3.1674 \(\int (A+B x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac{b^3 (d+e x)^{11} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac{b^2 (d+e x)^{10} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{2 b (d+e x)^9 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac{(d+e x)^8 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6}-\frac{(d+e x)^7 (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{b^4 B (d+e x)^{12}}{12 e^6} \]

[Out]

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

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Rubi [A]  time = 0.661421, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{b^3 (d+e x)^{11} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac{b^2 (d+e x)^{10} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{2 b (d+e x)^9 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac{(d+e x)^8 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6}-\frac{(d+e x)^7 (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{b^4 B (d+e x)^{12}}{12 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [B]  time = 0.251342, size = 737, normalized size = 3.58 \[ \frac{1}{9} b e^3 x^9 \left (6 a^2 b e^2 (A e+6 B d)+4 a^3 B e^3+12 a b^2 d e (2 A e+5 B d)+5 b^3 d^2 (3 A e+4 B d)\right )+\frac{1}{8} e^2 x^8 \left (18 a^2 b^2 d e^2 (2 A e+5 B d)+4 a^3 b e^3 (A e+6 B d)+a^4 B e^4+20 a b^3 d^2 e (3 A e+4 B d)+5 b^4 d^3 (4 A e+3 B d)\right )+\frac{1}{7} e x^7 \left (30 a^2 b^2 d^2 e^2 (3 A e+4 B d)+12 a^3 b d e^3 (2 A e+5 B d)+a^4 e^4 (A e+6 B d)+20 a b^3 d^3 e (4 A e+3 B d)+3 b^4 d^4 (5 A e+2 B d)\right )+\frac{1}{6} d x^6 \left (30 a^2 b^2 d^2 e^2 (4 A e+3 B d)+20 a^3 b d e^3 (3 A e+4 B d)+3 a^4 e^4 (2 A e+5 B d)+12 a b^3 d^3 e (5 A e+2 B d)+b^4 d^4 (6 A e+B d)\right )+\frac{1}{5} d^2 x^5 \left (A \left (90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4+24 a b^3 d^3 e+b^4 d^4\right )+4 a B d \left (15 a^2 b d e^2+5 a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )\right )+\frac{1}{4} a d^3 x^4 \left (4 A \left (15 a^2 b d e^2+5 a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+3 a B d \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )\right )+\frac{1}{3} a^2 d^4 x^3 \left (3 A \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+2 a B d (3 a e+2 b d)\right )+\frac{1}{10} b^2 e^4 x^{10} \left (6 a^2 B e^2+4 a b e (A e+6 B d)+3 b^2 d (2 A e+5 B d)\right )+\frac{1}{2} a^3 d^5 x^2 (6 a A e+a B d+4 A b d)+a^4 A d^6 x+\frac{1}{11} b^3 e^5 x^{11} (4 a B e+A b e+6 b B d)+\frac{1}{12} b^4 B e^6 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^6*x + (a^3*d^5*(4*A*b*d + a*B*d + 6*a*A*e)*x^2)/2 + (a^2*d^4*(2*a*B*d*(2*b*d + 3*a*e) + 3*A*(2*b^2*d^2
+ 8*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a*d^3*(3*a*B*d*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2) + 4*A*(b^3*d^3 + 9*a*b
^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3))*x^4)/4 + (d^2*(4*a*B*d*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a
^3*e^3) + A*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4))*x^5)/5 + (d*(3*a^4*
e^4*(5*B*d + 2*A*e) + 20*a^3*b*d*e^3*(4*B*d + 3*A*e) + 30*a^2*b^2*d^2*e^2*(3*B*d + 4*A*e) + 12*a*b^3*d^3*e*(2*
B*d + 5*A*e) + b^4*d^4*(B*d + 6*A*e))*x^6)/6 + (e*(a^4*e^4*(6*B*d + A*e) + 12*a^3*b*d*e^3*(5*B*d + 2*A*e) + 30
*a^2*b^2*d^2*e^2*(4*B*d + 3*A*e) + 20*a*b^3*d^3*e*(3*B*d + 4*A*e) + 3*b^4*d^4*(2*B*d + 5*A*e))*x^7)/7 + (e^2*(
a^4*B*e^4 + 4*a^3*b*e^3*(6*B*d + A*e) + 18*a^2*b^2*d*e^2*(5*B*d + 2*A*e) + 20*a*b^3*d^2*e*(4*B*d + 3*A*e) + 5*
b^4*d^3*(3*B*d + 4*A*e))*x^8)/8 + (b*e^3*(4*a^3*B*e^3 + 6*a^2*b*e^2*(6*B*d + A*e) + 12*a*b^2*d*e*(5*B*d + 2*A*
e) + 5*b^3*d^2*(4*B*d + 3*A*e))*x^9)/9 + (b^2*e^4*(6*a^2*B*e^2 + 4*a*b*e*(6*B*d + A*e) + 3*b^2*d*(5*B*d + 2*A*
e))*x^10)/10 + (b^3*e^5*(6*b*B*d + A*b*e + 4*a*B*e)*x^11)/11 + (b^4*B*e^6*x^12)/12

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Maple [B]  time = 0.002, size = 821, 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)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.12347, size = 1094, normalized size = 5.31 \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)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/12*B*b^4*e^6*x^12 + A*a^4*d^6*x + 1/11*(6*B*b^4*d*e^5 + (4*B*a*b^3 + A*b^4)*e^6)*x^11 + 1/10*(15*B*b^4*d^2*e
^4 + 6*(4*B*a*b^3 + A*b^4)*d*e^5 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*x^10 + 1/9*(20*B*b^4*d^3*e^3 + 15*(4*B*a*b
^3 + A*b^4)*d^2*e^4 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*x^9 + 1/8*(15*B*b^
4*d^4*e^2 + 20*(4*B*a*b^3 + A*b^4)*d^3*e^3 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 + 12*(2*B*a^3*b + 3*A*a^2*b^
2)*d*e^5 + (B*a^4 + 4*A*a^3*b)*e^6)*x^8 + 1/7*(6*B*b^4*d^5*e + A*a^4*e^6 + 15*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 40
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^3 + 30*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 6*(B*a^4 + 4*A*a^3*b)*d*e^5)*x^7 +
1/6*(B*b^4*d^6 + 6*A*a^4*d*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^5*e + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 40*(2*B
*a^3*b + 3*A*a^2*b^2)*d^3*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*x^6 + 1/5*(15*A*a^4*d^2*e^4 + (4*B*a*b^3 + A*b
^4)*d^6 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*e + 30*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^2 + 20*(B*a^4 + 4*A*a^3*b)*d
^3*e^3)*x^5 + 1/4*(20*A*a^4*d^3*e^3 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6 + 12*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*e + 1
5*(B*a^4 + 4*A*a^3*b)*d^4*e^2)*x^4 + 1/3*(15*A*a^4*d^4*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^6 + 6*(B*a^4 + 4*A*
a^3*b)*d^5*e)*x^3 + 1/2*(6*A*a^4*d^5*e + (B*a^4 + 4*A*a^3*b)*d^6)*x^2

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Fricas [B]  time = 1.55711, size = 2223, normalized size = 10.79 \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)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/12*x^12*e^6*b^4*B + 6/11*x^11*e^5*d*b^4*B + 4/11*x^11*e^6*b^3*a*B + 1/11*x^11*e^6*b^4*A + 3/2*x^10*e^4*d^2*b
^4*B + 12/5*x^10*e^5*d*b^3*a*B + 3/5*x^10*e^6*b^2*a^2*B + 3/5*x^10*e^5*d*b^4*A + 2/5*x^10*e^6*b^3*a*A + 20/9*x
^9*e^3*d^3*b^4*B + 20/3*x^9*e^4*d^2*b^3*a*B + 4*x^9*e^5*d*b^2*a^2*B + 4/9*x^9*e^6*b*a^3*B + 5/3*x^9*e^4*d^2*b^
4*A + 8/3*x^9*e^5*d*b^3*a*A + 2/3*x^9*e^6*b^2*a^2*A + 15/8*x^8*e^2*d^4*b^4*B + 10*x^8*e^3*d^3*b^3*a*B + 45/4*x
^8*e^4*d^2*b^2*a^2*B + 3*x^8*e^5*d*b*a^3*B + 1/8*x^8*e^6*a^4*B + 5/2*x^8*e^3*d^3*b^4*A + 15/2*x^8*e^4*d^2*b^3*
a*A + 9/2*x^8*e^5*d*b^2*a^2*A + 1/2*x^8*e^6*b*a^3*A + 6/7*x^7*e*d^5*b^4*B + 60/7*x^7*e^2*d^4*b^3*a*B + 120/7*x
^7*e^3*d^3*b^2*a^2*B + 60/7*x^7*e^4*d^2*b*a^3*B + 6/7*x^7*e^5*d*a^4*B + 15/7*x^7*e^2*d^4*b^4*A + 80/7*x^7*e^3*
d^3*b^3*a*A + 90/7*x^7*e^4*d^2*b^2*a^2*A + 24/7*x^7*e^5*d*b*a^3*A + 1/7*x^7*e^6*a^4*A + 1/6*x^6*d^6*b^4*B + 4*
x^6*e*d^5*b^3*a*B + 15*x^6*e^2*d^4*b^2*a^2*B + 40/3*x^6*e^3*d^3*b*a^3*B + 5/2*x^6*e^4*d^2*a^4*B + x^6*e*d^5*b^
4*A + 10*x^6*e^2*d^4*b^3*a*A + 20*x^6*e^3*d^3*b^2*a^2*A + 10*x^6*e^4*d^2*b*a^3*A + x^6*e^5*d*a^4*A + 4/5*x^5*d
^6*b^3*a*B + 36/5*x^5*e*d^5*b^2*a^2*B + 12*x^5*e^2*d^4*b*a^3*B + 4*x^5*e^3*d^3*a^4*B + 1/5*x^5*d^6*b^4*A + 24/
5*x^5*e*d^5*b^3*a*A + 18*x^5*e^2*d^4*b^2*a^2*A + 16*x^5*e^3*d^3*b*a^3*A + 3*x^5*e^4*d^2*a^4*A + 3/2*x^4*d^6*b^
2*a^2*B + 6*x^4*e*d^5*b*a^3*B + 15/4*x^4*e^2*d^4*a^4*B + x^4*d^6*b^3*a*A + 9*x^4*e*d^5*b^2*a^2*A + 15*x^4*e^2*
d^4*b*a^3*A + 5*x^4*e^3*d^3*a^4*A + 4/3*x^3*d^6*b*a^3*B + 2*x^3*e*d^5*a^4*B + 2*x^3*d^6*b^2*a^2*A + 8*x^3*e*d^
5*b*a^3*A + 5*x^3*e^2*d^4*a^4*A + 1/2*x^2*d^6*a^4*B + 2*x^2*d^6*b*a^3*A + 3*x^2*e*d^5*a^4*A + x*d^6*a^4*A

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Sympy [B]  time = 0.201574, size = 1035, normalized size = 5.02 \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)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**6*x + B*b**4*e**6*x**12/12 + x**11*(A*b**4*e**6/11 + 4*B*a*b**3*e**6/11 + 6*B*b**4*d*e**5/11) + x**1
0*(2*A*a*b**3*e**6/5 + 3*A*b**4*d*e**5/5 + 3*B*a**2*b**2*e**6/5 + 12*B*a*b**3*d*e**5/5 + 3*B*b**4*d**2*e**4/2)
+ x**9*(2*A*a**2*b**2*e**6/3 + 8*A*a*b**3*d*e**5/3 + 5*A*b**4*d**2*e**4/3 + 4*B*a**3*b*e**6/9 + 4*B*a**2*b**2
*d*e**5 + 20*B*a*b**3*d**2*e**4/3 + 20*B*b**4*d**3*e**3/9) + x**8*(A*a**3*b*e**6/2 + 9*A*a**2*b**2*d*e**5/2 +
15*A*a*b**3*d**2*e**4/2 + 5*A*b**4*d**3*e**3/2 + B*a**4*e**6/8 + 3*B*a**3*b*d*e**5 + 45*B*a**2*b**2*d**2*e**4/
4 + 10*B*a*b**3*d**3*e**3 + 15*B*b**4*d**4*e**2/8) + x**7*(A*a**4*e**6/7 + 24*A*a**3*b*d*e**5/7 + 90*A*a**2*b*
*2*d**2*e**4/7 + 80*A*a*b**3*d**3*e**3/7 + 15*A*b**4*d**4*e**2/7 + 6*B*a**4*d*e**5/7 + 60*B*a**3*b*d**2*e**4/7
+ 120*B*a**2*b**2*d**3*e**3/7 + 60*B*a*b**3*d**4*e**2/7 + 6*B*b**4*d**5*e/7) + x**6*(A*a**4*d*e**5 + 10*A*a**
3*b*d**2*e**4 + 20*A*a**2*b**2*d**3*e**3 + 10*A*a*b**3*d**4*e**2 + A*b**4*d**5*e + 5*B*a**4*d**2*e**4/2 + 40*B
*a**3*b*d**3*e**3/3 + 15*B*a**2*b**2*d**4*e**2 + 4*B*a*b**3*d**5*e + B*b**4*d**6/6) + x**5*(3*A*a**4*d**2*e**4
+ 16*A*a**3*b*d**3*e**3 + 18*A*a**2*b**2*d**4*e**2 + 24*A*a*b**3*d**5*e/5 + A*b**4*d**6/5 + 4*B*a**4*d**3*e**
3 + 12*B*a**3*b*d**4*e**2 + 36*B*a**2*b**2*d**5*e/5 + 4*B*a*b**3*d**6/5) + x**4*(5*A*a**4*d**3*e**3 + 15*A*a**
3*b*d**4*e**2 + 9*A*a**2*b**2*d**5*e + A*a*b**3*d**6 + 15*B*a**4*d**4*e**2/4 + 6*B*a**3*b*d**5*e + 3*B*a**2*b*
*2*d**6/2) + x**3*(5*A*a**4*d**4*e**2 + 8*A*a**3*b*d**5*e + 2*A*a**2*b**2*d**6 + 2*B*a**4*d**5*e + 4*B*a**3*b*
d**6/3) + x**2*(3*A*a**4*d**5*e + 2*A*a**3*b*d**6 + B*a**4*d**6/2)

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Giac [B]  time = 1.21243, size = 1316, normalized size = 6.39 \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)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/12*B*b^4*x^12*e^6 + 6/11*B*b^4*d*x^11*e^5 + 3/2*B*b^4*d^2*x^10*e^4 + 20/9*B*b^4*d^3*x^9*e^3 + 15/8*B*b^4*d^4
*x^8*e^2 + 6/7*B*b^4*d^5*x^7*e + 1/6*B*b^4*d^6*x^6 + 4/11*B*a*b^3*x^11*e^6 + 1/11*A*b^4*x^11*e^6 + 12/5*B*a*b^
3*d*x^10*e^5 + 3/5*A*b^4*d*x^10*e^5 + 20/3*B*a*b^3*d^2*x^9*e^4 + 5/3*A*b^4*d^2*x^9*e^4 + 10*B*a*b^3*d^3*x^8*e^
3 + 5/2*A*b^4*d^3*x^8*e^3 + 60/7*B*a*b^3*d^4*x^7*e^2 + 15/7*A*b^4*d^4*x^7*e^2 + 4*B*a*b^3*d^5*x^6*e + A*b^4*d^
5*x^6*e + 4/5*B*a*b^3*d^6*x^5 + 1/5*A*b^4*d^6*x^5 + 3/5*B*a^2*b^2*x^10*e^6 + 2/5*A*a*b^3*x^10*e^6 + 4*B*a^2*b^
2*d*x^9*e^5 + 8/3*A*a*b^3*d*x^9*e^5 + 45/4*B*a^2*b^2*d^2*x^8*e^4 + 15/2*A*a*b^3*d^2*x^8*e^4 + 120/7*B*a^2*b^2*
d^3*x^7*e^3 + 80/7*A*a*b^3*d^3*x^7*e^3 + 15*B*a^2*b^2*d^4*x^6*e^2 + 10*A*a*b^3*d^4*x^6*e^2 + 36/5*B*a^2*b^2*d^
5*x^5*e + 24/5*A*a*b^3*d^5*x^5*e + 3/2*B*a^2*b^2*d^6*x^4 + A*a*b^3*d^6*x^4 + 4/9*B*a^3*b*x^9*e^6 + 2/3*A*a^2*b
^2*x^9*e^6 + 3*B*a^3*b*d*x^8*e^5 + 9/2*A*a^2*b^2*d*x^8*e^5 + 60/7*B*a^3*b*d^2*x^7*e^4 + 90/7*A*a^2*b^2*d^2*x^7
*e^4 + 40/3*B*a^3*b*d^3*x^6*e^3 + 20*A*a^2*b^2*d^3*x^6*e^3 + 12*B*a^3*b*d^4*x^5*e^2 + 18*A*a^2*b^2*d^4*x^5*e^2
+ 6*B*a^3*b*d^5*x^4*e + 9*A*a^2*b^2*d^5*x^4*e + 4/3*B*a^3*b*d^6*x^3 + 2*A*a^2*b^2*d^6*x^3 + 1/8*B*a^4*x^8*e^6
+ 1/2*A*a^3*b*x^8*e^6 + 6/7*B*a^4*d*x^7*e^5 + 24/7*A*a^3*b*d*x^7*e^5 + 5/2*B*a^4*d^2*x^6*e^4 + 10*A*a^3*b*d^2
*x^6*e^4 + 4*B*a^4*d^3*x^5*e^3 + 16*A*a^3*b*d^3*x^5*e^3 + 15/4*B*a^4*d^4*x^4*e^2 + 15*A*a^3*b*d^4*x^4*e^2 + 2*
B*a^4*d^5*x^3*e + 8*A*a^3*b*d^5*x^3*e + 1/2*B*a^4*d^6*x^2 + 2*A*a^3*b*d^6*x^2 + 1/7*A*a^4*x^7*e^6 + A*a^4*d*x^
6*e^5 + 3*A*a^4*d^2*x^5*e^4 + 5*A*a^4*d^3*x^4*e^3 + 5*A*a^4*d^4*x^3*e^2 + 3*A*a^4*d^5*x^2*e + A*a^4*d^6*x