3.1052 \(\int (a+b x)^6 (A+B x) (d+e x)^7 \, dx\)
Optimal. Leaf size=292 \[ -\frac{b^5 (d+e x)^{14} (-6 a B e-A b e+7 b B d)}{14 e^8}+\frac{3 b^4 (d+e x)^{13} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac{5 b^3 (d+e x)^{12} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{12 e^8}+\frac{5 b^2 (d+e x)^{11} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{3 b (d+e x)^{10} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8}+\frac{(d+e x)^9 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac{(d+e x)^8 (b d-a e)^6 (B d-A e)}{8 e^8}+\frac{b^6 B (d+e x)^{15}}{15 e^8} \]
[Out]
-((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^8)/(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)
________________________________________________________________________________________
Rubi [A] time = 1.12258, antiderivative size = 292, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{77} \[ -\frac{b^5 (d+e x)^{14} (-6 a B e-A b e+7 b B d)}{14 e^8}+\frac{3 b^4 (d+e x)^{13} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac{5 b^3 (d+e x)^{12} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{12 e^8}+\frac{5 b^2 (d+e x)^{11} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{3 b (d+e x)^{10} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{10 e^8}+\frac{(d+e x)^9 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{9 e^8}-\frac{(d+e x)^8 (b d-a e)^6 (B d-A e)}{8 e^8}+\frac{b^6 B (d+e x)^{15}}{15 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^6*(A + B*x)*(d + e*x)^7,x]
[Out]
-((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^8)/(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 77
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*}
Mathematica [B] time = 0.416112, size = 1224, normalized size = 4.19 \[ \frac{1}{15} b^6 B e^7 x^{15}+\frac{1}{14} b^5 e^6 (7 b B d+A b e+6 a B e) x^{14}+\frac{1}{13} b^4 e^5 \left (7 d (3 B d+A e) b^2+6 a e (7 B d+A e) b+15 a^2 B e^2\right ) x^{13}+\frac{1}{12} b^3 e^4 \left (7 d^2 (5 B d+3 A e) b^3+42 a d e (3 B d+A e) b^2+15 a^2 e^2 (7 B d+A e) b+20 a^3 B e^3\right ) x^{12}+\frac{1}{11} b^2 e^3 \left (35 d^3 (B d+A e) b^4+42 a d^2 e (5 B d+3 A e) b^3+105 a^2 d e^2 (3 B d+A e) b^2+20 a^3 e^3 (7 B d+A e) b+15 a^4 B e^4\right ) x^{11}+\frac{1}{10} b e^2 \left (7 d^4 (3 B d+5 A e) b^5+210 a d^3 e (B d+A e) b^4+105 a^2 d^2 e^2 (5 B d+3 A e) b^3+140 a^3 d e^3 (3 B d+A e) b^2+15 a^4 e^4 (7 B d+A e) b+6 a^5 B e^5\right ) x^{10}+\frac{1}{9} e \left (7 d^5 (B d+3 A e) b^6+42 a d^4 e (3 B d+5 A e) b^5+525 a^2 d^3 e^2 (B d+A e) b^4+140 a^3 d^2 e^3 (5 B d+3 A e) b^3+105 a^4 d e^4 (3 B d+A e) b^2+6 a^5 e^5 (7 B d+A e) b+a^6 B e^6\right ) x^9+\frac{1}{8} \left (b^6 (B d+7 A e) d^6+42 a b^5 e (B d+3 A e) d^5+105 a^2 b^4 e^2 (3 B d+5 A e) d^4+700 a^3 b^3 e^3 (B d+A e) d^3+105 a^4 b^2 e^4 (5 B d+3 A e) d^2+42 a^5 b e^5 (3 B d+A e) d+a^6 e^6 (7 B d+A e)\right ) x^8+\frac{1}{7} d \left (3 a B d \left (2 b^5 d^5+35 a b^4 e d^4+140 a^2 b^3 e^2 d^3+175 a^3 b^2 e^3 d^2+70 a^4 b e^4 d+7 a^5 e^5\right )+A \left (b^6 d^6+42 a b^5 e d^5+315 a^2 b^4 e^2 d^4+700 a^3 b^3 e^3 d^3+525 a^4 b^2 e^4 d^2+126 a^5 b e^5 d+7 a^6 e^6\right )\right ) x^7+\frac{1}{6} a d^2 \left (5 a B d \left (3 b^4 d^4+28 a b^3 e d^3+63 a^2 b^2 e^2 d^2+42 a^3 b e^3 d+7 a^4 e^4\right )+3 A \left (2 b^5 d^5+35 a b^4 e d^4+140 a^2 b^3 e^2 d^3+175 a^3 b^2 e^3 d^2+70 a^4 b e^4 d+7 a^5 e^5\right )\right ) x^6+\frac{1}{5} a^2 d^3 \left (a B d \left (20 b^3 d^3+105 a b^2 e d^2+126 a^2 b e^2 d+35 a^3 e^3\right )+5 A \left (3 b^4 d^4+28 a b^3 e d^3+63 a^2 b^2 e^2 d^2+42 a^3 b e^3 d+7 a^4 e^4\right )\right ) x^5+\frac{1}{4} a^3 d^4 \left (3 a B d \left (5 b^2 d^2+14 a b e d+7 a^2 e^2\right )+A \left (20 b^3 d^3+105 a b^2 e d^2+126 a^2 b e^2 d+35 a^3 e^3\right )\right ) x^4+\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 e d+7 a^2 e^2\right )\right ) x^3+\frac{1}{2} a^5 d^6 (6 A b d+a B d+7 a A e) x^2+a^6 A d^7 x \]
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] time = 0.002, size = 1349, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^6*(B*x+A)*(e*x+d)^7,x)
[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*d^6*e)*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)*d^6*e+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)*d^6*e+(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)*d^6*e+(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)*d^6*e+(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)*d^6*e+(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)*d^6*e+(15*A*a^4*b^2+6*B*a^5*b)*d^7)*x^3+
1/2*(7*a^6*A*d^6*e+(6*A*a^5*b+B*a^6)*d^7)*x^2+a^6*A*d^7*x
________________________________________________________________________________________
Maxima [B] time = 1.90857, size = 1831, normalized size = 6.27 \begin{align*} \text{result too large to display} \end{align*}
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*e^7*x^15 + A*a^6*d^7*x + 1/14*(7*B*b^6*d*e^6 + (6*B*a*b^5 + A*b^6)*e^7)*x^14 + 1/13*(21*B*b^6*d^2*e
^5 + 7*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^13 + 1/12*(35*B*b^6*d^3*e^4 + 21*(6*B*a*
b^5 + A*b^6)*d^2*e^5 + 21*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^12 + 1/11*(35
*B*b^6*d^4*e^3 + 35*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 35*(4*B*a^3*b^3 + 3*A
*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^11 + 1/10*(21*B*b^6*d^5*e^2 + 35*(6*B*a*b^5 + A*b^6)*d^
4*e^3 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 105*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a^4*b^2 + 4*
A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^10 + 1/9*(7*B*b^6*d^6*e + 21*(6*B*a*b^5 + A*b^6)*d^5*e^2
+ 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 175*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 105*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^2*e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x^9 + 1/8*(B*b^6*d^7 + A*a^6*e^
7 + 7*(6*B*a*b^5 + A*b^6)*d^6*e + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 175*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e
^3 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 63*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 7*(B*a^6 + 6*A*a^5*b)*d*
e^6)*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 + 2*A*a*b^5)*d^6*e + 105*(4*B*a^3*b^
3 + 3*A*a^2*b^4)*d^5*e^2 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^3 + 105*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^4 + 2
1*(B*a^6 + 6*A*a^5*b)*d^2*e^5)*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
+ 3*A*a^2*b^4)*d^6*e + 105*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e^2 + 105*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e^3 + 35*(
B*a^6 + 6*A*a^5*b)*d^3*e^4)*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
+ 4*A*a^3*b^3)*d^6*e + 63*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*e^2 + 35*(B*a^6 + 6*A*a^5*b)*d^4*e^3)*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 + 5*A*a^4*b^2)*d^6*e + 21*(B*a^6 + 6*A*a^5*b)*
d^5*e^2)*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 + 6*A*a^5*b)*d^6*e)*x^3 + 1/
2*(7*A*a^6*d^6*e + (B*a^6 + 6*A*a^5*b)*d^7)*x^2
________________________________________________________________________________________
Fricas [B] time = 1.61466, size = 3834, normalized size = 13.13 \begin{align*} \text{result too large to display} \end{align*}
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]
1/15*x^15*e^7*b^6*B + 1/2*x^14*e^6*d*b^6*B + 3/7*x^14*e^7*b^5*a*B + 1/14*x^14*e^7*b^6*A + 21/13*x^13*e^5*d^2*b
^6*B + 42/13*x^13*e^6*d*b^5*a*B + 15/13*x^13*e^7*b^4*a^2*B + 7/13*x^13*e^6*d*b^6*A + 6/13*x^13*e^7*b^5*a*A + 3
5/12*x^12*e^4*d^3*b^6*B + 21/2*x^12*e^5*d^2*b^5*a*B + 35/4*x^12*e^6*d*b^4*a^2*B + 5/3*x^12*e^7*b^3*a^3*B + 7/4
*x^12*e^5*d^2*b^6*A + 7/2*x^12*e^6*d*b^5*a*A + 5/4*x^12*e^7*b^4*a^2*A + 35/11*x^11*e^3*d^4*b^6*B + 210/11*x^11
*e^4*d^3*b^5*a*B + 315/11*x^11*e^5*d^2*b^4*a^2*B + 140/11*x^11*e^6*d*b^3*a^3*B + 15/11*x^11*e^7*b^2*a^4*B + 35
/11*x^11*e^4*d^3*b^6*A + 126/11*x^11*e^5*d^2*b^5*a*A + 105/11*x^11*e^6*d*b^4*a^2*A + 20/11*x^11*e^7*b^3*a^3*A
+ 21/10*x^10*e^2*d^5*b^6*B + 21*x^10*e^3*d^4*b^5*a*B + 105/2*x^10*e^4*d^3*b^4*a^2*B + 42*x^10*e^5*d^2*b^3*a^3*
B + 21/2*x^10*e^6*d*b^2*a^4*B + 3/5*x^10*e^7*b*a^5*B + 7/2*x^10*e^3*d^4*b^6*A + 21*x^10*e^4*d^3*b^5*a*A + 63/2
*x^10*e^5*d^2*b^4*a^2*A + 14*x^10*e^6*d*b^3*a^3*A + 3/2*x^10*e^7*b^2*a^4*A + 7/9*x^9*e*d^6*b^6*B + 14*x^9*e^2*
d^5*b^5*a*B + 175/3*x^9*e^3*d^4*b^4*a^2*B + 700/9*x^9*e^4*d^3*b^3*a^3*B + 35*x^9*e^5*d^2*b^2*a^4*B + 14/3*x^9*
e^6*d*b*a^5*B + 1/9*x^9*e^7*a^6*B + 7/3*x^9*e^2*d^5*b^6*A + 70/3*x^9*e^3*d^4*b^5*a*A + 175/3*x^9*e^4*d^3*b^4*a
^2*A + 140/3*x^9*e^5*d^2*b^3*a^3*A + 35/3*x^9*e^6*d*b^2*a^4*A + 2/3*x^9*e^7*b*a^5*A + 1/8*x^8*d^7*b^6*B + 21/4
*x^8*e*d^6*b^5*a*B + 315/8*x^8*e^2*d^5*b^4*a^2*B + 175/2*x^8*e^3*d^4*b^3*a^3*B + 525/8*x^8*e^4*d^3*b^2*a^4*B +
63/4*x^8*e^5*d^2*b*a^5*B + 7/8*x^8*e^6*d*a^6*B + 7/8*x^8*e*d^6*b^6*A + 63/4*x^8*e^2*d^5*b^5*a*A + 525/8*x^8*e
^3*d^4*b^4*a^2*A + 175/2*x^8*e^4*d^3*b^3*a^3*A + 315/8*x^8*e^5*d^2*b^2*a^4*A + 21/4*x^8*e^6*d*b*a^5*A + 1/8*x^
8*e^7*a^6*A + 6/7*x^7*d^7*b^5*a*B + 15*x^7*e*d^6*b^4*a^2*B + 60*x^7*e^2*d^5*b^3*a^3*B + 75*x^7*e^3*d^4*b^2*a^4
*B + 30*x^7*e^4*d^3*b*a^5*B + 3*x^7*e^5*d^2*a^6*B + 1/7*x^7*d^7*b^6*A + 6*x^7*e*d^6*b^5*a*A + 45*x^7*e^2*d^5*b
^4*a^2*A + 100*x^7*e^3*d^4*b^3*a^3*A + 75*x^7*e^4*d^3*b^2*a^4*A + 18*x^7*e^5*d^2*b*a^5*A + x^7*e^6*d*a^6*A + 5
/2*x^6*d^7*b^4*a^2*B + 70/3*x^6*e*d^6*b^3*a^3*B + 105/2*x^6*e^2*d^5*b^2*a^4*B + 35*x^6*e^3*d^4*b*a^5*B + 35/6*
x^6*e^4*d^3*a^6*B + x^6*d^7*b^5*a*A + 35/2*x^6*e*d^6*b^4*a^2*A + 70*x^6*e^2*d^5*b^3*a^3*A + 175/2*x^6*e^3*d^4*
b^2*a^4*A + 35*x^6*e^4*d^3*b*a^5*A + 7/2*x^6*e^5*d^2*a^6*A + 4*x^5*d^7*b^3*a^3*B + 21*x^5*e*d^6*b^2*a^4*B + 12
6/5*x^5*e^2*d^5*b*a^5*B + 7*x^5*e^3*d^4*a^6*B + 3*x^5*d^7*b^4*a^2*A + 28*x^5*e*d^6*b^3*a^3*A + 63*x^5*e^2*d^5*
b^2*a^4*A + 42*x^5*e^3*d^4*b*a^5*A + 7*x^5*e^4*d^3*a^6*A + 15/4*x^4*d^7*b^2*a^4*B + 21/2*x^4*e*d^6*b*a^5*B + 2
1/4*x^4*e^2*d^5*a^6*B + 5*x^4*d^7*b^3*a^3*A + 105/4*x^4*e*d^6*b^2*a^4*A + 63/2*x^4*e^2*d^5*b*a^5*A + 35/4*x^4*
e^3*d^4*a^6*A + 2*x^3*d^7*b*a^5*B + 7/3*x^3*e*d^6*a^6*B + 5*x^3*d^7*b^2*a^4*A + 14*x^3*e*d^6*b*a^5*A + 7*x^3*e
^2*d^5*a^6*A + 1/2*x^2*d^7*a^6*B + 3*x^2*d^7*b*a^5*A + 7/2*x^2*e*d^6*a^6*A + x*d^7*a^6*A
________________________________________________________________________________________
Sympy [B] time = 0.237141, size = 1756, normalized size = 6.01 \begin{align*} \text{result too large to display} \end{align*}
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)
________________________________________________________________________________________
Giac [B] time = 1.73205, size = 2217, normalized size = 7.59 \begin{align*} \text{result too large to display} \end{align*}
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