3.11.52 \(\int (a+b x)^6 (A+B x) (d+e x)^7 \, dx\) [1052]

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

[Out]

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

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Rubi [A]
time = 0.74, 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)^{14} (-6 a B e-A b e+7 b B d)}{14 e^8}+\frac {3 b^4 (d+e x)^{13} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac {5 b^3 (d+e x)^{12} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{12 e^8}+\frac {5 b^2 (d+e x)^{11} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac {3 b (d+e x)^{10} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8}+\frac {(d+e x)^9 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac {(d+e x)^8 (b d-a e)^6 (B d-A e)}{8 e^8}+\frac {b^6 B (d+e x)^{15}}{15 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^7,x]

[Out]

-1/8*((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^8)/e^8 + ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^9)/(9*
e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^10)/(10*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)^11)/(11*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)^12
)/(12*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^13)/(13*e^8) - (b^5*(7*b*B*d - A*b*e -
6*a*B*e)*(d + e*x)^14)/(14*e^8) + (b^6*B*(d + e*x)^15)/(15*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)^7 \, dx &=\int \left (\frac {(-b d+a e)^6 (-B d+A e) (d+e x)^7}{e^7}+\frac {(-b d+a e)^5 (-7 b B d+6 A b e+a B e) (d+e x)^8}{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)^9}{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)^{10}}{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)^{11}}{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)^{12}}{e^7}+\frac {b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{13}}{e^7}+\frac {b^6 B (d+e x)^{14}}{e^7}\right ) \, dx\\ &=-\frac {(b d-a e)^6 (B d-A e) (d+e x)^8}{8 e^8}+\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^9}{9 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)^{10}}{10 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)^{11}}{11 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)^{12}}{12 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)^{13}}{13 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^{14}}{14 e^8}+\frac {b^6 B (d+e x)^{15}}{15 e^8}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1224\) vs. \(2(292)=584\).
time = 0.31, size = 1224, normalized size = 4.19 \begin {gather*} a^6 A d^7 x+\frac {1}{2} a^5 d^6 (6 A b d+a B d+7 a A e) x^2+\frac {1}{3} a^4 d^5 \left (a B d (6 b d+7 a e)+3 A \left (5 b^2 d^2+14 a b d e+7 a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 d^4 \left (3 a B d \left (5 b^2 d^2+14 a b d e+7 a^2 e^2\right )+A \left (20 b^3 d^3+105 a b^2 d^2 e+126 a^2 b d e^2+35 a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 d^3 \left (a B d \left (20 b^3 d^3+105 a b^2 d^2 e+126 a^2 b d e^2+35 a^3 e^3\right )+5 A \left (3 b^4 d^4+28 a b^3 d^3 e+63 a^2 b^2 d^2 e^2+42 a^3 b d e^3+7 a^4 e^4\right )\right ) x^5+\frac {1}{6} a d^2 \left (5 a B d \left (3 b^4 d^4+28 a b^3 d^3 e+63 a^2 b^2 d^2 e^2+42 a^3 b d e^3+7 a^4 e^4\right )+3 A \left (2 b^5 d^5+35 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+175 a^3 b^2 d^2 e^3+70 a^4 b d e^4+7 a^5 e^5\right )\right ) x^6+\frac {1}{7} d \left (3 a B d \left (2 b^5 d^5+35 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+175 a^3 b^2 d^2 e^3+70 a^4 b d e^4+7 a^5 e^5\right )+A \left (b^6 d^6+42 a b^5 d^5 e+315 a^2 b^4 d^4 e^2+700 a^3 b^3 d^3 e^3+525 a^4 b^2 d^2 e^4+126 a^5 b d e^5+7 a^6 e^6\right )\right ) x^7+\frac {1}{8} \left (700 a^3 b^3 d^3 e^3 (B d+A e)+42 a^5 b d e^5 (3 B d+A e)+a^6 e^6 (7 B d+A e)+42 a b^5 d^5 e (B d+3 A e)+105 a^4 b^2 d^2 e^4 (5 B d+3 A e)+105 a^2 b^4 d^4 e^2 (3 B d+5 A e)+b^6 d^6 (B d+7 A e)\right ) x^8+\frac {1}{9} e \left (a^6 B e^6+525 a^2 b^4 d^3 e^2 (B d+A e)+105 a^4 b^2 d e^4 (3 B d+A e)+6 a^5 b e^5 (7 B d+A e)+7 b^6 d^5 (B d+3 A e)+140 a^3 b^3 d^2 e^3 (5 B d+3 A e)+42 a b^5 d^4 e (3 B d+5 A e)\right ) x^9+\frac {1}{10} b e^2 \left (6 a^5 B e^5+210 a b^4 d^3 e (B d+A e)+140 a^3 b^2 d e^3 (3 B d+A e)+15 a^4 b e^4 (7 B d+A e)+105 a^2 b^3 d^2 e^2 (5 B d+3 A e)+7 b^5 d^4 (3 B d+5 A e)\right ) x^{10}+\frac {1}{11} b^2 e^3 \left (15 a^4 B e^4+35 b^4 d^3 (B d+A e)+105 a^2 b^2 d e^2 (3 B d+A e)+20 a^3 b e^3 (7 B d+A e)+42 a b^3 d^2 e (5 B d+3 A e)\right ) x^{11}+\frac {1}{12} b^3 e^4 \left (20 a^3 B e^3+42 a b^2 d e (3 B d+A e)+15 a^2 b e^2 (7 B d+A e)+7 b^3 d^2 (5 B d+3 A e)\right ) x^{12}+\frac {1}{13} b^4 e^5 \left (15 a^2 B e^2+7 b^2 d (3 B d+A e)+6 a b e (7 B d+A e)\right ) x^{13}+\frac {1}{14} b^5 e^6 (7 b B d+A b e+6 a B e) x^{14}+\frac {1}{15} b^6 B e^7 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^7,x]

[Out]

a^6*A*d^7*x + (a^5*d^6*(6*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^4*d^5*(a*B*d*(6*b*d + 7*a*e) + 3*A*(5*b^2*d^2 +
14*a*b*d*e + 7*a^2*e^2))*x^3)/3 + (a^3*d^4*(3*a*B*d*(5*b^2*d^2 + 14*a*b*d*e + 7*a^2*e^2) + A*(20*b^3*d^3 + 10
5*a*b^2*d^2*e + 126*a^2*b*d*e^2 + 35*a^3*e^3))*x^4)/4 + (a^2*d^3*(a*B*d*(20*b^3*d^3 + 105*a*b^2*d^2*e + 126*a^
2*b*d*e^2 + 35*a^3*e^3) + 5*A*(3*b^4*d^4 + 28*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 42*a^3*b*d*e^3 + 7*a^4*e^4))*
x^5)/5 + (a*d^2*(5*a*B*d*(3*b^4*d^4 + 28*a*b^3*d^3*e + 63*a^2*b^2*d^2*e^2 + 42*a^3*b*d*e^3 + 7*a^4*e^4) + 3*A*
(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 175*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5))*x^6)/6
+ (d*(3*a*B*d*(2*b^5*d^5 + 35*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 175*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5
*e^5) + A*(b^6*d^6 + 42*a*b^5*d^5*e + 315*a^2*b^4*d^4*e^2 + 700*a^3*b^3*d^3*e^3 + 525*a^4*b^2*d^2*e^4 + 126*a^
5*b*d*e^5 + 7*a^6*e^6))*x^7)/7 + ((700*a^3*b^3*d^3*e^3*(B*d + A*e) + 42*a^5*b*d*e^5*(3*B*d + A*e) + a^6*e^6*(7
*B*d + A*e) + 42*a*b^5*d^5*e*(B*d + 3*A*e) + 105*a^4*b^2*d^2*e^4*(5*B*d + 3*A*e) + 105*a^2*b^4*d^4*e^2*(3*B*d
+ 5*A*e) + b^6*d^6*(B*d + 7*A*e))*x^8)/8 + (e*(a^6*B*e^6 + 525*a^2*b^4*d^3*e^2*(B*d + A*e) + 105*a^4*b^2*d*e^4
*(3*B*d + A*e) + 6*a^5*b*e^5*(7*B*d + A*e) + 7*b^6*d^5*(B*d + 3*A*e) + 140*a^3*b^3*d^2*e^3*(5*B*d + 3*A*e) + 4
2*a*b^5*d^4*e*(3*B*d + 5*A*e))*x^9)/9 + (b*e^2*(6*a^5*B*e^5 + 210*a*b^4*d^3*e*(B*d + A*e) + 140*a^3*b^2*d*e^3*
(3*B*d + A*e) + 15*a^4*b*e^4*(7*B*d + A*e) + 105*a^2*b^3*d^2*e^2*(5*B*d + 3*A*e) + 7*b^5*d^4*(3*B*d + 5*A*e))*
x^10)/10 + (b^2*e^3*(15*a^4*B*e^4 + 35*b^4*d^3*(B*d + A*e) + 105*a^2*b^2*d*e^2*(3*B*d + A*e) + 20*a^3*b*e^3*(7
*B*d + A*e) + 42*a*b^3*d^2*e*(5*B*d + 3*A*e))*x^11)/11 + (b^3*e^4*(20*a^3*B*e^3 + 42*a*b^2*d*e*(3*B*d + A*e) +
15*a^2*b*e^2*(7*B*d + A*e) + 7*b^3*d^2*(5*B*d + 3*A*e))*x^12)/12 + (b^4*e^5*(15*a^2*B*e^2 + 7*b^2*d*(3*B*d +
A*e) + 6*a*b*e*(7*B*d + A*e))*x^13)/13 + (b^5*e^6*(7*b*B*d + A*b*e + 6*a*B*e)*x^14)/14 + (b^6*B*e^7*x^15)/15

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1348\) vs. \(2(276)=552\).
time = 0.09, size = 1349, normalized size = 4.62

method	result	size
default	\(\frac {b^{6} B \,e^{7} x^{15}}{15}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e^{7}+7 b^{6} B d \,e^{6}\right ) x^{14}}{14}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{7}+7 \left (b^{6} A +6 a \,b^{5} B \right ) d \,e^{6}+21 b^{6} B \,d^{2} e^{5}\right ) x^{13}}{13}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{7}+7 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d \,e^{6}+21 \left (b^{6} A +6 a \,b^{5} B \right ) d^{2} e^{5}+35 b^{6} B \,d^{3} e^{4}\right ) x^{12}}{12}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{7}+7 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d \,e^{6}+21 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2} e^{5}+35 \left (b^{6} A +6 a \,b^{5} B \right ) d^{3} e^{4}+35 b^{6} B \,d^{4} e^{3}\right ) x^{11}}{11}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{7}+7 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d \,e^{6}+21 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2} e^{5}+35 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{3} e^{4}+35 \left (b^{6} A +6 a \,b^{5} B \right ) d^{4} e^{3}+21 b^{6} B \,d^{5} e^{2}\right ) x^{10}}{10}+\frac {\left (\left (6 a^{5} b A +a^{6} B \right ) e^{7}+7 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d \,e^{6}+21 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2} e^{5}+35 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{3} e^{4}+35 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{4} e^{3}+21 \left (b^{6} A +6 a \,b^{5} B \right ) d^{5} e^{2}+7 b^{6} B e \,d^{6}\right ) x^{9}}{9}+\frac {\left (a^{6} A \,e^{7}+7 \left (6 a^{5} b A +a^{6} B \right ) d \,e^{6}+21 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2} e^{5}+35 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{3} e^{4}+35 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{4} e^{3}+21 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{5} e^{2}+7 \left (b^{6} A +6 a \,b^{5} B \right ) e \,d^{6}+b^{6} B \,d^{7}\right ) x^{8}}{8}+\frac {\left (7 a^{6} A d \,e^{6}+21 \left (6 a^{5} b A +a^{6} B \right ) d^{2} e^{5}+35 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{3} e^{4}+35 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{4} e^{3}+21 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{5} e^{2}+7 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e \,d^{6}+\left (b^{6} A +6 a \,b^{5} B \right ) d^{7}\right ) x^{7}}{7}+\frac {\left (21 a^{6} A \,d^{2} e^{5}+35 \left (6 a^{5} b A +a^{6} B \right ) d^{3} e^{4}+35 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{4} e^{3}+21 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{5} e^{2}+7 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e \,d^{6}+\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{6} A \,d^{3} e^{4}+35 \left (6 a^{5} b A +a^{6} B \right ) d^{4} e^{3}+21 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{5} e^{2}+7 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e \,d^{6}+\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{6} A \,d^{4} e^{3}+21 \left (6 a^{5} b A +a^{6} B \right ) d^{5} e^{2}+7 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e \,d^{6}+\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{6} A \,d^{5} e^{2}+7 \left (6 a^{5} b A +a^{6} B \right ) e \,d^{6}+\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{6} A e \,d^{6}+\left (6 a^{5} b A +a^{6} B \right ) d^{7}\right ) x^{2}}{2}+a^{6} A \,d^{7} x\)	\(1349\)
norman	\(\text {Expression too large to display}\)	\(1448\)
gosper	\(\text {Expression too large to display}\)	\(1713\)
risch	\(\text {Expression too large to display}\)	\(1713\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1384 vs. \(2 (295) = 590\).
time = 0.29, size = 1384, normalized size = 4.74 \begin {gather*} \frac {1}{15} \, B b^{6} x^{15} e^{7} + A a^{6} d^{7} x + \frac {1}{14} \, {\left (7 \, B b^{6} d e^{6} + 6 \, B a b^{5} e^{7} + A b^{6} e^{7}\right )} x^{14} + \frac {1}{13} \, {\left (21 \, B b^{6} d^{2} e^{5} + 15 \, B a^{2} b^{4} e^{7} + 6 \, A a b^{5} e^{7} + 7 \, {\left (6 \, B a b^{5} e^{6} + A b^{6} e^{6}\right )} d\right )} x^{13} + \frac {1}{12} \, {\left (35 \, B b^{6} d^{3} e^{4} + 20 \, B a^{3} b^{3} e^{7} + 15 \, A a^{2} b^{4} e^{7} + 21 \, {\left (6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} d^{2} + 21 \, {\left (5 \, B a^{2} b^{4} e^{6} + 2 \, A a b^{5} e^{6}\right )} d\right )} x^{12} + \frac {1}{11} \, {\left (35 \, B b^{6} d^{4} e^{3} + 15 \, B a^{4} b^{2} e^{7} + 20 \, A a^{3} b^{3} e^{7} + 35 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d^{3} + 63 \, {\left (5 \, B a^{2} b^{4} e^{5} + 2 \, A a b^{5} e^{5}\right )} d^{2} + 35 \, {\left (4 \, B a^{3} b^{3} e^{6} + 3 \, A a^{2} b^{4} e^{6}\right )} d\right )} x^{11} + \frac {1}{10} \, {\left (21 \, B b^{6} d^{5} e^{2} + 6 \, B a^{5} b e^{7} + 15 \, A a^{4} b^{2} e^{7} + 35 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{4} + 105 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d^{3} + 105 \, {\left (4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5}\right )} d^{2} + 35 \, {\left (3 \, B a^{4} b^{2} e^{6} + 4 \, A a^{3} b^{3} e^{6}\right )} d\right )} x^{10} + \frac {1}{9} \, {\left (7 \, B b^{6} d^{6} e + B a^{6} e^{7} + 6 \, A a^{5} b e^{7} + 21 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{5} + 105 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{4} + 175 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d^{3} + 105 \, {\left (3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5}\right )} d^{2} + 21 \, {\left (2 \, B a^{5} b e^{6} + 5 \, A a^{4} b^{2} e^{6}\right )} d\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{7} + A a^{6} e^{7} + 7 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{6} + 63 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{5} + 175 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{4} + 175 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d^{3} + 63 \, {\left (2 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} d^{2} + 7 \, {\left (B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (7 \, A a^{6} d e^{6} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{7} + 21 \, {\left (5 \, B a^{2} b^{4} e + 2 \, A a b^{5} e\right )} d^{6} + 105 \, {\left (4 \, B a^{3} b^{3} e^{2} + 3 \, A a^{2} b^{4} e^{2}\right )} d^{5} + 175 \, {\left (3 \, B a^{4} b^{2} e^{3} + 4 \, A a^{3} b^{3} e^{3}\right )} d^{4} + 105 \, {\left (2 \, B a^{5} b e^{4} + 5 \, A a^{4} b^{2} e^{4}\right )} d^{3} + 21 \, {\left (B a^{6} e^{5} + 6 \, A a^{5} b e^{5}\right )} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left (21 \, A a^{6} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{7} + 35 \, {\left (4 \, B a^{3} b^{3} e + 3 \, A a^{2} b^{4} e\right )} d^{6} + 105 \, {\left (3 \, B a^{4} b^{2} e^{2} + 4 \, A a^{3} b^{3} e^{2}\right )} d^{5} + 105 \, {\left (2 \, B a^{5} b e^{3} + 5 \, A a^{4} b^{2} e^{3}\right )} d^{4} + 35 \, {\left (B a^{6} e^{4} + 6 \, A a^{5} b e^{4}\right )} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (35 \, A a^{6} d^{3} e^{4} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{7} + 35 \, {\left (3 \, B a^{4} b^{2} e + 4 \, A a^{3} b^{3} e\right )} d^{6} + 63 \, {\left (2 \, B a^{5} b e^{2} + 5 \, A a^{4} b^{2} e^{2}\right )} d^{5} + 35 \, {\left (B a^{6} e^{3} + 6 \, A a^{5} b e^{3}\right )} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, A a^{6} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{7} + 21 \, {\left (2 \, B a^{5} b e + 5 \, A a^{4} b^{2} e\right )} d^{6} + 21 \, {\left (B a^{6} e^{2} + 6 \, A a^{5} b e^{2}\right )} d^{5}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, A a^{6} d^{5} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{7} + 7 \, {\left (B a^{6} e + 6 \, A a^{5} b e\right )} d^{6}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, A a^{6} d^{6} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{7}\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)^7,x, algorithm="maxima")

[Out]

1/15*B*b^6*x^15*e^7 + A*a^6*d^7*x + 1/14*(7*B*b^6*d*e^6 + 6*B*a*b^5*e^7 + A*b^6*e^7)*x^14 + 1/13*(21*B*b^6*d^2
*e^5 + 15*B*a^2*b^4*e^7 + 6*A*a*b^5*e^7 + 7*(6*B*a*b^5*e^6 + A*b^6*e^6)*d)*x^13 + 1/12*(35*B*b^6*d^3*e^4 + 20*
B*a^3*b^3*e^7 + 15*A*a^2*b^4*e^7 + 21*(6*B*a*b^5*e^5 + A*b^6*e^5)*d^2 + 21*(5*B*a^2*b^4*e^6 + 2*A*a*b^5*e^6)*d
)*x^12 + 1/11*(35*B*b^6*d^4*e^3 + 15*B*a^4*b^2*e^7 + 20*A*a^3*b^3*e^7 + 35*(6*B*a*b^5*e^4 + A*b^6*e^4)*d^3 + 6
3*(5*B*a^2*b^4*e^5 + 2*A*a*b^5*e^5)*d^2 + 35*(4*B*a^3*b^3*e^6 + 3*A*a^2*b^4*e^6)*d)*x^11 + 1/10*(21*B*b^6*d^5*
e^2 + 6*B*a^5*b*e^7 + 15*A*a^4*b^2*e^7 + 35*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^4 + 105*(5*B*a^2*b^4*e^4 + 2*A*a*b^5
*e^4)*d^3 + 105*(4*B*a^3*b^3*e^5 + 3*A*a^2*b^4*e^5)*d^2 + 35*(3*B*a^4*b^2*e^6 + 4*A*a^3*b^3*e^6)*d)*x^10 + 1/9
*(7*B*b^6*d^6*e + B*a^6*e^7 + 6*A*a^5*b*e^7 + 21*(6*B*a*b^5*e^2 + A*b^6*e^2)*d^5 + 105*(5*B*a^2*b^4*e^3 + 2*A*
a*b^5*e^3)*d^4 + 175*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d^3 + 105*(3*B*a^4*b^2*e^5 + 4*A*a^3*b^3*e^5)*d^2 + 2
1*(2*B*a^5*b*e^6 + 5*A*a^4*b^2*e^6)*d)*x^9 + 1/8*(B*b^6*d^7 + A*a^6*e^7 + 7*(6*B*a*b^5*e + A*b^6*e)*d^6 + 63*(
5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^5 + 175*(4*B*a^3*b^3*e^3 + 3*A*a^2*b^4*e^3)*d^4 + 175*(3*B*a^4*b^2*e^4 + 4*
A*a^3*b^3*e^4)*d^3 + 63*(2*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*d^2 + 7*(B*a^6*e^6 + 6*A*a^5*b*e^6)*d)*x^8 + 1/7*(7*
A*a^6*d*e^6 + (6*B*a*b^5 + A*b^6)*d^7 + 21*(5*B*a^2*b^4*e + 2*A*a*b^5*e)*d^6 + 105*(4*B*a^3*b^3*e^2 + 3*A*a^2*
b^4*e^2)*d^5 + 175*(3*B*a^4*b^2*e^3 + 4*A*a^3*b^3*e^3)*d^4 + 105*(2*B*a^5*b*e^4 + 5*A*a^4*b^2*e^4)*d^3 + 21*(B
*a^6*e^5 + 6*A*a^5*b*e^5)*d^2)*x^7 + 1/6*(21*A*a^6*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7 + 35*(4*B*a^3*b^3
*e + 3*A*a^2*b^4*e)*d^6 + 105*(3*B*a^4*b^2*e^2 + 4*A*a^3*b^3*e^2)*d^5 + 105*(2*B*a^5*b*e^3 + 5*A*a^4*b^2*e^3)*
d^4 + 35*(B*a^6*e^4 + 6*A*a^5*b*e^4)*d^3)*x^6 + 1/5*(35*A*a^6*d^3*e^4 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7 + 35
*(3*B*a^4*b^2*e + 4*A*a^3*b^3*e)*d^6 + 63*(2*B*a^5*b*e^2 + 5*A*a^4*b^2*e^2)*d^5 + 35*(B*a^6*e^3 + 6*A*a^5*b*e^
3)*d^4)*x^5 + 1/4*(35*A*a^6*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^7 + 21*(2*B*a^5*b*e + 5*A*a^4*b^2*e)*d^6
+ 21*(B*a^6*e^2 + 6*A*a^5*b*e^2)*d^5)*x^4 + 1/3*(21*A*a^6*d^5*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^7 + 7*(B*a^
6*e + 6*A*a^5*b*e)*d^6)*x^3 + 1/2*(7*A*a^6*d^6*e + (B*a^6 + 6*A*a^5*b)*d^7)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1351 vs. \(2 (295) = 590\).
time = 1.39, size = 1351, normalized size = 4.63 \begin {gather*} \frac {1}{8} \, B b^{6} d^{7} x^{8} + A a^{6} d^{7} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{7} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{7} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{7} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{7} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{7} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{7} x^{2} + \frac {1}{360360} \, {\left (24024 \, B b^{6} x^{15} + 45045 \, A a^{6} x^{8} + 25740 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{14} + 83160 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{13} + 150150 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{12} + 163800 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{11} + 108108 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{10} + 40040 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{9}\right )} e^{7} + \frac {1}{3432} \, {\left (1716 \, B b^{6} d x^{14} + 3432 \, A a^{6} d x^{7} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{13} + 6006 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{12} + 10920 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{11} + 12012 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{10} + 8008 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x^{9} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d x^{8}\right )} e^{6} + \frac {1}{3432} \, {\left (5544 \, B b^{6} d^{2} x^{13} + 12012 \, A a^{6} d^{2} x^{6} + 6006 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{12} + 19656 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{11} + 36036 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{10} + 40040 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x^{9} + 27027 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} x^{8} + 10296 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} x^{7}\right )} e^{5} + \frac {1}{792} \, {\left (2310 \, B b^{6} d^{3} x^{12} + 5544 \, A a^{6} d^{3} x^{5} + 2520 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{11} + 8316 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{10} + 15400 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x^{9} + 17325 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} x^{8} + 11880 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} x^{7} + 4620 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} x^{6}\right )} e^{4} + \frac {1}{264} \, {\left (840 \, B b^{6} d^{4} x^{11} + 2310 \, A a^{6} d^{4} x^{4} + 924 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{10} + 3080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x^{9} + 5775 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} x^{8} + 6600 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} x^{7} + 4620 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{4} x^{6} + 1848 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{4} x^{5}\right )} e^{3} + \frac {1}{120} \, {\left (252 \, B b^{6} d^{5} x^{10} + 840 \, A a^{6} d^{5} x^{3} + 280 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x^{9} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} x^{8} + 1800 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} x^{7} + 2100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{5} x^{6} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{5} x^{5} + 630 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{5} x^{4}\right )} e^{2} + \frac {1}{72} \, {\left (56 \, B b^{6} d^{6} x^{9} + 252 \, A a^{6} d^{6} x^{2} + 63 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} x^{8} + 216 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{6} x^{7} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{6} x^{6} + 504 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{6} x^{5} + 378 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{6} x^{4} + 168 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/8*B*b^6*d^7*x^8 + A*a^6*d^7*x + 1/7*(6*B*a*b^5 + A*b^6)*d^7*x^7 + 1/2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*x^6 + (4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*x^5 + 5/4*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^7*x^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^7*x^
3 + 1/2*(B*a^6 + 6*A*a^5*b)*d^7*x^2 + 1/360360*(24024*B*b^6*x^15 + 45045*A*a^6*x^8 + 25740*(6*B*a*b^5 + A*b^6)
*x^14 + 83160*(5*B*a^2*b^4 + 2*A*a*b^5)*x^13 + 150150*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^12 + 163800*(3*B*a^4*b^2 +
4*A*a^3*b^3)*x^11 + 108108*(2*B*a^5*b + 5*A*a^4*b^2)*x^10 + 40040*(B*a^6 + 6*A*a^5*b)*x^9)*e^7 + 1/3432*(1716
*B*b^6*d*x^14 + 3432*A*a^6*d*x^7 + 1848*(6*B*a*b^5 + A*b^6)*d*x^13 + 6006*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^12 + 1
0920*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*x^11 + 12012*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^10 + 8008*(2*B*a^5*b + 5*A*a^4
*b^2)*d*x^9 + 3003*(B*a^6 + 6*A*a^5*b)*d*x^8)*e^6 + 1/3432*(5544*B*b^6*d^2*x^13 + 12012*A*a^6*d^2*x^6 + 6006*(
6*B*a*b^5 + A*b^6)*d^2*x^12 + 19656*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^11 + 36036*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2
*x^10 + 40040*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x^9 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*x^8 + 10296*(B*a^6 + 6
*A*a^5*b)*d^2*x^7)*e^5 + 1/792*(2310*B*b^6*d^3*x^12 + 5544*A*a^6*d^3*x^5 + 2520*(6*B*a*b^5 + A*b^6)*d^3*x^11 +
8316*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^10 + 15400*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x^9 + 17325*(3*B*a^4*b^2 + 4*
A*a^3*b^3)*d^3*x^8 + 11880*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*x^7 + 4620*(B*a^6 + 6*A*a^5*b)*d^3*x^6)*e^4 + 1/264*(
840*B*b^6*d^4*x^11 + 2310*A*a^6*d^4*x^4 + 924*(6*B*a*b^5 + A*b^6)*d^4*x^10 + 3080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^
4*x^9 + 5775*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*x^8 + 6600*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*x^7 + 4620*(2*B*a^5*b
+ 5*A*a^4*b^2)*d^4*x^6 + 1848*(B*a^6 + 6*A*a^5*b)*d^4*x^5)*e^3 + 1/120*(252*B*b^6*d^5*x^10 + 840*A*a^6*d^5*x^3
+ 280*(6*B*a*b^5 + A*b^6)*d^5*x^9 + 945*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*x^8 + 1800*(4*B*a^3*b^3 + 3*A*a^2*b^4)*
d^5*x^7 + 2100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*x^6 + 1512*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*x^5 + 630*(B*a^6 + 6*A
*a^5*b)*d^5*x^4)*e^2 + 1/72*(56*B*b^6*d^6*x^9 + 252*A*a^6*d^6*x^2 + 63*(6*B*a*b^5 + A*b^6)*d^6*x^8 + 216*(5*B*
a^2*b^4 + 2*A*a*b^5)*d^6*x^7 + 420*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^6*x^6 + 504*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^6*x
^5 + 378*(2*B*a^5*b + 5*A*a^4*b^2)*d^6*x^4 + 168*(B*a^6 + 6*A*a^5*b)*d^6*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1756 vs. \(2 (296) = 592\).
time = 0.15, size = 1756, normalized size = 6.01 \begin {gather*} A a^{6} d^{7} x + \frac {B b^{6} e^{7} x^{15}}{15} + x^{14} \left (\frac {A b^{6} e^{7}}{14} + \frac {3 B a b^{5} e^{7}}{7} + \frac {B b^{6} d e^{6}}{2}\right ) + x^{13} \cdot \left (\frac {6 A a b^{5} e^{7}}{13} + \frac {7 A b^{6} d e^{6}}{13} + \frac {15 B a^{2} b^{4} e^{7}}{13} + \frac {42 B a b^{5} d e^{6}}{13} + \frac {21 B b^{6} d^{2} e^{5}}{13}\right ) + x^{12} \cdot \left (\frac {5 A a^{2} b^{4} e^{7}}{4} + \frac {7 A a b^{5} d e^{6}}{2} + \frac {7 A b^{6} d^{2} e^{5}}{4} + \frac {5 B a^{3} b^{3} e^{7}}{3} + \frac {35 B a^{2} b^{4} d e^{6}}{4} + \frac {21 B a b^{5} d^{2} e^{5}}{2} + \frac {35 B b^{6} d^{3} e^{4}}{12}\right ) + x^{11} \cdot \left (\frac {20 A a^{3} b^{3} e^{7}}{11} + \frac {105 A a^{2} b^{4} d e^{6}}{11} + \frac {126 A a b^{5} d^{2} e^{5}}{11} + \frac {35 A b^{6} d^{3} e^{4}}{11} + \frac {15 B a^{4} b^{2} e^{7}}{11} + \frac {140 B a^{3} b^{3} d e^{6}}{11} + \frac {315 B a^{2} b^{4} d^{2} e^{5}}{11} + \frac {210 B a b^{5} d^{3} e^{4}}{11} + \frac {35 B b^{6} d^{4} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a^{4} b^{2} e^{7}}{2} + 14 A a^{3} b^{3} d e^{6} + \frac {63 A a^{2} b^{4} d^{2} e^{5}}{2} + 21 A a b^{5} d^{3} e^{4} + \frac {7 A b^{6} d^{4} e^{3}}{2} + \frac {3 B a^{5} b e^{7}}{5} + \frac {21 B a^{4} b^{2} d e^{6}}{2} + 42 B a^{3} b^{3} d^{2} e^{5} + \frac {105 B a^{2} b^{4} d^{3} e^{4}}{2} + 21 B a b^{5} d^{4} e^{3} + \frac {21 B b^{6} d^{5} e^{2}}{10}\right ) + x^{9} \cdot \left (\frac {2 A a^{5} b e^{7}}{3} + \frac {35 A a^{4} b^{2} d e^{6}}{3} + \frac {140 A a^{3} b^{3} d^{2} e^{5}}{3} + \frac {175 A a^{2} b^{4} d^{3} e^{4}}{3} + \frac {70 A a b^{5} d^{4} e^{3}}{3} + \frac {7 A b^{6} d^{5} e^{2}}{3} + \frac {B a^{6} e^{7}}{9} + \frac {14 B a^{5} b d e^{6}}{3} + 35 B a^{4} b^{2} d^{2} e^{5} + \frac {700 B a^{3} b^{3} d^{3} e^{4}}{9} + \frac {175 B a^{2} b^{4} d^{4} e^{3}}{3} + 14 B a b^{5} d^{5} e^{2} + \frac {7 B b^{6} d^{6} e}{9}\right ) + x^{8} \left (\frac {A a^{6} e^{7}}{8} + \frac {21 A a^{5} b d e^{6}}{4} + \frac {315 A a^{4} b^{2} d^{2} e^{5}}{8} + \frac {175 A a^{3} b^{3} d^{3} e^{4}}{2} + \frac {525 A a^{2} b^{4} d^{4} e^{3}}{8} + \frac {63 A a b^{5} d^{5} e^{2}}{4} + \frac {7 A b^{6} d^{6} e}{8} + \frac {7 B a^{6} d e^{6}}{8} + \frac {63 B a^{5} b d^{2} e^{5}}{4} + \frac {525 B a^{4} b^{2} d^{3} e^{4}}{8} + \frac {175 B a^{3} b^{3} d^{4} e^{3}}{2} + \frac {315 B a^{2} b^{4} d^{5} e^{2}}{8} + \frac {21 B a b^{5} d^{6} e}{4} + \frac {B b^{6} d^{7}}{8}\right ) + x^{7} \left (A a^{6} d e^{6} + 18 A a^{5} b d^{2} e^{5} + 75 A a^{4} b^{2} d^{3} e^{4} + 100 A a^{3} b^{3} d^{4} e^{3} + 45 A a^{2} b^{4} d^{5} e^{2} + 6 A a b^{5} d^{6} e + \frac {A b^{6} d^{7}}{7} + 3 B a^{6} d^{2} e^{5} + 30 B a^{5} b d^{3} e^{4} + 75 B a^{4} b^{2} d^{4} e^{3} + 60 B a^{3} b^{3} d^{5} e^{2} + 15 B a^{2} b^{4} d^{6} e + \frac {6 B a b^{5} d^{7}}{7}\right ) + x^{6} \cdot \left (\frac {7 A a^{6} d^{2} e^{5}}{2} + 35 A a^{5} b d^{3} e^{4} + \frac {175 A a^{4} b^{2} d^{4} e^{3}}{2} + 70 A a^{3} b^{3} d^{5} e^{2} + \frac {35 A a^{2} b^{4} d^{6} e}{2} + A a b^{5} d^{7} + \frac {35 B a^{6} d^{3} e^{4}}{6} + 35 B a^{5} b d^{4} e^{3} + \frac {105 B a^{4} b^{2} d^{5} e^{2}}{2} + \frac {70 B a^{3} b^{3} d^{6} e}{3} + \frac {5 B a^{2} b^{4} d^{7}}{2}\right ) + x^{5} \cdot \left (7 A a^{6} d^{3} e^{4} + 42 A a^{5} b d^{4} e^{3} + 63 A a^{4} b^{2} d^{5} e^{2} + 28 A a^{3} b^{3} d^{6} e + 3 A a^{2} b^{4} d^{7} + 7 B a^{6} d^{4} e^{3} + \frac {126 B a^{5} b d^{5} e^{2}}{5} + 21 B a^{4} b^{2} d^{6} e + 4 B a^{3} b^{3} d^{7}\right ) + x^{4} \cdot \left (\frac {35 A a^{6} d^{4} e^{3}}{4} + \frac {63 A a^{5} b d^{5} e^{2}}{2} + \frac {105 A a^{4} b^{2} d^{6} e}{4} + 5 A a^{3} b^{3} d^{7} + \frac {21 B a^{6} d^{5} e^{2}}{4} + \frac {21 B a^{5} b d^{6} e}{2} + \frac {15 B a^{4} b^{2} d^{7}}{4}\right ) + x^{3} \cdot \left (7 A a^{6} d^{5} e^{2} + 14 A a^{5} b d^{6} e + 5 A a^{4} b^{2} d^{7} + \frac {7 B a^{6} d^{6} e}{3} + 2 B a^{5} b d^{7}\right ) + x^{2} \cdot \left (\frac {7 A a^{6} d^{6} e}{2} + 3 A a^{5} b d^{7} + \frac {B a^{6} d^{7}}{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)**7,x)

[Out]

A*a**6*d**7*x + B*b**6*e**7*x**15/15 + x**14*(A*b**6*e**7/14 + 3*B*a*b**5*e**7/7 + B*b**6*d*e**6/2) + x**13*(6
*A*a*b**5*e**7/13 + 7*A*b**6*d*e**6/13 + 15*B*a**2*b**4*e**7/13 + 42*B*a*b**5*d*e**6/13 + 21*B*b**6*d**2*e**5/
13) + x**12*(5*A*a**2*b**4*e**7/4 + 7*A*a*b**5*d*e**6/2 + 7*A*b**6*d**2*e**5/4 + 5*B*a**3*b**3*e**7/3 + 35*B*a
**2*b**4*d*e**6/4 + 21*B*a*b**5*d**2*e**5/2 + 35*B*b**6*d**3*e**4/12) + x**11*(20*A*a**3*b**3*e**7/11 + 105*A*
a**2*b**4*d*e**6/11 + 126*A*a*b**5*d**2*e**5/11 + 35*A*b**6*d**3*e**4/11 + 15*B*a**4*b**2*e**7/11 + 140*B*a**3
*b**3*d*e**6/11 + 315*B*a**2*b**4*d**2*e**5/11 + 210*B*a*b**5*d**3*e**4/11 + 35*B*b**6*d**4*e**3/11) + x**10*(
3*A*a**4*b**2*e**7/2 + 14*A*a**3*b**3*d*e**6 + 63*A*a**2*b**4*d**2*e**5/2 + 21*A*a*b**5*d**3*e**4 + 7*A*b**6*d
**4*e**3/2 + 3*B*a**5*b*e**7/5 + 21*B*a**4*b**2*d*e**6/2 + 42*B*a**3*b**3*d**2*e**5 + 105*B*a**2*b**4*d**3*e**
4/2 + 21*B*a*b**5*d**4*e**3 + 21*B*b**6*d**5*e**2/10) + x**9*(2*A*a**5*b*e**7/3 + 35*A*a**4*b**2*d*e**6/3 + 14
0*A*a**3*b**3*d**2*e**5/3 + 175*A*a**2*b**4*d**3*e**4/3 + 70*A*a*b**5*d**4*e**3/3 + 7*A*b**6*d**5*e**2/3 + B*a
**6*e**7/9 + 14*B*a**5*b*d*e**6/3 + 35*B*a**4*b**2*d**2*e**5 + 700*B*a**3*b**3*d**3*e**4/9 + 175*B*a**2*b**4*d
**4*e**3/3 + 14*B*a*b**5*d**5*e**2 + 7*B*b**6*d**6*e/9) + x**8*(A*a**6*e**7/8 + 21*A*a**5*b*d*e**6/4 + 315*A*a
**4*b**2*d**2*e**5/8 + 175*A*a**3*b**3*d**3*e**4/2 + 525*A*a**2*b**4*d**4*e**3/8 + 63*A*a*b**5*d**5*e**2/4 + 7
*A*b**6*d**6*e/8 + 7*B*a**6*d*e**6/8 + 63*B*a**5*b*d**2*e**5/4 + 525*B*a**4*b**2*d**3*e**4/8 + 175*B*a**3*b**3
*d**4*e**3/2 + 315*B*a**2*b**4*d**5*e**2/8 + 21*B*a*b**5*d**6*e/4 + B*b**6*d**7/8) + x**7*(A*a**6*d*e**6 + 18*
A*a**5*b*d**2*e**5 + 75*A*a**4*b**2*d**3*e**4 + 100*A*a**3*b**3*d**4*e**3 + 45*A*a**2*b**4*d**5*e**2 + 6*A*a*b
**5*d**6*e + A*b**6*d**7/7 + 3*B*a**6*d**2*e**5 + 30*B*a**5*b*d**3*e**4 + 75*B*a**4*b**2*d**4*e**3 + 60*B*a**3
*b**3*d**5*e**2 + 15*B*a**2*b**4*d**6*e + 6*B*a*b**5*d**7/7) + x**6*(7*A*a**6*d**2*e**5/2 + 35*A*a**5*b*d**3*e
**4 + 175*A*a**4*b**2*d**4*e**3/2 + 70*A*a**3*b**3*d**5*e**2 + 35*A*a**2*b**4*d**6*e/2 + A*a*b**5*d**7 + 35*B*
a**6*d**3*e**4/6 + 35*B*a**5*b*d**4*e**3 + 105*B*a**4*b**2*d**5*e**2/2 + 70*B*a**3*b**3*d**6*e/3 + 5*B*a**2*b*
*4*d**7/2) + x**5*(7*A*a**6*d**3*e**4 + 42*A*a**5*b*d**4*e**3 + 63*A*a**4*b**2*d**5*e**2 + 28*A*a**3*b**3*d**6
*e + 3*A*a**2*b**4*d**7 + 7*B*a**6*d**4*e**3 + 126*B*a**5*b*d**5*e**2/5 + 21*B*a**4*b**2*d**6*e + 4*B*a**3*b**
3*d**7) + x**4*(35*A*a**6*d**4*e**3/4 + 63*A*a**5*b*d**5*e**2/2 + 105*A*a**4*b**2*d**6*e/4 + 5*A*a**3*b**3*d**
7 + 21*B*a**6*d**5*e**2/4 + 21*B*a**5*b*d**6*e/2 + 15*B*a**4*b**2*d**7/4) + x**3*(7*A*a**6*d**5*e**2 + 14*A*a*
*5*b*d**6*e + 5*A*a**4*b**2*d**7 + 7*B*a**6*d**6*e/3 + 2*B*a**5*b*d**7) + x**2*(7*A*a**6*d**6*e/2 + 3*A*a**5*b
*d**7 + B*a**6*d**7/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (295) = 590\).
time = 1.44, size = 1642, normalized size = 5.62 \begin {gather*} \frac {1}{15} \, B b^{6} x^{15} e^{7} + \frac {1}{2} \, B b^{6} d x^{14} e^{6} + \frac {21}{13} \, B b^{6} d^{2} x^{13} e^{5} + \frac {35}{12} \, B b^{6} d^{3} x^{12} e^{4} + \frac {35}{11} \, B b^{6} d^{4} x^{11} e^{3} + \frac {21}{10} \, B b^{6} d^{5} x^{10} e^{2} + \frac {7}{9} \, B b^{6} d^{6} x^{9} e + \frac {1}{8} \, B b^{6} d^{7} x^{8} + \frac {3}{7} \, B a b^{5} x^{14} e^{7} + \frac {1}{14} \, A b^{6} x^{14} e^{7} + \frac {42}{13} \, B a b^{5} d x^{13} e^{6} + \frac {7}{13} \, A b^{6} d x^{13} e^{6} + \frac {21}{2} \, B a b^{5} d^{2} x^{12} e^{5} + \frac {7}{4} \, A b^{6} d^{2} x^{12} e^{5} + \frac {210}{11} \, B a b^{5} d^{3} x^{11} e^{4} + \frac {35}{11} \, A b^{6} d^{3} x^{11} e^{4} + 21 \, B a b^{5} d^{4} x^{10} e^{3} + \frac {7}{2} \, A b^{6} d^{4} x^{10} e^{3} + 14 \, B a b^{5} d^{5} x^{9} e^{2} + \frac {7}{3} \, A b^{6} d^{5} x^{9} e^{2} + \frac {21}{4} \, B a b^{5} d^{6} x^{8} e + \frac {7}{8} \, A b^{6} d^{6} x^{8} e + \frac {6}{7} \, B a b^{5} d^{7} x^{7} + \frac {1}{7} \, A b^{6} d^{7} x^{7} + \frac {15}{13} \, B a^{2} b^{4} x^{13} e^{7} + \frac {6}{13} \, A a b^{5} x^{13} e^{7} + \frac {35}{4} \, B a^{2} b^{4} d x^{12} e^{6} + \frac {7}{2} \, A a b^{5} d x^{12} e^{6} + \frac {315}{11} \, B a^{2} b^{4} d^{2} x^{11} e^{5} + \frac {126}{11} \, A a b^{5} d^{2} x^{11} e^{5} + \frac {105}{2} \, B a^{2} b^{4} d^{3} x^{10} e^{4} + 21 \, A a b^{5} d^{3} x^{10} e^{4} + \frac {175}{3} \, B a^{2} b^{4} d^{4} x^{9} e^{3} + \frac {70}{3} \, A a b^{5} d^{4} x^{9} e^{3} + \frac {315}{8} \, B a^{2} b^{4} d^{5} x^{8} e^{2} + \frac {63}{4} \, A a b^{5} d^{5} x^{8} e^{2} + 15 \, B a^{2} b^{4} d^{6} x^{7} e + 6 \, A a b^{5} d^{6} x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d^{7} x^{6} + A a b^{5} d^{7} x^{6} + \frac {5}{3} \, B a^{3} b^{3} x^{12} e^{7} + \frac {5}{4} \, A a^{2} b^{4} x^{12} e^{7} + \frac {140}{11} \, B a^{3} b^{3} d x^{11} e^{6} + \frac {105}{11} \, A a^{2} b^{4} d x^{11} e^{6} + 42 \, B a^{3} b^{3} d^{2} x^{10} e^{5} + \frac {63}{2} \, A a^{2} b^{4} d^{2} x^{10} e^{5} + \frac {700}{9} \, B a^{3} b^{3} d^{3} x^{9} e^{4} + \frac {175}{3} \, A a^{2} b^{4} d^{3} x^{9} e^{4} + \frac {175}{2} \, B a^{3} b^{3} d^{4} x^{8} e^{3} + \frac {525}{8} \, A a^{2} b^{4} d^{4} x^{8} e^{3} + 60 \, B a^{3} b^{3} d^{5} x^{7} e^{2} + 45 \, A a^{2} b^{4} d^{5} x^{7} e^{2} + \frac {70}{3} \, B a^{3} b^{3} d^{6} x^{6} e + \frac {35}{2} \, A a^{2} b^{4} d^{6} x^{6} e + 4 \, B a^{3} b^{3} d^{7} x^{5} + 3 \, A a^{2} b^{4} d^{7} x^{5} + \frac {15}{11} \, B a^{4} b^{2} x^{11} e^{7} + \frac {20}{11} \, A a^{3} b^{3} x^{11} e^{7} + \frac {21}{2} \, B a^{4} b^{2} d x^{10} e^{6} + 14 \, A a^{3} b^{3} d x^{10} e^{6} + 35 \, B a^{4} b^{2} d^{2} x^{9} e^{5} + \frac {140}{3} \, A a^{3} b^{3} d^{2} x^{9} e^{5} + \frac {525}{8} \, B a^{4} b^{2} d^{3} x^{8} e^{4} + \frac {175}{2} \, A a^{3} b^{3} d^{3} x^{8} e^{4} + 75 \, B a^{4} b^{2} d^{4} x^{7} e^{3} + 100 \, A a^{3} b^{3} d^{4} x^{7} e^{3} + \frac {105}{2} \, B a^{4} b^{2} d^{5} x^{6} e^{2} + 70 \, A a^{3} b^{3} d^{5} x^{6} e^{2} + 21 \, B a^{4} b^{2} d^{6} x^{5} e + 28 \, A a^{3} b^{3} d^{6} x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d^{7} x^{4} + 5 \, A a^{3} b^{3} d^{7} x^{4} + \frac {3}{5} \, B a^{5} b x^{10} e^{7} + \frac {3}{2} \, A a^{4} b^{2} x^{10} e^{7} + \frac {14}{3} \, B a^{5} b d x^{9} e^{6} + \frac {35}{3} \, A a^{4} b^{2} d x^{9} e^{6} + \frac {63}{4} \, B a^{5} b d^{2} x^{8} e^{5} + \frac {315}{8} \, A a^{4} b^{2} d^{2} x^{8} e^{5} + 30 \, B a^{5} b d^{3} x^{7} e^{4} + 75 \, A a^{4} b^{2} d^{3} x^{7} e^{4} + 35 \, B a^{5} b d^{4} x^{6} e^{3} + \frac {175}{2} \, A a^{4} b^{2} d^{4} x^{6} e^{3} + \frac {126}{5} \, B a^{5} b d^{5} x^{5} e^{2} + 63 \, A a^{4} b^{2} d^{5} x^{5} e^{2} + \frac {21}{2} \, B a^{5} b d^{6} x^{4} e + \frac {105}{4} \, A a^{4} b^{2} d^{6} x^{4} e + 2 \, B a^{5} b d^{7} x^{3} + 5 \, A a^{4} b^{2} d^{7} x^{3} + \frac {1}{9} \, B a^{6} x^{9} e^{7} + \frac {2}{3} \, A a^{5} b x^{9} e^{7} + \frac {7}{8} \, B a^{6} d x^{8} e^{6} + \frac {21}{4} \, A a^{5} b d x^{8} e^{6} + 3 \, B a^{6} d^{2} x^{7} e^{5} + 18 \, A a^{5} b d^{2} x^{7} e^{5} + \frac {35}{6} \, B a^{6} d^{3} x^{6} e^{4} + 35 \, A a^{5} b d^{3} x^{6} e^{4} + 7 \, B a^{6} d^{4} x^{5} e^{3} + 42 \, A a^{5} b d^{4} x^{5} e^{3} + \frac {21}{4} \, B a^{6} d^{5} x^{4} e^{2} + \frac {63}{2} \, A a^{5} b d^{5} x^{4} e^{2} + \frac {7}{3} \, B a^{6} d^{6} x^{3} e + 14 \, A a^{5} b d^{6} x^{3} e + \frac {1}{2} \, B a^{6} d^{7} x^{2} + 3 \, A a^{5} b d^{7} x^{2} + \frac {1}{8} \, A a^{6} x^{8} e^{7} + A a^{6} d x^{7} e^{6} + \frac {7}{2} \, A a^{6} d^{2} x^{6} e^{5} + 7 \, A a^{6} d^{3} x^{5} e^{4} + \frac {35}{4} \, A a^{6} d^{4} x^{4} e^{3} + 7 \, A a^{6} d^{5} x^{3} e^{2} + \frac {7}{2} \, A a^{6} d^{6} x^{2} e + A a^{6} d^{7} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^7,x, algorithm="giac")

[Out]

1/15*B*b^6*x^15*e^7 + 1/2*B*b^6*d*x^14*e^6 + 21/13*B*b^6*d^2*x^13*e^5 + 35/12*B*b^6*d^3*x^12*e^4 + 35/11*B*b^6
*d^4*x^11*e^3 + 21/10*B*b^6*d^5*x^10*e^2 + 7/9*B*b^6*d^6*x^9*e + 1/8*B*b^6*d^7*x^8 + 3/7*B*a*b^5*x^14*e^7 + 1/
14*A*b^6*x^14*e^7 + 42/13*B*a*b^5*d*x^13*e^6 + 7/13*A*b^6*d*x^13*e^6 + 21/2*B*a*b^5*d^2*x^12*e^5 + 7/4*A*b^6*d
^2*x^12*e^5 + 210/11*B*a*b^5*d^3*x^11*e^4 + 35/11*A*b^6*d^3*x^11*e^4 + 21*B*a*b^5*d^4*x^10*e^3 + 7/2*A*b^6*d^4
*x^10*e^3 + 14*B*a*b^5*d^5*x^9*e^2 + 7/3*A*b^6*d^5*x^9*e^2 + 21/4*B*a*b^5*d^6*x^8*e + 7/8*A*b^6*d^6*x^8*e + 6/
7*B*a*b^5*d^7*x^7 + 1/7*A*b^6*d^7*x^7 + 15/13*B*a^2*b^4*x^13*e^7 + 6/13*A*a*b^5*x^13*e^7 + 35/4*B*a^2*b^4*d*x^
12*e^6 + 7/2*A*a*b^5*d*x^12*e^6 + 315/11*B*a^2*b^4*d^2*x^11*e^5 + 126/11*A*a*b^5*d^2*x^11*e^5 + 105/2*B*a^2*b^
4*d^3*x^10*e^4 + 21*A*a*b^5*d^3*x^10*e^4 + 175/3*B*a^2*b^4*d^4*x^9*e^3 + 70/3*A*a*b^5*d^4*x^9*e^3 + 315/8*B*a^
2*b^4*d^5*x^8*e^2 + 63/4*A*a*b^5*d^5*x^8*e^2 + 15*B*a^2*b^4*d^6*x^7*e + 6*A*a*b^5*d^6*x^7*e + 5/2*B*a^2*b^4*d^
7*x^6 + A*a*b^5*d^7*x^6 + 5/3*B*a^3*b^3*x^12*e^7 + 5/4*A*a^2*b^4*x^12*e^7 + 140/11*B*a^3*b^3*d*x^11*e^6 + 105/
11*A*a^2*b^4*d*x^11*e^6 + 42*B*a^3*b^3*d^2*x^10*e^5 + 63/2*A*a^2*b^4*d^2*x^10*e^5 + 700/9*B*a^3*b^3*d^3*x^9*e^
4 + 175/3*A*a^2*b^4*d^3*x^9*e^4 + 175/2*B*a^3*b^3*d^4*x^8*e^3 + 525/8*A*a^2*b^4*d^4*x^8*e^3 + 60*B*a^3*b^3*d^5
*x^7*e^2 + 45*A*a^2*b^4*d^5*x^7*e^2 + 70/3*B*a^3*b^3*d^6*x^6*e + 35/2*A*a^2*b^4*d^6*x^6*e + 4*B*a^3*b^3*d^7*x^
5 + 3*A*a^2*b^4*d^7*x^5 + 15/11*B*a^4*b^2*x^11*e^7 + 20/11*A*a^3*b^3*x^11*e^7 + 21/2*B*a^4*b^2*d*x^10*e^6 + 14
*A*a^3*b^3*d*x^10*e^6 + 35*B*a^4*b^2*d^2*x^9*e^5 + 140/3*A*a^3*b^3*d^2*x^9*e^5 + 525/8*B*a^4*b^2*d^3*x^8*e^4 +
175/2*A*a^3*b^3*d^3*x^8*e^4 + 75*B*a^4*b^2*d^4*x^7*e^3 + 100*A*a^3*b^3*d^4*x^7*e^3 + 105/2*B*a^4*b^2*d^5*x^6*
e^2 + 70*A*a^3*b^3*d^5*x^6*e^2 + 21*B*a^4*b^2*d^6*x^5*e + 28*A*a^3*b^3*d^6*x^5*e + 15/4*B*a^4*b^2*d^7*x^4 + 5*
A*a^3*b^3*d^7*x^4 + 3/5*B*a^5*b*x^10*e^7 + 3/2*A*a^4*b^2*x^10*e^7 + 14/3*B*a^5*b*d*x^9*e^6 + 35/3*A*a^4*b^2*d*
x^9*e^6 + 63/4*B*a^5*b*d^2*x^8*e^5 + 315/8*A*a^4*b^2*d^2*x^8*e^5 + 30*B*a^5*b*d^3*x^7*e^4 + 75*A*a^4*b^2*d^3*x
^7*e^4 + 35*B*a^5*b*d^4*x^6*e^3 + 175/2*A*a^4*b^2*d^4*x^6*e^3 + 126/5*B*a^5*b*d^5*x^5*e^2 + 63*A*a^4*b^2*d^5*x
^5*e^2 + 21/2*B*a^5*b*d^6*x^4*e + 105/4*A*a^4*b^2*d^6*x^4*e + 2*B*a^5*b*d^7*x^3 + 5*A*a^4*b^2*d^7*x^3 + 1/9*B*
a^6*x^9*e^7 + 2/3*A*a^5*b*x^9*e^7 + 7/8*B*a^6*d*x^8*e^6 + 21/4*A*a^5*b*d*x^8*e^6 + 3*B*a^6*d^2*x^7*e^5 + 18*A*
a^5*b*d^2*x^7*e^5 + 35/6*B*a^6*d^3*x^6*e^4 + 35*A*a^5*b*d^3*x^6*e^4 + 7*B*a^6*d^4*x^5*e^3 + 42*A*a^5*b*d^4*x^5
*e^3 + 21/4*B*a^6*d^5*x^4*e^2 + 63/2*A*a^5*b*d^5*x^4*e^2 + 7/3*B*a^6*d^6*x^3*e + 14*A*a^5*b*d^6*x^3*e + 1/2*B*
a^6*d^7*x^2 + 3*A*a^5*b*d^7*x^2 + 1/8*A*a^6*x^8*e^7 + A*a^6*d*x^7*e^6 + 7/2*A*a^6*d^2*x^6*e^5 + 7*A*a^6*d^3*x^
5*e^4 + 35/4*A*a^6*d^4*x^4*e^3 + 7*A*a^6*d^5*x^3*e^2 + 7/2*A*a^6*d^6*x^2*e + A*a^6*d^7*x

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Mupad [B]
time = 0.49, size = 1431, normalized size = 4.90 \begin {gather*} x^6\,\left (\frac {35\,B\,a^6\,d^3\,e^4}{6}+\frac {7\,A\,a^6\,d^2\,e^5}{2}+35\,B\,a^5\,b\,d^4\,e^3+35\,A\,a^5\,b\,d^3\,e^4+\frac {105\,B\,a^4\,b^2\,d^5\,e^2}{2}+\frac {175\,A\,a^4\,b^2\,d^4\,e^3}{2}+\frac {70\,B\,a^3\,b^3\,d^6\,e}{3}+70\,A\,a^3\,b^3\,d^5\,e^2+\frac {5\,B\,a^2\,b^4\,d^7}{2}+\frac {35\,A\,a^2\,b^4\,d^6\,e}{2}+A\,a\,b^5\,d^7\right )+x^{10}\,\left (\frac {3\,B\,a^5\,b\,e^7}{5}+\frac {21\,B\,a^4\,b^2\,d\,e^6}{2}+\frac {3\,A\,a^4\,b^2\,e^7}{2}+42\,B\,a^3\,b^3\,d^2\,e^5+14\,A\,a^3\,b^3\,d\,e^6+\frac {105\,B\,a^2\,b^4\,d^3\,e^4}{2}+\frac {63\,A\,a^2\,b^4\,d^2\,e^5}{2}+21\,B\,a\,b^5\,d^4\,e^3+21\,A\,a\,b^5\,d^3\,e^4+\frac {21\,B\,b^6\,d^5\,e^2}{10}+\frac {7\,A\,b^6\,d^4\,e^3}{2}\right )+x^5\,\left (7\,B\,a^6\,d^4\,e^3+7\,A\,a^6\,d^3\,e^4+\frac {126\,B\,a^5\,b\,d^5\,e^2}{5}+42\,A\,a^5\,b\,d^4\,e^3+21\,B\,a^4\,b^2\,d^6\,e+63\,A\,a^4\,b^2\,d^5\,e^2+4\,B\,a^3\,b^3\,d^7+28\,A\,a^3\,b^3\,d^6\,e+3\,A\,a^2\,b^4\,d^7\right )+x^{11}\,\left (\frac {15\,B\,a^4\,b^2\,e^7}{11}+\frac {140\,B\,a^3\,b^3\,d\,e^6}{11}+\frac {20\,A\,a^3\,b^3\,e^7}{11}+\frac {315\,B\,a^2\,b^4\,d^2\,e^5}{11}+\frac {105\,A\,a^2\,b^4\,d\,e^6}{11}+\frac {210\,B\,a\,b^5\,d^3\,e^4}{11}+\frac {126\,A\,a\,b^5\,d^2\,e^5}{11}+\frac {35\,B\,b^6\,d^4\,e^3}{11}+\frac {35\,A\,b^6\,d^3\,e^4}{11}\right )+x^3\,\left (\frac {7\,B\,a^6\,d^6\,e}{3}+7\,A\,a^6\,d^5\,e^2+2\,B\,a^5\,b\,d^7+14\,A\,a^5\,b\,d^6\,e+5\,A\,a^4\,b^2\,d^7\right )+x^8\,\left (\frac {7\,B\,a^6\,d\,e^6}{8}+\frac {A\,a^6\,e^7}{8}+\frac {63\,B\,a^5\,b\,d^2\,e^5}{4}+\frac {21\,A\,a^5\,b\,d\,e^6}{4}+\frac {525\,B\,a^4\,b^2\,d^3\,e^4}{8}+\frac {315\,A\,a^4\,b^2\,d^2\,e^5}{8}+\frac {175\,B\,a^3\,b^3\,d^4\,e^3}{2}+\frac {175\,A\,a^3\,b^3\,d^3\,e^4}{2}+\frac {315\,B\,a^2\,b^4\,d^5\,e^2}{8}+\frac {525\,A\,a^2\,b^4\,d^4\,e^3}{8}+\frac {21\,B\,a\,b^5\,d^6\,e}{4}+\frac {63\,A\,a\,b^5\,d^5\,e^2}{4}+\frac {B\,b^6\,d^7}{8}+\frac {7\,A\,b^6\,d^6\,e}{8}\right )+x^{13}\,\left (\frac {15\,B\,a^2\,b^4\,e^7}{13}+\frac {42\,B\,a\,b^5\,d\,e^6}{13}+\frac {6\,A\,a\,b^5\,e^7}{13}+\frac {21\,B\,b^6\,d^2\,e^5}{13}+\frac {7\,A\,b^6\,d\,e^6}{13}\right )+x^4\,\left (\frac {21\,B\,a^6\,d^5\,e^2}{4}+\frac {35\,A\,a^6\,d^4\,e^3}{4}+\frac {21\,B\,a^5\,b\,d^6\,e}{2}+\frac {63\,A\,a^5\,b\,d^5\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^7}{4}+\frac {105\,A\,a^4\,b^2\,d^6\,e}{4}+5\,A\,a^3\,b^3\,d^7\right )+x^{12}\,\left (\frac {5\,B\,a^3\,b^3\,e^7}{3}+\frac {35\,B\,a^2\,b^4\,d\,e^6}{4}+\frac {5\,A\,a^2\,b^4\,e^7}{4}+\frac {21\,B\,a\,b^5\,d^2\,e^5}{2}+\frac {7\,A\,a\,b^5\,d\,e^6}{2}+\frac {35\,B\,b^6\,d^3\,e^4}{12}+\frac {7\,A\,b^6\,d^2\,e^5}{4}\right )+x^7\,\left (3\,B\,a^6\,d^2\,e^5+A\,a^6\,d\,e^6+30\,B\,a^5\,b\,d^3\,e^4+18\,A\,a^5\,b\,d^2\,e^5+75\,B\,a^4\,b^2\,d^4\,e^3+75\,A\,a^4\,b^2\,d^3\,e^4+60\,B\,a^3\,b^3\,d^5\,e^2+100\,A\,a^3\,b^3\,d^4\,e^3+15\,B\,a^2\,b^4\,d^6\,e+45\,A\,a^2\,b^4\,d^5\,e^2+\frac {6\,B\,a\,b^5\,d^7}{7}+6\,A\,a\,b^5\,d^6\,e+\frac {A\,b^6\,d^7}{7}\right )+x^9\,\left (\frac {B\,a^6\,e^7}{9}+\frac {14\,B\,a^5\,b\,d\,e^6}{3}+\frac {2\,A\,a^5\,b\,e^7}{3}+35\,B\,a^4\,b^2\,d^2\,e^5+\frac {35\,A\,a^4\,b^2\,d\,e^6}{3}+\frac {700\,B\,a^3\,b^3\,d^3\,e^4}{9}+\frac {140\,A\,a^3\,b^3\,d^2\,e^5}{3}+\frac {175\,B\,a^2\,b^4\,d^4\,e^3}{3}+\frac {175\,A\,a^2\,b^4\,d^3\,e^4}{3}+14\,B\,a\,b^5\,d^5\,e^2+\frac {70\,A\,a\,b^5\,d^4\,e^3}{3}+\frac {7\,B\,b^6\,d^6\,e}{9}+\frac {7\,A\,b^6\,d^5\,e^2}{3}\right )+\frac {a^5\,d^6\,x^2\,\left (7\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^6\,x^{14}\,\left (A\,b\,e+6\,B\,a\,e+7\,B\,b\,d\right )}{14}+A\,a^6\,d^7\,x+\frac {B\,b^6\,e^7\,x^{15}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^6*(d + e*x)^7,x)

[Out]

x^6*(A*a*b^5*d^7 + (5*B*a^2*b^4*d^7)/2 + (7*A*a^6*d^2*e^5)/2 + (35*B*a^6*d^3*e^4)/6 + (35*A*a^2*b^4*d^6*e)/2 +
35*A*a^5*b*d^3*e^4 + (70*B*a^3*b^3*d^6*e)/3 + 35*B*a^5*b*d^4*e^3 + 70*A*a^3*b^3*d^5*e^2 + (175*A*a^4*b^2*d^4*
e^3)/2 + (105*B*a^4*b^2*d^5*e^2)/2) + x^10*((3*B*a^5*b*e^7)/5 + (3*A*a^4*b^2*e^7)/2 + (7*A*b^6*d^4*e^3)/2 + (2
1*B*b^6*d^5*e^2)/10 + 21*A*a*b^5*d^3*e^4 + 14*A*a^3*b^3*d*e^6 + 21*B*a*b^5*d^4*e^3 + (21*B*a^4*b^2*d*e^6)/2 +
(63*A*a^2*b^4*d^2*e^5)/2 + (105*B*a^2*b^4*d^3*e^4)/2 + 42*B*a^3*b^3*d^2*e^5) + x^5*(3*A*a^2*b^4*d^7 + 4*B*a^3*
b^3*d^7 + 7*A*a^6*d^3*e^4 + 7*B*a^6*d^4*e^3 + 28*A*a^3*b^3*d^6*e + 42*A*a^5*b*d^4*e^3 + 21*B*a^4*b^2*d^6*e + (
126*B*a^5*b*d^5*e^2)/5 + 63*A*a^4*b^2*d^5*e^2) + x^11*((20*A*a^3*b^3*e^7)/11 + (15*B*a^4*b^2*e^7)/11 + (35*A*b
^6*d^3*e^4)/11 + (35*B*b^6*d^4*e^3)/11 + (126*A*a*b^5*d^2*e^5)/11 + (105*A*a^2*b^4*d*e^6)/11 + (210*B*a*b^5*d^
3*e^4)/11 + (140*B*a^3*b^3*d*e^6)/11 + (315*B*a^2*b^4*d^2*e^5)/11) + x^3*(2*B*a^5*b*d^7 + (7*B*a^6*d^6*e)/3 +
5*A*a^4*b^2*d^7 + 7*A*a^6*d^5*e^2 + 14*A*a^5*b*d^6*e) + x^8*((A*a^6*e^7)/8 + (B*b^6*d^7)/8 + (7*A*b^6*d^6*e)/8
+ (7*B*a^6*d*e^6)/8 + (63*A*a*b^5*d^5*e^2)/4 + (63*B*a^5*b*d^2*e^5)/4 + (525*A*a^2*b^4*d^4*e^3)/8 + (175*A*a^
3*b^3*d^3*e^4)/2 + (315*A*a^4*b^2*d^2*e^5)/8 + (315*B*a^2*b^4*d^5*e^2)/8 + (175*B*a^3*b^3*d^4*e^3)/2 + (525*B*
a^4*b^2*d^3*e^4)/8 + (21*A*a^5*b*d*e^6)/4 + (21*B*a*b^5*d^6*e)/4) + x^13*((6*A*a*b^5*e^7)/13 + (7*A*b^6*d*e^6)
/13 + (15*B*a^2*b^4*e^7)/13 + (21*B*b^6*d^2*e^5)/13 + (42*B*a*b^5*d*e^6)/13) + x^4*(5*A*a^3*b^3*d^7 + (15*B*a^
4*b^2*d^7)/4 + (35*A*a^6*d^4*e^3)/4 + (21*B*a^6*d^5*e^2)/4 + (105*A*a^4*b^2*d^6*e)/4 + (63*A*a^5*b*d^5*e^2)/2
+ (21*B*a^5*b*d^6*e)/2) + x^12*((5*A*a^2*b^4*e^7)/4 + (5*B*a^3*b^3*e^7)/3 + (7*A*b^6*d^2*e^5)/4 + (35*B*b^6*d^
3*e^4)/12 + (21*B*a*b^5*d^2*e^5)/2 + (35*B*a^2*b^4*d*e^6)/4 + (7*A*a*b^5*d*e^6)/2) + x^7*((A*b^6*d^7)/7 + (6*B
*a*b^5*d^7)/7 + A*a^6*d*e^6 + 3*B*a^6*d^2*e^5 + 18*A*a^5*b*d^2*e^5 + 15*B*a^2*b^4*d^6*e + 30*B*a^5*b*d^3*e^4 +
45*A*a^2*b^4*d^5*e^2 + 100*A*a^3*b^3*d^4*e^3 + 75*A*a^4*b^2*d^3*e^4 + 60*B*a^3*b^3*d^5*e^2 + 75*B*a^4*b^2*d^4
*e^3 + 6*A*a*b^5*d^6*e) + x^9*((B*a^6*e^7)/9 + (2*A*a^5*b*e^7)/3 + (7*B*b^6*d^6*e)/9 + (7*A*b^6*d^5*e^2)/3 + (
70*A*a*b^5*d^4*e^3)/3 + (35*A*a^4*b^2*d*e^6)/3 + 14*B*a*b^5*d^5*e^2 + (175*A*a^2*b^4*d^3*e^4)/3 + (140*A*a^3*b
^3*d^2*e^5)/3 + (175*B*a^2*b^4*d^4*e^3)/3 + (700*B*a^3*b^3*d^3*e^4)/9 + 35*B*a^4*b^2*d^2*e^5 + (14*B*a^5*b*d*e
^6)/3) + (a^5*d^6*x^2*(7*A*a*e + 6*A*b*d + B*a*d))/2 + (b^5*e^6*x^14*(A*b*e + 6*B*a*e + 7*B*b*d))/14 + A*a^6*d
^7*x + (B*b^6*e^7*x^15)/15

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