3.1054 \(\int (a+b x)^6 (A+B x) (d+e x)^5 \, dx\)

Optimal. Leaf size=240 \[ \frac{e^4 (a+b x)^{12} (-6 a B e+A b e+5 b B d)}{12 b^7}+\frac{5 e^3 (a+b x)^{11} (b d-a e) (-3 a B e+A b e+2 b B d)}{11 b^7}+\frac{e^2 (a+b x)^{10} (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac{5 e (a+b x)^9 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{9 b^7}+\frac{(a+b x)^8 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{8 b^7}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^5}{7 b^7}+\frac{B e^5 (a+b x)^{13}}{13 b^7} \]

[Out]

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

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Rubi [A]  time = 0.794018, antiderivative size = 240, 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} \[ \frac{e^4 (a+b x)^{12} (-6 a B e+A b e+5 b B d)}{12 b^7}+\frac{5 e^3 (a+b x)^{11} (b d-a e) (-3 a B e+A b e+2 b B d)}{11 b^7}+\frac{e^2 (a+b x)^{10} (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac{5 e (a+b x)^9 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{9 b^7}+\frac{(a+b x)^8 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{8 b^7}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^5}{7 b^7}+\frac{B e^5 (a+b x)^{13}}{13 b^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^5,x]

[Out]

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

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

Mathematica [B]  time = 0.303627, size = 907, normalized size = 3.78 \[ \frac{1}{13} b^6 B e^5 x^{13}+\frac{1}{12} b^5 e^4 (5 b B d+A b e+6 a B e) x^{12}+\frac{1}{11} b^4 e^3 \left (5 d (2 B d+A e) b^2+6 a e (5 B d+A e) b+15 a^2 B e^2\right ) x^{11}+\frac{1}{2} b^3 e^2 \left (2 d^2 (B d+A e) b^3+6 a d e (2 B d+A e) b^2+3 a^2 e^2 (5 B d+A e) b+4 a^3 B e^3\right ) x^{10}+\frac{5}{9} b^2 e \left (d^3 (B d+2 A e) b^4+12 a d^2 e (B d+A e) b^3+15 a^2 d e^2 (2 B d+A e) b^2+4 a^3 e^3 (5 B d+A e) b+3 a^4 B e^4\right ) x^9+\frac{1}{8} b \left (d^4 (B d+5 A e) b^5+30 a d^3 e (B d+2 A e) b^4+150 a^2 d^2 e^2 (B d+A e) b^3+100 a^3 d e^3 (2 B d+A e) b^2+15 a^4 e^4 (5 B d+A e) b+6 a^5 B e^5\right ) x^8+\frac{1}{7} \left (a B \left (6 b^5 d^5+75 a b^4 e d^4+200 a^2 b^3 e^2 d^3+150 a^3 b^2 e^3 d^2+30 a^4 b e^4 d+a^5 e^5\right )+A b \left (b^5 d^5+30 a b^4 e d^4+150 a^2 b^3 e^2 d^3+200 a^3 b^2 e^3 d^2+75 a^4 b e^4 d+6 a^5 e^5\right )\right ) x^7+\frac{1}{6} a \left (5 a B d \left (3 b^4 d^4+20 a b^3 e d^3+30 a^2 b^2 e^2 d^2+12 a^3 b e^3 d+a^4 e^4\right )+A \left (6 b^5 d^5+75 a b^4 e d^4+200 a^2 b^3 e^2 d^3+150 a^3 b^2 e^3 d^2+30 a^4 b e^4 d+a^5 e^5\right )\right ) x^6+a^2 d \left (a B d \left (4 b^3 d^3+15 a b^2 e d^2+12 a^2 b e^2 d+2 a^3 e^3\right )+A \left (3 b^4 d^4+20 a b^3 e d^3+30 a^2 b^2 e^2 d^2+12 a^3 b e^3 d+a^4 e^4\right )\right ) x^5+\frac{5}{4} a^3 d^2 \left (a B d \left (3 b^2 d^2+6 a b e d+2 a^2 e^2\right )+A \left (4 b^3 d^3+15 a b^2 e d^2+12 a^2 b e^2 d+2 a^3 e^3\right )\right ) x^4+\frac{1}{3} a^4 d^3 \left (a B d (6 b d+5 a e)+5 A \left (3 b^2 d^2+6 a b e d+2 a^2 e^2\right )\right ) x^3+\frac{1}{2} a^5 d^4 (6 A b d+a B d+5 a A e) x^2+a^6 A d^5 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^5,x]

[Out]

a^6*A*d^5*x + (a^5*d^4*(6*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^4*d^3*(a*B*d*(6*b*d + 5*a*e) + 5*A*(3*b^2*d^2 +
6*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (5*a^3*d^2*(a*B*d*(3*b^2*d^2 + 6*a*b*d*e + 2*a^2*e^2) + A*(4*b^3*d^3 + 15*a*
b^2*d^2*e + 12*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^2*d*(a*B*d*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 +
2*a^3*e^3) + A*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a*(5*a*B*d
*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4) + A*(6*b^5*d^5 + 75*a*b^4*d^4*e
+ 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5))*x^6)/6 + ((a*B*(6*b^5*d^5 + 75*a*b^4*
d^4*e + 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5) + A*b*(b^5*d^5 + 30*a*b^4*d^4*e
+ 150*a^2*b^3*d^3*e^2 + 200*a^3*b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5))*x^7)/7 + (b*(6*a^5*B*e^5 + 150*a^2*
b^3*d^2*e^2*(B*d + A*e) + 100*a^3*b^2*d*e^3*(2*B*d + A*e) + 15*a^4*b*e^4*(5*B*d + A*e) + 30*a*b^4*d^3*e*(B*d +
2*A*e) + b^5*d^4*(B*d + 5*A*e))*x^8)/8 + (5*b^2*e*(3*a^4*B*e^4 + 12*a*b^3*d^2*e*(B*d + A*e) + 15*a^2*b^2*d*e^
2*(2*B*d + A*e) + 4*a^3*b*e^3*(5*B*d + A*e) + b^4*d^3*(B*d + 2*A*e))*x^9)/9 + (b^3*e^2*(4*a^3*B*e^3 + 2*b^3*d^
2*(B*d + A*e) + 6*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*(5*B*d + A*e))*x^10)/2 + (b^4*e^3*(15*a^2*B*e^2 + 5*b^
2*d*(2*B*d + A*e) + 6*a*b*e*(5*B*d + A*e))*x^11)/11 + (b^5*e^4*(5*b*B*d + A*b*e + 6*a*B*e)*x^12)/12 + (b^6*B*e
^5*x^13)/13

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Maple [B]  time = 0.001, size = 997, normalized size = 4.2 \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)^5,x)

[Out]

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

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Maxima [B]  time = 1.25704, size = 1346, normalized size = 5.61 \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)^5,x, algorithm="maxima")

[Out]

1/13*B*b^6*e^5*x^13 + A*a^6*d^5*x + 1/12*(5*B*b^6*d*e^4 + (6*B*a*b^5 + A*b^6)*e^5)*x^12 + 1/11*(10*B*b^6*d^2*e
^3 + 5*(6*B*a*b^5 + A*b^6)*d*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^5)*x^11 + 1/2*(2*B*b^6*d^3*e^2 + 2*(6*B*a*b^5
+ A*b^6)*d^2*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^10 + 5/9*(B*b^6*d^4
*e + 2*(6*B*a*b^5 + A*b^6)*d^3*e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^3 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^4
+ (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^5)*x^9 + 1/8*(B*b^6*d^5 + 5*(6*B*a*b^5 + A*b^6)*d^4*e + 30*(5*B*a^2*b^4 + 2*A
*a*b^5)*d^3*e^2 + 50*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 25*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 + 3*(2*B*a^5*b
+ 5*A*a^4*b^2)*e^5)*x^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e + 50*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^3*e^2 + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^3 + 15*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^4 + (B*a^6
+ 6*A*a^5*b)*e^5)*x^7 + 1/6*(A*a^6*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5 + 25*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4
*e + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^2 + 30*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^3 + 5*(B*a^6 + 6*A*a^5*b)*d*e
^4)*x^6 + (A*a^6*d*e^4 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e + 6*(2*B*a^5*b
+ 5*A*a^4*b^2)*d^3*e^2 + 2*(B*a^6 + 6*A*a^5*b)*d^2*e^3)*x^5 + 5/4*(2*A*a^6*d^2*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^
3)*d^5 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e + 2*(B*a^6 + 6*A*a^5*b)*d^3*e^2)*x^4 + 1/3*(10*A*a^6*d^3*e^2 + 3*(2
*B*a^5*b + 5*A*a^4*b^2)*d^5 + 5*(B*a^6 + 6*A*a^5*b)*d^4*e)*x^3 + 1/2*(5*A*a^6*d^4*e + (B*a^6 + 6*A*a^5*b)*d^5)
*x^2

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Fricas [B]  time = 1.60298, size = 2763, normalized size = 11.51 \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)^5,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*b^6*B + 5/12*x^12*e^4*d*b^6*B + 1/2*x^12*e^5*b^5*a*B + 1/12*x^12*e^5*b^6*A + 10/11*x^11*e^3*d^2*
b^6*B + 30/11*x^11*e^4*d*b^5*a*B + 15/11*x^11*e^5*b^4*a^2*B + 5/11*x^11*e^4*d*b^6*A + 6/11*x^11*e^5*b^5*a*A +
x^10*e^2*d^3*b^6*B + 6*x^10*e^3*d^2*b^5*a*B + 15/2*x^10*e^4*d*b^4*a^2*B + 2*x^10*e^5*b^3*a^3*B + x^10*e^3*d^2*
b^6*A + 3*x^10*e^4*d*b^5*a*A + 3/2*x^10*e^5*b^4*a^2*A + 5/9*x^9*e*d^4*b^6*B + 20/3*x^9*e^2*d^3*b^5*a*B + 50/3*
x^9*e^3*d^2*b^4*a^2*B + 100/9*x^9*e^4*d*b^3*a^3*B + 5/3*x^9*e^5*b^2*a^4*B + 10/9*x^9*e^2*d^3*b^6*A + 20/3*x^9*
e^3*d^2*b^5*a*A + 25/3*x^9*e^4*d*b^4*a^2*A + 20/9*x^9*e^5*b^3*a^3*A + 1/8*x^8*d^5*b^6*B + 15/4*x^8*e*d^4*b^5*a
*B + 75/4*x^8*e^2*d^3*b^4*a^2*B + 25*x^8*e^3*d^2*b^3*a^3*B + 75/8*x^8*e^4*d*b^2*a^4*B + 3/4*x^8*e^5*b*a^5*B +
5/8*x^8*e*d^4*b^6*A + 15/2*x^8*e^2*d^3*b^5*a*A + 75/4*x^8*e^3*d^2*b^4*a^2*A + 25/2*x^8*e^4*d*b^3*a^3*A + 15/8*
x^8*e^5*b^2*a^4*A + 6/7*x^7*d^5*b^5*a*B + 75/7*x^7*e*d^4*b^4*a^2*B + 200/7*x^7*e^2*d^3*b^3*a^3*B + 150/7*x^7*e
^3*d^2*b^2*a^4*B + 30/7*x^7*e^4*d*b*a^5*B + 1/7*x^7*e^5*a^6*B + 1/7*x^7*d^5*b^6*A + 30/7*x^7*e*d^4*b^5*a*A + 1
50/7*x^7*e^2*d^3*b^4*a^2*A + 200/7*x^7*e^3*d^2*b^3*a^3*A + 75/7*x^7*e^4*d*b^2*a^4*A + 6/7*x^7*e^5*b*a^5*A + 5/
2*x^6*d^5*b^4*a^2*B + 50/3*x^6*e*d^4*b^3*a^3*B + 25*x^6*e^2*d^3*b^2*a^4*B + 10*x^6*e^3*d^2*b*a^5*B + 5/6*x^6*e
^4*d*a^6*B + x^6*d^5*b^5*a*A + 25/2*x^6*e*d^4*b^4*a^2*A + 100/3*x^6*e^2*d^3*b^3*a^3*A + 25*x^6*e^3*d^2*b^2*a^4
*A + 5*x^6*e^4*d*b*a^5*A + 1/6*x^6*e^5*a^6*A + 4*x^5*d^5*b^3*a^3*B + 15*x^5*e*d^4*b^2*a^4*B + 12*x^5*e^2*d^3*b
*a^5*B + 2*x^5*e^3*d^2*a^6*B + 3*x^5*d^5*b^4*a^2*A + 20*x^5*e*d^4*b^3*a^3*A + 30*x^5*e^2*d^3*b^2*a^4*A + 12*x^
5*e^3*d^2*b*a^5*A + x^5*e^4*d*a^6*A + 15/4*x^4*d^5*b^2*a^4*B + 15/2*x^4*e*d^4*b*a^5*B + 5/2*x^4*e^2*d^3*a^6*B
+ 5*x^4*d^5*b^3*a^3*A + 75/4*x^4*e*d^4*b^2*a^4*A + 15*x^4*e^2*d^3*b*a^5*A + 5/2*x^4*e^3*d^2*a^6*A + 2*x^3*d^5*
b*a^5*B + 5/3*x^3*e*d^4*a^6*B + 5*x^3*d^5*b^2*a^4*A + 10*x^3*e*d^4*b*a^5*A + 10/3*x^3*e^2*d^3*a^6*A + 1/2*x^2*
d^5*a^6*B + 3*x^2*d^5*b*a^5*A + 5/2*x^2*e*d^4*a^6*A + x*d^5*a^6*A

________________________________________________________________________________________

Sympy [B]  time = 0.192459, size = 1278, normalized size = 5.32 \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)**5,x)

[Out]

A*a**6*d**5*x + B*b**6*e**5*x**13/13 + x**12*(A*b**6*e**5/12 + B*a*b**5*e**5/2 + 5*B*b**6*d*e**4/12) + x**11*(
6*A*a*b**5*e**5/11 + 5*A*b**6*d*e**4/11 + 15*B*a**2*b**4*e**5/11 + 30*B*a*b**5*d*e**4/11 + 10*B*b**6*d**2*e**3
/11) + x**10*(3*A*a**2*b**4*e**5/2 + 3*A*a*b**5*d*e**4 + A*b**6*d**2*e**3 + 2*B*a**3*b**3*e**5 + 15*B*a**2*b**
4*d*e**4/2 + 6*B*a*b**5*d**2*e**3 + B*b**6*d**3*e**2) + x**9*(20*A*a**3*b**3*e**5/9 + 25*A*a**2*b**4*d*e**4/3
+ 20*A*a*b**5*d**2*e**3/3 + 10*A*b**6*d**3*e**2/9 + 5*B*a**4*b**2*e**5/3 + 100*B*a**3*b**3*d*e**4/9 + 50*B*a**
2*b**4*d**2*e**3/3 + 20*B*a*b**5*d**3*e**2/3 + 5*B*b**6*d**4*e/9) + x**8*(15*A*a**4*b**2*e**5/8 + 25*A*a**3*b*
*3*d*e**4/2 + 75*A*a**2*b**4*d**2*e**3/4 + 15*A*a*b**5*d**3*e**2/2 + 5*A*b**6*d**4*e/8 + 3*B*a**5*b*e**5/4 + 7
5*B*a**4*b**2*d*e**4/8 + 25*B*a**3*b**3*d**2*e**3 + 75*B*a**2*b**4*d**3*e**2/4 + 15*B*a*b**5*d**4*e/4 + B*b**6
*d**5/8) + x**7*(6*A*a**5*b*e**5/7 + 75*A*a**4*b**2*d*e**4/7 + 200*A*a**3*b**3*d**2*e**3/7 + 150*A*a**2*b**4*d
**3*e**2/7 + 30*A*a*b**5*d**4*e/7 + A*b**6*d**5/7 + B*a**6*e**5/7 + 30*B*a**5*b*d*e**4/7 + 150*B*a**4*b**2*d**
2*e**3/7 + 200*B*a**3*b**3*d**3*e**2/7 + 75*B*a**2*b**4*d**4*e/7 + 6*B*a*b**5*d**5/7) + x**6*(A*a**6*e**5/6 +
5*A*a**5*b*d*e**4 + 25*A*a**4*b**2*d**2*e**3 + 100*A*a**3*b**3*d**3*e**2/3 + 25*A*a**2*b**4*d**4*e/2 + A*a*b**
5*d**5 + 5*B*a**6*d*e**4/6 + 10*B*a**5*b*d**2*e**3 + 25*B*a**4*b**2*d**3*e**2 + 50*B*a**3*b**3*d**4*e/3 + 5*B*
a**2*b**4*d**5/2) + x**5*(A*a**6*d*e**4 + 12*A*a**5*b*d**2*e**3 + 30*A*a**4*b**2*d**3*e**2 + 20*A*a**3*b**3*d*
*4*e + 3*A*a**2*b**4*d**5 + 2*B*a**6*d**2*e**3 + 12*B*a**5*b*d**3*e**2 + 15*B*a**4*b**2*d**4*e + 4*B*a**3*b**3
*d**5) + x**4*(5*A*a**6*d**2*e**3/2 + 15*A*a**5*b*d**3*e**2 + 75*A*a**4*b**2*d**4*e/4 + 5*A*a**3*b**3*d**5 + 5
*B*a**6*d**3*e**2/2 + 15*B*a**5*b*d**4*e/2 + 15*B*a**4*b**2*d**5/4) + x**3*(10*A*a**6*d**3*e**2/3 + 10*A*a**5*
b*d**4*e + 5*A*a**4*b**2*d**5 + 5*B*a**6*d**4*e/3 + 2*B*a**5*b*d**5) + x**2*(5*A*a**6*d**4*e/2 + 3*A*a**5*b*d*
*5 + B*a**6*d**5/2)

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Giac [B]  time = 2.60102, size = 1625, normalized size = 6.77 \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)^5,x, algorithm="giac")

[Out]

1/13*B*b^6*x^13*e^5 + 5/12*B*b^6*d*x^12*e^4 + 10/11*B*b^6*d^2*x^11*e^3 + B*b^6*d^3*x^10*e^2 + 5/9*B*b^6*d^4*x^
9*e + 1/8*B*b^6*d^5*x^8 + 1/2*B*a*b^5*x^12*e^5 + 1/12*A*b^6*x^12*e^5 + 30/11*B*a*b^5*d*x^11*e^4 + 5/11*A*b^6*d
*x^11*e^4 + 6*B*a*b^5*d^2*x^10*e^3 + A*b^6*d^2*x^10*e^3 + 20/3*B*a*b^5*d^3*x^9*e^2 + 10/9*A*b^6*d^3*x^9*e^2 +
15/4*B*a*b^5*d^4*x^8*e + 5/8*A*b^6*d^4*x^8*e + 6/7*B*a*b^5*d^5*x^7 + 1/7*A*b^6*d^5*x^7 + 15/11*B*a^2*b^4*x^11*
e^5 + 6/11*A*a*b^5*x^11*e^5 + 15/2*B*a^2*b^4*d*x^10*e^4 + 3*A*a*b^5*d*x^10*e^4 + 50/3*B*a^2*b^4*d^2*x^9*e^3 +
20/3*A*a*b^5*d^2*x^9*e^3 + 75/4*B*a^2*b^4*d^3*x^8*e^2 + 15/2*A*a*b^5*d^3*x^8*e^2 + 75/7*B*a^2*b^4*d^4*x^7*e +
30/7*A*a*b^5*d^4*x^7*e + 5/2*B*a^2*b^4*d^5*x^6 + A*a*b^5*d^5*x^6 + 2*B*a^3*b^3*x^10*e^5 + 3/2*A*a^2*b^4*x^10*e
^5 + 100/9*B*a^3*b^3*d*x^9*e^4 + 25/3*A*a^2*b^4*d*x^9*e^4 + 25*B*a^3*b^3*d^2*x^8*e^3 + 75/4*A*a^2*b^4*d^2*x^8*
e^3 + 200/7*B*a^3*b^3*d^3*x^7*e^2 + 150/7*A*a^2*b^4*d^3*x^7*e^2 + 50/3*B*a^3*b^3*d^4*x^6*e + 25/2*A*a^2*b^4*d^
4*x^6*e + 4*B*a^3*b^3*d^5*x^5 + 3*A*a^2*b^4*d^5*x^5 + 5/3*B*a^4*b^2*x^9*e^5 + 20/9*A*a^3*b^3*x^9*e^5 + 75/8*B*
a^4*b^2*d*x^8*e^4 + 25/2*A*a^3*b^3*d*x^8*e^4 + 150/7*B*a^4*b^2*d^2*x^7*e^3 + 200/7*A*a^3*b^3*d^2*x^7*e^3 + 25*
B*a^4*b^2*d^3*x^6*e^2 + 100/3*A*a^3*b^3*d^3*x^6*e^2 + 15*B*a^4*b^2*d^4*x^5*e + 20*A*a^3*b^3*d^4*x^5*e + 15/4*B
*a^4*b^2*d^5*x^4 + 5*A*a^3*b^3*d^5*x^4 + 3/4*B*a^5*b*x^8*e^5 + 15/8*A*a^4*b^2*x^8*e^5 + 30/7*B*a^5*b*d*x^7*e^4
+ 75/7*A*a^4*b^2*d*x^7*e^4 + 10*B*a^5*b*d^2*x^6*e^3 + 25*A*a^4*b^2*d^2*x^6*e^3 + 12*B*a^5*b*d^3*x^5*e^2 + 30*
A*a^4*b^2*d^3*x^5*e^2 + 15/2*B*a^5*b*d^4*x^4*e + 75/4*A*a^4*b^2*d^4*x^4*e + 2*B*a^5*b*d^5*x^3 + 5*A*a^4*b^2*d^
5*x^3 + 1/7*B*a^6*x^7*e^5 + 6/7*A*a^5*b*x^7*e^5 + 5/6*B*a^6*d*x^6*e^4 + 5*A*a^5*b*d*x^6*e^4 + 2*B*a^6*d^2*x^5*
e^3 + 12*A*a^5*b*d^2*x^5*e^3 + 5/2*B*a^6*d^3*x^4*e^2 + 15*A*a^5*b*d^3*x^4*e^2 + 5/3*B*a^6*d^4*x^3*e + 10*A*a^5
*b*d^4*x^3*e + 1/2*B*a^6*d^5*x^2 + 3*A*a^5*b*d^5*x^2 + 1/6*A*a^6*x^6*e^5 + A*a^6*d*x^5*e^4 + 5/2*A*a^6*d^2*x^4
*e^3 + 10/3*A*a^6*d^3*x^3*e^2 + 5/2*A*a^6*d^4*x^2*e + A*a^6*d^5*x