3.1053 \(\int (a+b x)^6 (A+B x) (d+e x)^6 \, dx\)

Optimal. Leaf size=290 \[ \frac{e^5 (a+b x)^{13} (-7 a B e+A b e+6 b B d)}{13 b^8}+\frac{e^4 (a+b x)^{12} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{4 b^8}+\frac{5 e^3 (a+b x)^{11} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{11 b^8}+\frac{e^2 (a+b x)^{10} (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{2 b^8}+\frac{e (a+b x)^9 (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{3 b^8}+\frac{(a+b x)^8 (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{8 b^8}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^6}{7 b^8}+\frac{B e^6 (a+b x)^{14}}{14 b^8} \]

[Out]

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

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Rubi [A]  time = 0.977626, antiderivative size = 290, 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} \[ \frac{e^5 (a+b x)^{13} (-7 a B e+A b e+6 b B d)}{13 b^8}+\frac{e^4 (a+b x)^{12} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{4 b^8}+\frac{5 e^3 (a+b x)^{11} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{11 b^8}+\frac{e^2 (a+b x)^{10} (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{2 b^8}+\frac{e (a+b x)^9 (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{3 b^8}+\frac{(a+b x)^8 (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{8 b^8}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^6}{7 b^8}+\frac{B e^6 (a+b x)^{14}}{14 b^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.349332, size = 1069, normalized size = 3.69 \[ \frac{1}{14} b^6 B e^6 x^{14}+\frac{1}{13} b^5 e^5 (6 b B d+A b e+6 a B e) x^{13}+\frac{1}{4} b^4 e^4 \left (d (5 B d+2 A e) b^2+2 a e (6 B d+A e) b+5 a^2 B e^2\right ) x^{12}+\frac{1}{11} b^3 e^3 \left (5 d^2 (4 B d+3 A e) b^3+18 a d e (5 B d+2 A e) b^2+15 a^2 e^2 (6 B d+A e) b+20 a^3 B e^3\right ) x^{11}+\frac{1}{2} b^2 e^2 \left (d^3 (3 B d+4 A e) b^4+6 a d^2 e (4 B d+3 A e) b^3+9 a^2 d e^2 (5 B d+2 A e) b^2+4 a^3 e^3 (6 B d+A e) b+3 a^4 B e^4\right ) x^{10}+\frac{1}{3} b e \left (d^4 (2 B d+5 A e) b^5+10 a d^3 e (3 B d+4 A e) b^4+25 a^2 d^2 e^2 (4 B d+3 A e) b^3+20 a^3 d e^3 (5 B d+2 A e) b^2+5 a^4 e^4 (6 B d+A e) b+2 a^5 B e^5\right ) x^9+\frac{1}{8} \left (d^5 (B d+6 A e) b^6+18 a d^4 e (2 B d+5 A e) b^5+75 a^2 d^3 e^2 (3 B d+4 A e) b^4+100 a^3 d^2 e^3 (4 B d+3 A e) b^3+45 a^4 d e^4 (5 B d+2 A e) b^2+6 a^5 e^5 (6 B d+A e) b+a^6 B e^6\right ) x^8+\frac{1}{7} \left (6 a B d \left (b^5 d^5+15 a b^4 e d^4+50 a^2 b^3 e^2 d^3+50 a^3 b^2 e^3 d^2+15 a^4 b e^4 d+a^5 e^5\right )+A \left (b^6 d^6+36 a b^5 e d^5+225 a^2 b^4 e^2 d^4+400 a^3 b^3 e^3 d^3+225 a^4 b^2 e^4 d^2+36 a^5 b e^5 d+a^6 e^6\right )\right ) x^7+\frac{1}{2} a d \left (5 a B d \left (b^4 d^4+8 a b^3 e d^3+15 a^2 b^2 e^2 d^2+8 a^3 b e^3 d+a^4 e^4\right )+2 A \left (b^5 d^5+15 a b^4 e d^4+50 a^2 b^3 e^2 d^3+50 a^3 b^2 e^3 d^2+15 a^4 b e^4 d+a^5 e^5\right )\right ) x^6+a^2 d^2 \left (2 a B d \left (2 b^3 d^3+9 a b^2 e d^2+9 a^2 b e^2 d+2 a^3 e^3\right )+3 A \left (b^4 d^4+8 a b^3 e d^3+15 a^2 b^2 e^2 d^2+8 a^3 b e^3 d+a^4 e^4\right )\right ) x^5+\frac{1}{4} a^3 d^3 \left (3 a B d \left (5 b^2 d^2+12 a b e d+5 a^2 e^2\right )+10 A \left (2 b^3 d^3+9 a b^2 e d^2+9 a^2 b e^2 d+2 a^3 e^3\right )\right ) x^4+a^4 d^4 \left (2 a B d (b d+a e)+A \left (5 b^2 d^2+12 a b e d+5 a^2 e^2\right )\right ) x^3+\frac{1}{2} a^5 d^5 (a B d+6 A (b d+a e)) x^2+a^6 A d^6 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^6*A*d^6*x + (a^5*d^5*(a*B*d + 6*A*(b*d + a*e))*x^2)/2 + a^4*d^4*(2*a*B*d*(b*d + a*e) + A*(5*b^2*d^2 + 12*a*b
*d*e + 5*a^2*e^2))*x^3 + (a^3*d^3*(3*a*B*d*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2) + 10*A*(2*b^3*d^3 + 9*a*b^2*d^
2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^2*d^2*(2*a*B*d*(2*b^3*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3
*e^3) + 3*A*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a*d*(5*a*B*d*(b^4
*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4) + 2*A*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2*
b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5))*x^6)/2 + ((6*a*B*d*(b^5*d^5 + 15*a*b^4*d^4*e + 5
0*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5) + A*(b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b^4
*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6))*x^7)/7 + ((a^6*B*e^6 + 6*a^5
*b*e^5*(6*B*d + A*e) + 45*a^4*b^2*d*e^4*(5*B*d + 2*A*e) + 100*a^3*b^3*d^2*e^3*(4*B*d + 3*A*e) + 75*a^2*b^4*d^3
*e^2*(3*B*d + 4*A*e) + 18*a*b^5*d^4*e*(2*B*d + 5*A*e) + b^6*d^5*(B*d + 6*A*e))*x^8)/8 + (b*e*(2*a^5*B*e^5 + 5*
a^4*b*e^4*(6*B*d + A*e) + 20*a^3*b^2*d*e^3*(5*B*d + 2*A*e) + 25*a^2*b^3*d^2*e^2*(4*B*d + 3*A*e) + 10*a*b^4*d^3
*e*(3*B*d + 4*A*e) + b^5*d^4*(2*B*d + 5*A*e))*x^9)/3 + (b^2*e^2*(3*a^4*B*e^4 + 4*a^3*b*e^3*(6*B*d + A*e) + 9*a
^2*b^2*d*e^2*(5*B*d + 2*A*e) + 6*a*b^3*d^2*e*(4*B*d + 3*A*e) + b^4*d^3*(3*B*d + 4*A*e))*x^10)/2 + (b^3*e^3*(20
*a^3*B*e^3 + 15*a^2*b*e^2*(6*B*d + A*e) + 18*a*b^2*d*e*(5*B*d + 2*A*e) + 5*b^3*d^2*(4*B*d + 3*A*e))*x^11)/11 +
(b^4*e^4*(5*a^2*B*e^2 + 2*a*b*e*(6*B*d + A*e) + b^2*d*(5*B*d + 2*A*e))*x^12)/4 + (b^5*e^5*(6*b*B*d + A*b*e +
6*a*B*e)*x^13)/13 + (b^6*B*e^6*x^14)/14

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Maple [B]  time = 0.003, size = 1173, normalized size = 4. \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)^6,x)

[Out]

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

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Maxima [B]  time = 1.20245, size = 1582, normalized size = 5.46 \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)^6,x, algorithm="maxima")

[Out]

1/14*B*b^6*e^6*x^14 + A*a^6*d^6*x + 1/13*(6*B*b^6*d*e^5 + (6*B*a*b^5 + A*b^6)*e^6)*x^13 + 1/4*(5*B*b^6*d^2*e^4
+ 2*(6*B*a*b^5 + A*b^6)*d*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^6)*x^12 + 1/11*(20*B*b^6*d^3*e^3 + 15*(6*B*a*b^5
+ A*b^6)*d^2*e^4 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^5 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^6)*x^11 + 1/2*(3*B*b^6
*d^4*e^2 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 + 6*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*d*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^6)*x^10 + 1/3*(2*B*b^6*d^5*e + 5*(6*B*a*b^5 + A*b^6)*d^4*e^2 + 20*(5*B*
a^2*b^4 + 2*A*a*b^5)*d^3*e^3 + 25*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^4 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 +
(2*B*a^5*b + 5*A*a^4*b^2)*e^6)*x^9 + 1/8*(B*b^6*d^6 + 6*(6*B*a*b^5 + A*b^6)*d^5*e + 45*(5*B*a^2*b^4 + 2*A*a*b
^5)*d^4*e^2 + 100*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 75*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 + 18*(2*B*a^5*b
+ 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*x^8 + 1/7*(A*a^6*e^6 + (6*B*a*b^5 + A*b^6)*d^6 + 18*(5*B*a^2*
b^4 + 2*A*a*b^5)*d^5*e + 75*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^2 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^3 + 45
*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 + 6*A*a^5*b)*d*e^5)*x^7 + 1/2*(2*A*a^6*d*e^5 + (5*B*a^2*b^4 + 2*
A*a*b^5)*d^6 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e + 25*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^2 + 20*(2*B*a^5*b +
5*A*a^4*b^2)*d^3*e^3 + 5*(B*a^6 + 6*A*a^5*b)*d^2*e^4)*x^6 + (3*A*a^6*d^2*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^
6 + 6*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e + 9*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e^2 + 4*(B*a^6 + 6*A*a^5*b)*d^3*e^3)
*x^5 + 1/4*(20*A*a^6*d^3*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^6 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*e + 15*(B*
a^6 + 6*A*a^5*b)*d^4*e^2)*x^4 + (5*A*a^6*d^4*e^2 + (2*B*a^5*b + 5*A*a^4*b^2)*d^6 + 2*(B*a^6 + 6*A*a^5*b)*d^5*e
)*x^3 + 1/2*(6*A*a^6*d^5*e + (B*a^6 + 6*A*a^5*b)*d^6)*x^2

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Fricas [B]  time = 1.6374, size = 3262, normalized size = 11.25 \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)^6,x, algorithm="fricas")

[Out]

1/14*x^14*e^6*b^6*B + 6/13*x^13*e^5*d*b^6*B + 6/13*x^13*e^6*b^5*a*B + 1/13*x^13*e^6*b^6*A + 5/4*x^12*e^4*d^2*b
^6*B + 3*x^12*e^5*d*b^5*a*B + 5/4*x^12*e^6*b^4*a^2*B + 1/2*x^12*e^5*d*b^6*A + 1/2*x^12*e^6*b^5*a*A + 20/11*x^1
1*e^3*d^3*b^6*B + 90/11*x^11*e^4*d^2*b^5*a*B + 90/11*x^11*e^5*d*b^4*a^2*B + 20/11*x^11*e^6*b^3*a^3*B + 15/11*x
^11*e^4*d^2*b^6*A + 36/11*x^11*e^5*d*b^5*a*A + 15/11*x^11*e^6*b^4*a^2*A + 3/2*x^10*e^2*d^4*b^6*B + 12*x^10*e^3
*d^3*b^5*a*B + 45/2*x^10*e^4*d^2*b^4*a^2*B + 12*x^10*e^5*d*b^3*a^3*B + 3/2*x^10*e^6*b^2*a^4*B + 2*x^10*e^3*d^3
*b^6*A + 9*x^10*e^4*d^2*b^5*a*A + 9*x^10*e^5*d*b^4*a^2*A + 2*x^10*e^6*b^3*a^3*A + 2/3*x^9*e*d^5*b^6*B + 10*x^9
*e^2*d^4*b^5*a*B + 100/3*x^9*e^3*d^3*b^4*a^2*B + 100/3*x^9*e^4*d^2*b^3*a^3*B + 10*x^9*e^5*d*b^2*a^4*B + 2/3*x^
9*e^6*b*a^5*B + 5/3*x^9*e^2*d^4*b^6*A + 40/3*x^9*e^3*d^3*b^5*a*A + 25*x^9*e^4*d^2*b^4*a^2*A + 40/3*x^9*e^5*d*b
^3*a^3*A + 5/3*x^9*e^6*b^2*a^4*A + 1/8*x^8*d^6*b^6*B + 9/2*x^8*e*d^5*b^5*a*B + 225/8*x^8*e^2*d^4*b^4*a^2*B + 5
0*x^8*e^3*d^3*b^3*a^3*B + 225/8*x^8*e^4*d^2*b^2*a^4*B + 9/2*x^8*e^5*d*b*a^5*B + 1/8*x^8*e^6*a^6*B + 3/4*x^8*e*
d^5*b^6*A + 45/4*x^8*e^2*d^4*b^5*a*A + 75/2*x^8*e^3*d^3*b^4*a^2*A + 75/2*x^8*e^4*d^2*b^3*a^3*A + 45/4*x^8*e^5*
d*b^2*a^4*A + 3/4*x^8*e^6*b*a^5*A + 6/7*x^7*d^6*b^5*a*B + 90/7*x^7*e*d^5*b^4*a^2*B + 300/7*x^7*e^2*d^4*b^3*a^3
*B + 300/7*x^7*e^3*d^3*b^2*a^4*B + 90/7*x^7*e^4*d^2*b*a^5*B + 6/7*x^7*e^5*d*a^6*B + 1/7*x^7*d^6*b^6*A + 36/7*x
^7*e*d^5*b^5*a*A + 225/7*x^7*e^2*d^4*b^4*a^2*A + 400/7*x^7*e^3*d^3*b^3*a^3*A + 225/7*x^7*e^4*d^2*b^2*a^4*A + 3
6/7*x^7*e^5*d*b*a^5*A + 1/7*x^7*e^6*a^6*A + 5/2*x^6*d^6*b^4*a^2*B + 20*x^6*e*d^5*b^3*a^3*B + 75/2*x^6*e^2*d^4*
b^2*a^4*B + 20*x^6*e^3*d^3*b*a^5*B + 5/2*x^6*e^4*d^2*a^6*B + x^6*d^6*b^5*a*A + 15*x^6*e*d^5*b^4*a^2*A + 50*x^6
*e^2*d^4*b^3*a^3*A + 50*x^6*e^3*d^3*b^2*a^4*A + 15*x^6*e^4*d^2*b*a^5*A + x^6*e^5*d*a^6*A + 4*x^5*d^6*b^3*a^3*B
+ 18*x^5*e*d^5*b^2*a^4*B + 18*x^5*e^2*d^4*b*a^5*B + 4*x^5*e^3*d^3*a^6*B + 3*x^5*d^6*b^4*a^2*A + 24*x^5*e*d^5*
b^3*a^3*A + 45*x^5*e^2*d^4*b^2*a^4*A + 24*x^5*e^3*d^3*b*a^5*A + 3*x^5*e^4*d^2*a^6*A + 15/4*x^4*d^6*b^2*a^4*B +
9*x^4*e*d^5*b*a^5*B + 15/4*x^4*e^2*d^4*a^6*B + 5*x^4*d^6*b^3*a^3*A + 45/2*x^4*e*d^5*b^2*a^4*A + 45/2*x^4*e^2*
d^4*b*a^5*A + 5*x^4*e^3*d^3*a^6*A + 2*x^3*d^6*b*a^5*B + 2*x^3*e*d^5*a^6*B + 5*x^3*d^6*b^2*a^4*A + 12*x^3*e*d^5
*b*a^5*A + 5*x^3*e^2*d^4*a^6*A + 1/2*x^2*d^6*a^6*B + 3*x^2*d^6*b*a^5*A + 3*x^2*e*d^5*a^6*A + x*d^6*a^6*A

________________________________________________________________________________________

Sympy [B]  time = 0.217191, size = 1504, normalized size = 5.19 \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)**6,x)

[Out]

A*a**6*d**6*x + B*b**6*e**6*x**14/14 + x**13*(A*b**6*e**6/13 + 6*B*a*b**5*e**6/13 + 6*B*b**6*d*e**5/13) + x**1
2*(A*a*b**5*e**6/2 + A*b**6*d*e**5/2 + 5*B*a**2*b**4*e**6/4 + 3*B*a*b**5*d*e**5 + 5*B*b**6*d**2*e**4/4) + x**1
1*(15*A*a**2*b**4*e**6/11 + 36*A*a*b**5*d*e**5/11 + 15*A*b**6*d**2*e**4/11 + 20*B*a**3*b**3*e**6/11 + 90*B*a**
2*b**4*d*e**5/11 + 90*B*a*b**5*d**2*e**4/11 + 20*B*b**6*d**3*e**3/11) + x**10*(2*A*a**3*b**3*e**6 + 9*A*a**2*b
**4*d*e**5 + 9*A*a*b**5*d**2*e**4 + 2*A*b**6*d**3*e**3 + 3*B*a**4*b**2*e**6/2 + 12*B*a**3*b**3*d*e**5 + 45*B*a
**2*b**4*d**2*e**4/2 + 12*B*a*b**5*d**3*e**3 + 3*B*b**6*d**4*e**2/2) + x**9*(5*A*a**4*b**2*e**6/3 + 40*A*a**3*
b**3*d*e**5/3 + 25*A*a**2*b**4*d**2*e**4 + 40*A*a*b**5*d**3*e**3/3 + 5*A*b**6*d**4*e**2/3 + 2*B*a**5*b*e**6/3
+ 10*B*a**4*b**2*d*e**5 + 100*B*a**3*b**3*d**2*e**4/3 + 100*B*a**2*b**4*d**3*e**3/3 + 10*B*a*b**5*d**4*e**2 +
2*B*b**6*d**5*e/3) + x**8*(3*A*a**5*b*e**6/4 + 45*A*a**4*b**2*d*e**5/4 + 75*A*a**3*b**3*d**2*e**4/2 + 75*A*a**
2*b**4*d**3*e**3/2 + 45*A*a*b**5*d**4*e**2/4 + 3*A*b**6*d**5*e/4 + B*a**6*e**6/8 + 9*B*a**5*b*d*e**5/2 + 225*B
*a**4*b**2*d**2*e**4/8 + 50*B*a**3*b**3*d**3*e**3 + 225*B*a**2*b**4*d**4*e**2/8 + 9*B*a*b**5*d**5*e/2 + B*b**6
*d**6/8) + x**7*(A*a**6*e**6/7 + 36*A*a**5*b*d*e**5/7 + 225*A*a**4*b**2*d**2*e**4/7 + 400*A*a**3*b**3*d**3*e**
3/7 + 225*A*a**2*b**4*d**4*e**2/7 + 36*A*a*b**5*d**5*e/7 + A*b**6*d**6/7 + 6*B*a**6*d*e**5/7 + 90*B*a**5*b*d**
2*e**4/7 + 300*B*a**4*b**2*d**3*e**3/7 + 300*B*a**3*b**3*d**4*e**2/7 + 90*B*a**2*b**4*d**5*e/7 + 6*B*a*b**5*d*
*6/7) + x**6*(A*a**6*d*e**5 + 15*A*a**5*b*d**2*e**4 + 50*A*a**4*b**2*d**3*e**3 + 50*A*a**3*b**3*d**4*e**2 + 15
*A*a**2*b**4*d**5*e + A*a*b**5*d**6 + 5*B*a**6*d**2*e**4/2 + 20*B*a**5*b*d**3*e**3 + 75*B*a**4*b**2*d**4*e**2/
2 + 20*B*a**3*b**3*d**5*e + 5*B*a**2*b**4*d**6/2) + x**5*(3*A*a**6*d**2*e**4 + 24*A*a**5*b*d**3*e**3 + 45*A*a*
*4*b**2*d**4*e**2 + 24*A*a**3*b**3*d**5*e + 3*A*a**2*b**4*d**6 + 4*B*a**6*d**3*e**3 + 18*B*a**5*b*d**4*e**2 +
18*B*a**4*b**2*d**5*e + 4*B*a**3*b**3*d**6) + x**4*(5*A*a**6*d**3*e**3 + 45*A*a**5*b*d**4*e**2/2 + 45*A*a**4*b
**2*d**5*e/2 + 5*A*a**3*b**3*d**6 + 15*B*a**6*d**4*e**2/4 + 9*B*a**5*b*d**5*e + 15*B*a**4*b**2*d**6/4) + x**3*
(5*A*a**6*d**4*e**2 + 12*A*a**5*b*d**5*e + 5*A*a**4*b**2*d**6 + 2*B*a**6*d**5*e + 2*B*a**5*b*d**6) + x**2*(3*A
*a**6*d**5*e + 3*A*a**5*b*d**6 + B*a**6*d**6/2)

________________________________________________________________________________________

Giac [B]  time = 1.93722, size = 1922, normalized size = 6.63 \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)^6,x, algorithm="giac")

[Out]

1/14*B*b^6*x^14*e^6 + 6/13*B*b^6*d*x^13*e^5 + 5/4*B*b^6*d^2*x^12*e^4 + 20/11*B*b^6*d^3*x^11*e^3 + 3/2*B*b^6*d^
4*x^10*e^2 + 2/3*B*b^6*d^5*x^9*e + 1/8*B*b^6*d^6*x^8 + 6/13*B*a*b^5*x^13*e^6 + 1/13*A*b^6*x^13*e^6 + 3*B*a*b^5
*d*x^12*e^5 + 1/2*A*b^6*d*x^12*e^5 + 90/11*B*a*b^5*d^2*x^11*e^4 + 15/11*A*b^6*d^2*x^11*e^4 + 12*B*a*b^5*d^3*x^
10*e^3 + 2*A*b^6*d^3*x^10*e^3 + 10*B*a*b^5*d^4*x^9*e^2 + 5/3*A*b^6*d^4*x^9*e^2 + 9/2*B*a*b^5*d^5*x^8*e + 3/4*A
*b^6*d^5*x^8*e + 6/7*B*a*b^5*d^6*x^7 + 1/7*A*b^6*d^6*x^7 + 5/4*B*a^2*b^4*x^12*e^6 + 1/2*A*a*b^5*x^12*e^6 + 90/
11*B*a^2*b^4*d*x^11*e^5 + 36/11*A*a*b^5*d*x^11*e^5 + 45/2*B*a^2*b^4*d^2*x^10*e^4 + 9*A*a*b^5*d^2*x^10*e^4 + 10
0/3*B*a^2*b^4*d^3*x^9*e^3 + 40/3*A*a*b^5*d^3*x^9*e^3 + 225/8*B*a^2*b^4*d^4*x^8*e^2 + 45/4*A*a*b^5*d^4*x^8*e^2
+ 90/7*B*a^2*b^4*d^5*x^7*e + 36/7*A*a*b^5*d^5*x^7*e + 5/2*B*a^2*b^4*d^6*x^6 + A*a*b^5*d^6*x^6 + 20/11*B*a^3*b^
3*x^11*e^6 + 15/11*A*a^2*b^4*x^11*e^6 + 12*B*a^3*b^3*d*x^10*e^5 + 9*A*a^2*b^4*d*x^10*e^5 + 100/3*B*a^3*b^3*d^2
*x^9*e^4 + 25*A*a^2*b^4*d^2*x^9*e^4 + 50*B*a^3*b^3*d^3*x^8*e^3 + 75/2*A*a^2*b^4*d^3*x^8*e^3 + 300/7*B*a^3*b^3*
d^4*x^7*e^2 + 225/7*A*a^2*b^4*d^4*x^7*e^2 + 20*B*a^3*b^3*d^5*x^6*e + 15*A*a^2*b^4*d^5*x^6*e + 4*B*a^3*b^3*d^6*
x^5 + 3*A*a^2*b^4*d^6*x^5 + 3/2*B*a^4*b^2*x^10*e^6 + 2*A*a^3*b^3*x^10*e^6 + 10*B*a^4*b^2*d*x^9*e^5 + 40/3*A*a^
3*b^3*d*x^9*e^5 + 225/8*B*a^4*b^2*d^2*x^8*e^4 + 75/2*A*a^3*b^3*d^2*x^8*e^4 + 300/7*B*a^4*b^2*d^3*x^7*e^3 + 400
/7*A*a^3*b^3*d^3*x^7*e^3 + 75/2*B*a^4*b^2*d^4*x^6*e^2 + 50*A*a^3*b^3*d^4*x^6*e^2 + 18*B*a^4*b^2*d^5*x^5*e + 24
*A*a^3*b^3*d^5*x^5*e + 15/4*B*a^4*b^2*d^6*x^4 + 5*A*a^3*b^3*d^6*x^4 + 2/3*B*a^5*b*x^9*e^6 + 5/3*A*a^4*b^2*x^9*
e^6 + 9/2*B*a^5*b*d*x^8*e^5 + 45/4*A*a^4*b^2*d*x^8*e^5 + 90/7*B*a^5*b*d^2*x^7*e^4 + 225/7*A*a^4*b^2*d^2*x^7*e^
4 + 20*B*a^5*b*d^3*x^6*e^3 + 50*A*a^4*b^2*d^3*x^6*e^3 + 18*B*a^5*b*d^4*x^5*e^2 + 45*A*a^4*b^2*d^4*x^5*e^2 + 9*
B*a^5*b*d^5*x^4*e + 45/2*A*a^4*b^2*d^5*x^4*e + 2*B*a^5*b*d^6*x^3 + 5*A*a^4*b^2*d^6*x^3 + 1/8*B*a^6*x^8*e^6 + 3
/4*A*a^5*b*x^8*e^6 + 6/7*B*a^6*d*x^7*e^5 + 36/7*A*a^5*b*d*x^7*e^5 + 5/2*B*a^6*d^2*x^6*e^4 + 15*A*a^5*b*d^2*x^6
*e^4 + 4*B*a^6*d^3*x^5*e^3 + 24*A*a^5*b*d^3*x^5*e^3 + 15/4*B*a^6*d^4*x^4*e^2 + 45/2*A*a^5*b*d^4*x^4*e^2 + 2*B*
a^6*d^5*x^3*e + 12*A*a^5*b*d^5*x^3*e + 1/2*B*a^6*d^6*x^2 + 3*A*a^5*b*d^6*x^2 + 1/7*A*a^6*x^7*e^6 + A*a^6*d*x^6
*e^5 + 3*A*a^6*d^2*x^5*e^4 + 5*A*a^6*d^3*x^4*e^3 + 5*A*a^6*d^4*x^3*e^2 + 3*A*a^6*d^5*x^2*e + A*a^6*d^6*x