + \frac {24 B a^{5} b d e^{7}}{5} + 42 B a^{4} b^{2} d^{2} e^{6} + 112 B a^{3} b^{3} d^{3} e^{5} + 105 B a^{2} b^{4} d^{4} e^{4} + \frac {168 B a b^{5} d^{5} e^{3}}{5} + \frac {14 B b^{6} d^{6} e^{2}}{5}\right ) + x^{9} \left (\frac {A a^{6} e^{8}}{9} + \frac {16 A a^{5} b d e^{7}}{3} + \frac {140 A a^{4} b^{2} d^{2} e^{6}}{3} + \frac {1120 A a^{3} b^{3} d^{3} e^{5}}{9} + \frac {350 A a^{2} b^{4} d^{4} e^{4}}{3} + \frac {112 A a b^{5} d^{5} e^{3}}{3} + \frac {28 A b^{6} d^{6} e^{2}}{9} + \frac {8 B a^{6} d e^{7}}{9} + \frac {56 B a^{5} b d^{2} e^{6}}{3} + \frac {280 B a^{4} b^{2} d^{3} e^{5}}{3} + \frac {1400 B a^{3} b^{3} d^{4} e^{4}}{9} + \frac {280 B a^{2} b^{4} d^{5} e^{3}}{3} + \frac {56 B a b^{5} d^{6} e^{2}}{3} + \frac {8 B b^{6} d^{7} e}{9}\right ) + x^{8} \left (A a^{6} d e^{7} + 21 A a^{5} b d^{2} e^{6} + 105 A a^{4} b^{2} d^{3} e^{5} + 175 A a^{3} b^{3} d^{4} e^{4} + 105 A a^{2} b^{4} d^{5} e^{3} + 21 A a b^{5} d^{6} e^{2} + A b^{6} d^{7} e + \frac {7 B a^{6} d^{2} e^{6}}{2} + 42 B a^{5} b d^{3} e^{5} + \frac {525 B a^{4} b^{2} d^{4} e^{4}}{4} + 140 B a^{3} b^{3} d^{5} e^{3} + \frac {105 B a^{2} b^{4} d^{6} e^{2}}{2} + 6 B a b^{5} d^{7} e + \frac {B b^{6} d^{8}}{8}\right ) + x^{7} \cdot \left (4 A a^{6} d^{2} e^{6} + 48 A a^{5} b d^{3} e^{5} + 150 A a^{4} b^{2} d^{4} e^{4} + 160 A a^{3} b^{3} d^{5} e^{3} + 60 A a^{2} b^{4} d^{6} e^{2} + \frac {48 A a b^{5} d^{7} e}{7} + \frac {A b^{6} d^{8}}{7} + 8 B a^{6} d^{3} e^{5} + 60 B a^{5} b d^{4} e^{4} + 120 B a^{4} b^{2} d^{5} e^{3} + 80 B a^{3} b^{3} d^{6} e^{2} + \frac {120 B a^{2} b^{4} d^{7} e}{7} + \frac {6 B a b^{5} d^{8}}{7}\right ) + x^{6} \cdot \left (\frac {28 A a^{6} d^{3} e^{5}}{3} + 70 A a^{5} b d^{4} e^{4} + 140 A a^{4} b^{2} d^{5} e^{3} + \frac {280 A a^{3} b^{3} d^{6} e^{2}}{3} + 20 A a^{2} b^{4} d^{7} e + A a b^{5} d^{8} + \frac {35 B a^{6} d^{4} e^{4}}{3} + 56 B a^{5} b d^{5} e^{3} + 70 B a^{4} b^{2} d^{6} e^{2} + \frac {80 B a^{3} b^{3} d^{7} e}{3} + \frac {5 B a^{2} b^{4} d^{8}}{2}\right ) + x^{5} \cdot \left (14 A a^{6} d^{4} e^{4} + \frac {336 A a^{5} b d^{5} e^{3}}{5} + 84 A a^{4} b^{2} d^{6} e^{2} + 32 A a^{3} b^{3} d^{7} e + 3 A a^{2} b^{4} d^{8} + \frac {56 B a^{6} d^{5} e^{3}}{5} + \frac {168 B a^{5} b d^{6} e^{2}}{5} + 24 B a^{4} b^{2} d^{7} e + 4 B a^{3} b^{3} d^{8}\right ) + x^{4} \cdot \left (14 A a^{6} d^{5} e^{3} + 42 A a^{5} b d^{6} e^{2} + 30 A a^{4} b^{2} d^{7} e + 5 A a^{3} b^{3} d^{8} + 7 B a^{6} d^{6} e^{2} + 12 B a^{5} b d^{7} e + \frac {15 B a^{4} b^{2} d^{8}}{4}\right ) + x^{3} \cdot \left (\frac {28 A a^{6} d^{6} e^{2}}{3} + 16 A a^{5} b d^{7} e + 5 A a^{4} b^{2} d^{8} + \frac {8 B a^{6} d^{7} e}{3} + 2 B a^{5} b d^{8}\right ) + x^{2} \cdot \left (4 A a^{6} d^{7} e + 3 A a^{5} b d^{8} + \frac {B a^{6} d^{8}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**8,x)

[Out]

A*a**6*d**8*x + B*b**6*e**8*x**16/16 + x**15*(A*b**6*e**8/15 + 2*B*a*b**5*e**8/5 + 8*B*b**6*d*e**7/15) + x**14
*(3*A*a*b**5*e**8/7 + 4*A*b**6*d*e**7/7 + 15*B*a**2*b**4*e**8/14 + 24*B*a*b**5*d*e**7/7 + 2*B*b**6*d**2*e**6)
+ x**13*(15*A*a**2*b**4*e**8/13 + 48*A*a*b**5*d*e**7/13 + 28*A*b**6*d**2*e**6/13 + 20*B*a**3*b**3*e**8/13 + 12
0*B*a**2*b**4*d*e**7/13 + 168*B*a*b**5*d**2*e**6/13 + 56*B*b**6*d**3*e**5/13) + x**12*(5*A*a**3*b**3*e**8/3 +
10*A*a**2*b**4*d*e**7 + 14*A*a*b**5*d**2*e**6 + 14*A*b**6*d**3*e**5/3 + 5*B*a**4*b**2*e**8/4 + 40*B*a**3*b**3*
d*e**7/3 + 35*B*a**2*b**4*d**2*e**6 + 28*B*a*b**5*d**3*e**5 + 35*B*b**6*d**4*e**4/6) + x**11*(15*A*a**4*b**2*e
**8/11 + 160*A*a**3*b**3*d*e**7/11 + 420*A*a**2*b**4*d**2*e**6/11 + 336*A*a*b**5*d**3*e**5/11 + 70*A*b**6*d**4
*e**4/11 + 6*B*a**5*b*e**8/11 + 120*B*a**4*b**2*d*e**7/11 + 560*B*a**3*b**3*d**2*e**6/11 + 840*B*a**2*b**4*d**
3*e**5/11 + 420*B*a*b**5*d**4*e**4/11 + 56*B*b**6*d**5*e**3/11) + x**10*(3*A*a**5*b*e**8/5 + 12*A*a**4*b**2*d*
e**7 + 56*A*a**3*b**3*d**2*e**6 + 84*A*a**2*b**4*d**3*e**5 + 42*A*a*b**5*d**4*e**4 + 28*A*b**6*d**5*e**3/5 + B
*a**6*e**8/10 + 24*B*a**5*b*d*e**7/5 + 42*B*a**4*b**2*d**2*e**6 + 112*B*a**3*b**3*d**3*e**5 + 105*B*a**2*b**4*
d**4*e**4 + 168*B*a*b**5*d**5*e**3/5 + 14*B*b**6*d**6*e**2/5) + x**9*(A*a**6*e**8/9 + 16*A*a**5*b*d*e**7/3 + 1
40*A*a**4*b**2*d**2*e**6/3 + 1120*A*a**3*b**3*d**3*e**5/9 + 350*A*a**2*b**4*d**4*e**4/3 + 112*A*a*b**5*d**5*e*
*3/3 + 28*A*b**6*d**6*e**2/9 + 8*B*a**6*d*e**7/9 + 56*B*a**5*b*d**2*e**6/3 + 280*B*a**4*b**2*d**3*e**5/3 + 140
0*B*a**3*b**3*d**4*e**4/9 + 280*B*a**2*b**4*d**5*e**3/3 + 56*B*a*b**5*d**6*e**2/3 + 8*B*b**6*d**7*e/9) + x**8*
(A*a**6*d*e**7 + 21*A*a**5*b*d**2*e**6 + 105*A*a**4*b**2*d**3*e**5 + 175*A*a**3*b**3*d**4*e**4 + 105*A*a**2*b*
*4*d**5*e**3 + 21*A*a*b**5*d**6*e**2 + A*b**6*d**7*e + 7*B*a**6*d**2*e**6/2 + 42*B*a**5*b*d**3*e**5 + 525*B*a*
*4*b**2*d**4*e**4/4 + 140*B*a**3*b**3*d**5*e**3 + 105*B*a**2*b**4*d**6*e**2/2 + 6*B*a*b**5*d**7*e + B*b**6*d**
8/8) + x**7*(4*A*a**6*d**2*e**6 + 48*A*a**5*b*d**3*e**5 + 150*A*a**4*b**2*d**4*e**4 + 160*A*a**3*b**3*d**5*e**
3 + 60*A*a**2*b**4*d**6*e**2 + 48*A*a*b**5*d**7*e/7 + A*b**6*d**8/7 + 8*B*a**6*d**3*e**5 + 60*B*a**5*b*d**4*e*
*4 + 120*B*a**4*b**2*d**5*e**3 + 80*B*a**3*b**3*d**6*e**2 + 120*B*a**2*b**4*d**7*e/7 + 6*B*a*b**5*d**8/7) + x*
*6*(28*A*a**6*d**3*e**5/3 + 70*A*a**5*b*d**4*e**4 + 140*A*a**4*b**2*d**5*e**3 + 280*A*a**3*b**3*d**6*e**2/3 +
20*A*a**2*b**4*d**7*e + A*a*b**5*d**8 + 35*B*a**6*d**4*e**4/3 + 56*B*a**5*b*d**5*e**3 + 70*B*a**4*b**2*d**6*e*
*2 + 80*B*a**3*b**3*d**7*e/3 + 5*B*a**2*b**4*d**8/2) + x**5*(14*A*a**6*d**4*e**4 + 336*A*a**5*b*d**5*e**3/5 +
84*A*a**4*b**2*d**6*e**2 + 32*A*a**3*b**3*d**7*e + 3*A*a**2*b**4*d**8 + 56*B*a**6*d**5*e**3/5 + 168*B*a**5*b*d
**6*e**2/5 + 24*B*a**4*b**2*d**7*e + 4*B*a**3*b**3*d**8) + x**4*(14*A*a**6*d**5*e**3 + 42*A*a**5*b*d**6*e**2 +
30*A*a**4*b**2*d**7*e + 5*A*a**3*b**3*d**8 + 7*B*a**6*d**6*e**2 + 12*B*a**5*b*d**7*e + 15*B*a**4*b**2*d**8/4)
+ x**3*(28*A*a**6*d**6*e**2/3 + 16*A*a**5*b*d**7*e + 5*A*a**4*b**2*d**8 + 8*B*a**6*d**7*e/3 + 2*B*a**5*b*d**8
) + x**2*(4*A*a**6*d**7*e + 3*A*a**5*b*d**8 + B*a**6*d**8/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1859 vs. \(2 (295) = 590\).
time = 2.14, size = 1859, normalized size = 6.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^8,x, algorithm="giac")

[Out]

1/16*B*b^6*x^16*e^8 + 8/15*B*b^6*d*x^15*e^7 + 2*B*b^6*d^2*x^14*e^6 + 56/13*B*b^6*d^3*x^13*e^5 + 35/6*B*b^6*d^4
*x^12*e^4 + 56/11*B*b^6*d^5*x^11*e^3 + 14/5*B*b^6*d^6*x^10*e^2 + 8/9*B*b^6*d^7*x^9*e + 1/8*B*b^6*d^8*x^8 + 2/5
*B*a*b^5*x^15*e^8 + 1/15*A*b^6*x^15*e^8 + 24/7*B*a*b^5*d*x^14*e^7 + 4/7*A*b^6*d*x^14*e^7 + 168/13*B*a*b^5*d^2*
x^13*e^6 + 28/13*A*b^6*d^2*x^13*e^6 + 28*B*a*b^5*d^3*x^12*e^5 + 14/3*A*b^6*d^3*x^12*e^5 + 420/11*B*a*b^5*d^4*x
^11*e^4 + 70/11*A*b^6*d^4*x^11*e^4 + 168/5*B*a*b^5*d^5*x^10*e^3 + 28/5*A*b^6*d^5*x^10*e^3 + 56/3*B*a*b^5*d^6*x
^9*e^2 + 28/9*A*b^6*d^6*x^9*e^2 + 6*B*a*b^5*d^7*x^8*e + A*b^6*d^7*x^8*e + 6/7*B*a*b^5*d^8*x^7 + 1/7*A*b^6*d^8*
x^7 + 15/14*B*a^2*b^4*x^14*e^8 + 3/7*A*a*b^5*x^14*e^8 + 120/13*B*a^2*b^4*d*x^13*e^7 + 48/13*A*a*b^5*d*x^13*e^7
+ 35*B*a^2*b^4*d^2*x^12*e^6 + 14*A*a*b^5*d^2*x^12*e^6 + 840/11*B*a^2*b^4*d^3*x^11*e^5 + 336/11*A*a*b^5*d^3*x^
11*e^5 + 105*B*a^2*b^4*d^4*x^10*e^4 + 42*A*a*b^5*d^4*x^10*e^4 + 280/3*B*a^2*b^4*d^5*x^9*e^3 + 112/3*A*a*b^5*d^
5*x^9*e^3 + 105/2*B*a^2*b^4*d^6*x^8*e^2 + 21*A*a*b^5*d^6*x^8*e^2 + 120/7*B*a^2*b^4*d^7*x^7*e + 48/7*A*a*b^5*d^
7*x^7*e + 5/2*B*a^2*b^4*d^8*x^6 + A*a*b^5*d^8*x^6 + 20/13*B*a^3*b^3*x^13*e^8 + 15/13*A*a^2*b^4*x^13*e^8 + 40/3
*B*a^3*b^3*d*x^12*e^7 + 10*A*a^2*b^4*d*x^12*e^7 + 560/11*B*a^3*b^3*d^2*x^11*e^6 + 420/11*A*a^2*b^4*d^2*x^11*e^
6 + 112*B*a^3*b^3*d^3*x^10*e^5 + 84*A*a^2*b^4*d^3*x^10*e^5 + 1400/9*B*a^3*b^3*d^4*x^9*e^4 + 350/3*A*a^2*b^4*d^
4*x^9*e^4 + 140*B*a^3*b^3*d^5*x^8*e^3 + 105*A*a^2*b^4*d^5*x^8*e^3 + 80*B*a^3*b^3*d^6*x^7*e^2 + 60*A*a^2*b^4*d^
6*x^7*e^2 + 80/3*B*a^3*b^3*d^7*x^6*e + 20*A*a^2*b^4*d^7*x^6*e + 4*B*a^3*b^3*d^8*x^5 + 3*A*a^2*b^4*d^8*x^5 + 5/
4*B*a^4*b^2*x^12*e^8 + 5/3*A*a^3*b^3*x^12*e^8 + 120/11*B*a^4*b^2*d*x^11*e^7 + 160/11*A*a^3*b^3*d*x^11*e^7 + 42
*B*a^4*b^2*d^2*x^10*e^6 + 56*A*a^3*b^3*d^2*x^10*e^6 + 280/3*B*a^4*b^2*d^3*x^9*e^5 + 1120/9*A*a^3*b^3*d^3*x^9*e
^5 + 525/4*B*a^4*b^2*d^4*x^8*e^4 + 175*A*a^3*b^3*d^4*x^8*e^4 + 120*B*a^4*b^2*d^5*x^7*e^3 + 160*A*a^3*b^3*d^5*x
^7*e^3 + 70*B*a^4*b^2*d^6*x^6*e^2 + 280/3*A*a^3*b^3*d^6*x^6*e^2 + 24*B*a^4*b^2*d^7*x^5*e + 32*A*a^3*b^3*d^7*x^
5*e + 15/4*B*a^4*b^2*d^8*x^4 + 5*A*a^3*b^3*d^8*x^4 + 6/11*B*a^5*b*x^11*e^8 + 15/11*A*a^4*b^2*x^11*e^8 + 24/5*B
*a^5*b*d*x^10*e^7 + 12*A*a^4*b^2*d*x^10*e^7 + 56/3*B*a^5*b*d^2*x^9*e^6 + 140/3*A*a^4*b^2*d^2*x^9*e^6 + 42*B*a^
5*b*d^3*x^8*e^5 + 105*A*a^4*b^2*d^3*x^8*e^5 + 60*B*a^5*b*d^4*x^7*e^4 + 150*A*a^4*b^2*d^4*x^7*e^4 + 56*B*a^5*b*
d^5*x^6*e^3 + 140*A*a^4*b^2*d^5*x^6*e^3 + 168/5*B*a^5*b*d^6*x^5*e^2 + 84*A*a^4*b^2*d^6*x^5*e^2 + 12*B*a^5*b*d^
7*x^4*e + 30*A*a^4*b^2*d^7*x^4*e + 2*B*a^5*b*d^8*x^3 + 5*A*a^4*b^2*d^8*x^3 + 1/10*B*a^6*x^10*e^8 + 3/5*A*a^5*b
*x^10*e^8 + 8/9*B*a^6*d*x^9*e^7 + 16/3*A*a^5*b*d*x^9*e^7 + 7/2*B*a^6*d^2*x^8*e^6 + 21*A*a^5*b*d^2*x^8*e^6 + 8*
B*a^6*d^3*x^7*e^5 + 48*A*a^5*b*d^3*x^7*e^5 + 35/3*B*a^6*d^4*x^6*e^4 + 70*A*a^5*b*d^4*x^6*e^4 + 56/5*B*a^6*d^5*
x^5*e^3 + 336/5*A*a^5*b*d^5*x^5*e^3 + 7*B*a^6*d^6*x^4*e^2 + 42*A*a^5*b*d^6*x^4*e^2 + 8/3*B*a^6*d^7*x^3*e + 16*
A*a^5*b*d^7*x^3*e + 1/2*B*a^6*d^8*x^2 + 3*A*a^5*b*d^8*x^2 + 1/9*A*a^6*x^9*e^8 + A*a^6*d*x^8*e^7 + 4*A*a^6*d^2*
x^7*e^6 + 28/3*A*a^6*d^3*x^6*e^5 + 14*A*a^6*d^4*x^5*e^4 + 14*A*a^6*d^5*x^4*e^3 + 28/3*A*a^6*d^6*x^3*e^2 + 4*A*
a^6*d^7*x^2*e + A*a^6*d^8*x

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Mupad [B]
time = 1.61, size = 1625, normalized size = 5.57 \begin {gather*} x^6\,\left (\frac {35\,B\,a^6\,d^4\,e^4}{3}+\frac {28\,A\,a^6\,d^3\,e^5}{3}+56\,B\,a^5\,b\,d^5\,e^3+70\,A\,a^5\,b\,d^4\,e^4+70\,B\,a^4\,b^2\,d^6\,e^2+140\,A\,a^4\,b^2\,d^5\,e^3+\frac {80\,B\,a^3\,b^3\,d^7\,e}{3}+\frac {280\,A\,a^3\,b^3\,d^6\,e^2}{3}+\frac {5\,B\,a^2\,b^4\,d^8}{2}+20\,A\,a^2\,b^4\,d^7\,e+A\,a\,b^5\,d^8\right )+x^{11}\,\left (\frac {6\,B\,a^5\,b\,e^8}{11}+\frac {120\,B\,a^4\,b^2\,d\,e^7}{11}+\frac {15\,A\,a^4\,b^2\,e^8}{11}+\frac {560\,B\,a^3\,b^3\,d^2\,e^6}{11}+\frac {160\,A\,a^3\,b^3\,d\,e^7}{11}+\frac {840\,B\,a^2\,b^4\,d^3\,e^5}{11}+\frac {420\,A\,a^2\,b^4\,d^2\,e^6}{11}+\frac {420\,B\,a\,b^5\,d^4\,e^4}{11}+\frac {336\,A\,a\,b^5\,d^3\,e^5}{11}+\frac {56\,B\,b^6\,d^5\,e^3}{11}+\frac {70\,A\,b^6\,d^4\,e^4}{11}\right )+x^5\,\left (\frac {56\,B\,a^6\,d^5\,e^3}{5}+14\,A\,a^6\,d^4\,e^4+\frac {168\,B\,a^5\,b\,d^6\,e^2}{5}+\frac {336\,A\,a^5\,b\,d^5\,e^3}{5}+24\,B\,a^4\,b^2\,d^7\,e+84\,A\,a^4\,b^2\,d^6\,e^2+4\,B\,a^3\,b^3\,d^8+32\,A\,a^3\,b^3\,d^7\,e+3\,A\,a^2\,b^4\,d^8\right )+x^{12}\,\left (\frac {5\,B\,a^4\,b^2\,e^8}{4}+\frac {40\,B\,a^3\,b^3\,d\,e^7}{3}+\frac {5\,A\,a^3\,b^3\,e^8}{3}+35\,B\,a^2\,b^4\,d^2\,e^6+10\,A\,a^2\,b^4\,d\,e^7+28\,B\,a\,b^5\,d^3\,e^5+14\,A\,a\,b^5\,d^2\,e^6+\frac {35\,B\,b^6\,d^4\,e^4}{6}+\frac {14\,A\,b^6\,d^3\,e^5}{3}\right )+x^7\,\left (8\,B\,a^6\,d^3\,e^5+4\,A\,a^6\,d^2\,e^6+60\,B\,a^5\,b\,d^4\,e^4+48\,A\,a^5\,b\,d^3\,e^5+120\,B\,a^4\,b^2\,d^5\,e^3+150\,A\,a^4\,b^2\,d^4\,e^4+80\,B\,a^3\,b^3\,d^6\,e^2+160\,A\,a^3\,b^3\,d^5\,e^3+\frac {120\,B\,a^2\,b^4\,d^7\,e}{7}+60\,A\,a^2\,b^4\,d^6\,e^2+\frac {6\,B\,a\,b^5\,d^8}{7}+\frac {48\,A\,a\,b^5\,d^7\,e}{7}+\frac {A\,b^6\,d^8}{7}\right )+x^{10}\,\left (\frac {B\,a^6\,e^8}{10}+\frac {24\,B\,a^5\,b\,d\,e^7}{5}+\frac {3\,A\,a^5\,b\,e^8}{5}+42\,B\,a^4\,b^2\,d^2\,e^6+12\,A\,a^4\,b^2\,d\,e^7+112\,B\,a^3\,b^3\,d^3\,e^5+56\,A\,a^3\,b^3\,d^2\,e^6+105\,B\,a^2\,b^4\,d^4\,e^4+84\,A\,a^2\,b^4\,d^3\,e^5+\frac {168\,B\,a\,b^5\,d^5\,e^3}{5}+42\,A\,a\,b^5\,d^4\,e^4+\frac {14\,B\,b^6\,d^6\,e^2}{5}+\frac {28\,A\,b^6\,d^5\,e^3}{5}\right )+x^3\,\left (\frac {8\,B\,a^6\,d^7\,e}{3}+\frac {28\,A\,a^6\,d^6\,e^2}{3}+2\,B\,a^5\,b\,d^8+16\,A\,a^5\,b\,d^7\,e+5\,A\,a^4\,b^2\,d^8\right )+x^{14}\,\left (\frac {15\,B\,a^2\,b^4\,e^8}{14}+\frac {24\,B\,a\,b^5\,d\,e^7}{7}+\frac {3\,A\,a\,b^5\,e^8}{7}+2\,B\,b^6\,d^2\,e^6+\frac {4\,A\,b^6\,d\,e^7}{7}\right )+x^8\,\left (\frac {7\,B\,a^6\,d^2\,e^6}{2}+A\,a^6\,d\,e^7+42\,B\,a^5\,b\,d^3\,e^5+21\,A\,a^5\,b\,d^2\,e^6+\frac {525\,B\,a^4\,b^2\,d^4\,e^4}{4}+105\,A\,a^4\,b^2\,d^3\,e^5+140\,B\,a^3\,b^3\,d^5\,e^3+175\,A\,a^3\,b^3\,d^4\,e^4+\frac {105\,B\,a^2\,b^4\,d^6\,e^2}{2}+105\,A\,a^2\,b^4\,d^5\,e^3+6\,B\,a\,b^5\,d^7\,e+21\,A\,a\,b^5\,d^6\,e^2+\frac {B\,b^6\,d^8}{8}+A\,b^6\,d^7\,e\right )+x^9\,\left (\frac {8\,B\,a^6\,d\,e^7}{9}+\frac {A\,a^6\,e^8}{9}+\frac {56\,B\,a^5\,b\,d^2\,e^6}{3}+\frac {16\,A\,a^5\,b\,d\,e^7}{3}+\frac {280\,B\,a^4\,b^2\,d^3\,e^5}{3}+\frac {140\,A\,a^4\,b^2\,d^2\,e^6}{3}+\frac {1400\,B\,a^3\,b^3\,d^4\,e^4}{9}+\frac {1120\,A\,a^3\,b^3\,d^3\,e^5}{9}+\frac {280\,B\,a^2\,b^4\,d^5\,e^3}{3}+\frac {350\,A\,a^2\,b^4\,d^4\,e^4}{3}+\frac {56\,B\,a\,b^5\,d^6\,e^2}{3}+\frac {112\,A\,a\,b^5\,d^5\,e^3}{3}+\frac {8\,B\,b^6\,d^7\,e}{9}+\frac {28\,A\,b^6\,d^6\,e^2}{9}\right )+x^4\,\left (7\,B\,a^6\,d^6\,e^2+14\,A\,a^6\,d^5\,e^3+12\,B\,a^5\,b\,d^7\,e+42\,A\,a^5\,b\,d^6\,e^2+\frac {15\,B\,a^4\,b^2\,d^8}{4}+30\,A\,a^4\,b^2\,d^7\,e+5\,A\,a^3\,b^3\,d^8\right )+x^{13}\,\left (\frac {20\,B\,a^3\,b^3\,e^8}{13}+\frac {120\,B\,a^2\,b^4\,d\,e^7}{13}+\frac {15\,A\,a^2\,b^4\,e^8}{13}+\frac {168\,B\,a\,b^5\,d^2\,e^6}{13}+\frac {48\,A\,a\,b^5\,d\,e^7}{13}+\frac {56\,B\,b^6\,d^3\,e^5}{13}+\frac {28\,A\,b^6\,d^2\,e^6}{13}\right )+\frac {a^5\,d^7\,x^2\,\left (8\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^7\,x^{15}\,\left (A\,b\,e+6\,B\,a\,e+8\,B\,b\,d\right )}{15}+A\,a^6\,d^8\,x+\frac {B\,b^6\,e^8\,x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^6*(d + e*x)^8,x)

[Out]

x^6*(A*a*b^5*d^8 + (5*B*a^2*b^4*d^8)/2 + (28*A*a^6*d^3*e^5)/3 + (35*B*a^6*d^4*e^4)/3 + 20*A*a^2*b^4*d^7*e + 70
*A*a^5*b*d^4*e^4 + (80*B*a^3*b^3*d^7*e)/3 + 56*B*a^5*b*d^5*e^3 + (280*A*a^3*b^3*d^6*e^2)/3 + 140*A*a^4*b^2*d^5
*e^3 + 70*B*a^4*b^2*d^6*e^2) + x^11*((6*B*a^5*b*e^8)/11 + (15*A*a^4*b^2*e^8)/11 + (70*A*b^6*d^4*e^4)/11 + (56*
B*b^6*d^5*e^3)/11 + (336*A*a*b^5*d^3*e^5)/11 + (160*A*a^3*b^3*d*e^7)/11 + (420*B*a*b^5*d^4*e^4)/11 + (120*B*a^
4*b^2*d*e^7)/11 + (420*A*a^2*b^4*d^2*e^6)/11 + (840*B*a^2*b^4*d^3*e^5)/11 + (560*B*a^3*b^3*d^2*e^6)/11) + x^5*
(3*A*a^2*b^4*d^8 + 4*B*a^3*b^3*d^8 + 14*A*a^6*d^4*e^4 + (56*B*a^6*d^5*e^3)/5 + 32*A*a^3*b^3*d^7*e + (336*A*a^5
*b*d^5*e^3)/5 + 24*B*a^4*b^2*d^7*e + (168*B*a^5*b*d^6*e^2)/5 + 84*A*a^4*b^2*d^6*e^2) + x^12*((5*A*a^3*b^3*e^8)
/3 + (5*B*a^4*b^2*e^8)/4 + (14*A*b^6*d^3*e^5)/3 + (35*B*b^6*d^4*e^4)/6 + 14*A*a*b^5*d^2*e^6 + 10*A*a^2*b^4*d*e
^7 + 28*B*a*b^5*d^3*e^5 + (40*B*a^3*b^3*d*e^7)/3 + 35*B*a^2*b^4*d^2*e^6) + x^7*((A*b^6*d^8)/7 + (6*B*a*b^5*d^8
)/7 + 4*A*a^6*d^2*e^6 + 8*B*a^6*d^3*e^5 + 48*A*a^5*b*d^3*e^5 + (120*B*a^2*b^4*d^7*e)/7 + 60*B*a^5*b*d^4*e^4 +
60*A*a^2*b^4*d^6*e^2 + 160*A*a^3*b^3*d^5*e^3 + 150*A*a^4*b^2*d^4*e^4 + 80*B*a^3*b^3*d^6*e^2 + 120*B*a^4*b^2*d^
5*e^3 + (48*A*a*b^5*d^7*e)/7) + x^10*((B*a^6*e^8)/10 + (3*A*a^5*b*e^8)/5 + (28*A*b^6*d^5*e^3)/5 + (14*B*b^6*d^
6*e^2)/5 + 42*A*a*b^5*d^4*e^4 + 12*A*a^4*b^2*d*e^7 + (168*B*a*b^5*d^5*e^3)/5 + 84*A*a^2*b^4*d^3*e^5 + 56*A*a^3
*b^3*d^2*e^6 + 105*B*a^2*b^4*d^4*e^4 + 112*B*a^3*b^3*d^3*e^5 + 42*B*a^4*b^2*d^2*e^6 + (24*B*a^5*b*d*e^7)/5) +
x^3*(2*B*a^5*b*d^8 + (8*B*a^6*d^7*e)/3 + 5*A*a^4*b^2*d^8 + (28*A*a^6*d^6*e^2)/3 + 16*A*a^5*b*d^7*e) + x^14*((3
*A*a*b^5*e^8)/7 + (4*A*b^6*d*e^7)/7 + (15*B*a^2*b^4*e^8)/14 + 2*B*b^6*d^2*e^6 + (24*B*a*b^5*d*e^7)/7) + x^8*((
B*b^6*d^8)/8 + A*a^6*d*e^7 + A*b^6*d^7*e + (7*B*a^6*d^2*e^6)/2 + 21*A*a*b^5*d^6*e^2 + 21*A*a^5*b*d^2*e^6 + 42*
B*a^5*b*d^3*e^5 + 105*A*a^2*b^4*d^5*e^3 + 175*A*a^3*b^3*d^4*e^4 + 105*A*a^4*b^2*d^3*e^5 + (105*B*a^2*b^4*d^6*e
^2)/2 + 140*B*a^3*b^3*d^5*e^3 + (525*B*a^4*b^2*d^4*e^4)/4 + 6*B*a*b^5*d^7*e) + x^9*((A*a^6*e^8)/9 + (8*B*a^6*d
*e^7)/9 + (8*B*b^6*d^7*e)/9 + (28*A*b^6*d^6*e^2)/9 + (112*A*a*b^5*d^5*e^3)/3 + (56*B*a*b^5*d^6*e^2)/3 + (56*B*
a^5*b*d^2*e^6)/3 + (350*A*a^2*b^4*d^4*e^4)/3 + (1120*A*a^3*b^3*d^3*e^5)/9 + (140*A*a^4*b^2*d^2*e^6)/3 + (280*B
*a^2*b^4*d^5*e^3)/3 + (1400*B*a^3*b^3*d^4*e^4)/9 + (280*B*a^4*b^2*d^3*e^5)/3 + (16*A*a^5*b*d*e^7)/3) + x^4*(5*
A*a^3*b^3*d^8 + (15*B*a^4*b^2*d^8)/4 + 14*A*a^6*d^5*e^3 + 7*B*a^6*d^6*e^2 + 30*A*a^4*b^2*d^7*e + 42*A*a^5*b*d^
6*e^2 + 12*B*a^5*b*d^7*e) + x^13*((15*A*a^2*b^4*e^8)/13 + (20*B*a^3*b^3*e^8)/13 + (28*A*b^6*d^2*e^6)/13 + (56*
B*b^6*d^3*e^5)/13 + (168*B*a*b^5*d^2*e^6)/13 + (120*B*a^2*b^4*d*e^7)/13 + (48*A*a*b^5*d*e^7)/13) + (a^5*d^7*x^
2*(8*A*a*e + 6*A*b*d + B*a*d))/2 + (b^5*e^7*x^15*(A*b*e + 6*B*a*e + 8*B*b*d))/15 + A*a^6*d^8*x + (B*b^6*e^8*x^
16)/16

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