3.2333 \(\int (A+B x) (d+e x)^5 (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=555 \[ -\frac{(d+e x)^9 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{9 e^8}-\frac{3 c (d+e x)^{11} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{11 e^8}-\frac{(d+e x)^{10} \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{10 e^8}-\frac{3 (d+e x)^8 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{8 e^8}-\frac{(d+e x)^7 \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{7 e^8}-\frac{(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8}-\frac{c^2 (d+e x)^{12} (-A c e-3 b B e+7 B c d)}{12 e^8}+\frac{B c^3 (d+e x)^{13}}{13 e^8} \]

[Out]

-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^6)/(6*e^8) - ((c*d^2 - b*d*e + a*e^2)^2*(3*A*e*(2*c*d - b*e)
- B*(7*c*d^2 - e*(4*b*d - a*e)))*(d + e*x)^7)/(7*e^8) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b
*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^8)/(8*e^8) - ((A
*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30
*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e*x)^9)/(9*e^8) - ((B*(35*c^3*d^3
- b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d
- a*e)))*(d + e*x)^10)/(10*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e)))*(d
+ e*x)^11)/(11*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^12)/(12*e^8) + (B*c^3*(d + e*x)^13)/(13*e^8)

________________________________________________________________________________________

Rubi [A]  time = 1.74659, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{(d+e x)^9 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{9 e^8}-\frac{3 c (d+e x)^{11} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{11 e^8}-\frac{(d+e x)^{10} \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{10 e^8}-\frac{3 (d+e x)^8 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{8 e^8}+\frac{(d+e x)^7 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{7 e^8}-\frac{(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8}-\frac{c^2 (d+e x)^{12} (-A c e-3 b B e+7 B c d)}{12 e^8}+\frac{B c^3 (d+e x)^{13}}{13 e^8} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^3,x]

[Out]

-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^6)/(6*e^8) + ((c*d^2 - b*d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*
b*d - a*e) - 3*A*e*(2*c*d - b*e))*(d + e*x)^7)/(7*e^8) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b
*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^8)/(8*e^8) - ((A
*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30
*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e*x)^9)/(9*e^8) - ((B*(35*c^3*d^3
- b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d
- a*e)))*(d + e*x)^10)/(10*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e)))*(d
+ e*x)^11)/(11*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^12)/(12*e^8) + (B*c^3*(d + e*x)^13)/(13*e^8)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{e^7}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) (d+e x)^6}{e^7}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^7}{e^7}+\frac{\left (-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^8}{e^7}+\frac{\left (-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^9}{e^7}+\frac{3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{10}}{e^7}+\frac{c^2 (-7 B c d+3 b B e+A c e) (d+e x)^{11}}{e^7}+\frac{B c^3 (d+e x)^{12}}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^6}{6 e^8}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right ) (d+e x)^7}{7 e^8}-\frac{3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^8}{8 e^8}-\frac{\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^9}{9 e^8}-\frac{\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^{10}}{10 e^8}-\frac{3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{11}}{11 e^8}-\frac{c^2 (7 B c d-3 b B e-A c e) (d+e x)^{12}}{12 e^8}+\frac{B c^3 (d+e x)^{13}}{13 e^8}\\ \end{align*}

Mathematica [B]  time = 0.664119, size = 1178, normalized size = 2.12 \[ \frac{1}{13} B c^3 e^5 x^{13}+\frac{1}{12} c^2 e^4 (5 B c d+3 b B e+A c e) x^{12}+\frac{1}{11} c e^3 \left (A c e (5 c d+3 b e)+B \left (10 c^2 d^2+3 b^2 e^2+3 c e (5 b d+a e)\right )\right ) x^{11}+\frac{1}{10} e^2 \left (A c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (5 b d+a e)\right )+B \left (10 c^3 d^3+15 c^2 e (2 b d+a e) d+b^3 e^3+3 b c e^2 (5 b d+2 a e)\right )\right ) x^{10}+\frac{1}{9} e \left (A e \left (10 c^3 d^3+15 c^2 e (2 b d+a e) d+b^3 e^3+3 b c e^2 (5 b d+2 a e)\right )+B \left (5 c^3 d^4+30 c^2 e (b d+a e) d^2+b^2 e^3 (5 b d+3 a e)+3 c e^2 \left (10 b^2 d^2+10 a b e d+a^2 e^2\right )\right )\right ) x^9+\frac{1}{8} \left (A e \left (5 c^3 d^4+30 c^2 e (b d+a e) d^2+b^2 e^3 (5 b d+3 a e)+3 c e^2 \left (10 b^2 d^2+10 a b e d+a^2 e^2\right )\right )+B \left (c^3 d^5+15 c^2 e (b d+2 a e) d^3+15 c e^2 \left (2 b^2 d^2+4 a b e d+a^2 e^2\right ) d+b e^3 \left (10 b^2 d^2+15 a b e d+3 a^2 e^2\right )\right )\right ) x^8+\frac{1}{7} \left (10 d^2 e^2 (B d+A e) b^3+15 d e \left (B c d^3+2 A c e d^2+2 a B e^2 d+a A e^3\right ) b^2+3 \left (A e \left (5 c^2 d^4+20 a c e^2 d^2+a^2 e^4\right )+B \left (c^2 d^5+20 a c e^2 d^3+5 a^2 e^4 d\right )\right ) b+a B e \left (15 c^2 d^4+30 a c e^2 d^2+a^2 e^4\right )+A c d \left (c^2 d^4+30 a c e^2 d^2+15 a^2 e^4\right )\right ) x^7+\frac{1}{6} \left (5 b^3 e (B d+2 A e) d^3+3 b^2 \left (B c d^3+5 A c e d^2+10 a B e^2 d+10 a A e^3\right ) d^2+3 b \left (10 a B d e \left (c d^2+a e^2\right )+A \left (c^2 d^4+20 a c e^2 d^2+5 a^2 e^4\right )\right ) d+a \left (A e \left (15 c^2 d^4+30 a c e^2 d^2+a^2 e^4\right )+B \left (3 c^2 d^5+30 a c e^2 d^3+5 a^2 e^4 d\right )\right )\right ) x^6+\frac{1}{5} d \left (b^3 (B d+5 A e) d^3+3 b^2 \left (A c d^2+5 a B e d+10 a A e^2\right ) d^2+6 a b \left (B c d^3+5 A c e d^2+5 a B e^2 d+5 a A e^3\right ) d+a \left (5 a B d e \left (3 c d^2+2 a e^2\right )+A \left (3 c^2 d^4+30 a c e^2 d^2+5 a^2 e^4\right )\right )\right ) x^5+\frac{1}{4} d^2 \left (A \left (b^3 d^3+15 a b^2 e d^2+6 a b \left (c d^2+5 a e^2\right ) d+5 a^2 e \left (3 c d^2+2 a e^2\right )\right )+a B d \left (3 b^2 d^2+15 a b e d+a \left (3 c d^2+10 a e^2\right )\right )\right ) x^4+\frac{1}{3} a d^3 \left (a B d (3 b d+5 a e)+A \left (3 b^2 d^2+15 a b e d+a \left (3 c d^2+10 a e^2\right )\right )\right ) x^3+\frac{1}{2} a^2 d^4 (3 A b d+a B d+5 a A e) x^2+a^3 A d^5 x \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^3,x]

[Out]

a^3*A*d^5*x + (a^2*d^4*(3*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a*d^3*(a*B*d*(3*b*d + 5*a*e) + A*(3*b^2*d^2 + 15*
a*b*d*e + a*(3*c*d^2 + 10*a*e^2)))*x^3)/3 + (d^2*(A*(b^3*d^3 + 15*a*b^2*d^2*e + 5*a^2*e*(3*c*d^2 + 2*a*e^2) +
6*a*b*d*(c*d^2 + 5*a*e^2)) + a*B*d*(3*b^2*d^2 + 15*a*b*d*e + a*(3*c*d^2 + 10*a*e^2)))*x^4)/4 + (d*(b^3*d^3*(B*
d + 5*A*e) + 3*b^2*d^2*(A*c*d^2 + 5*a*B*d*e + 10*a*A*e^2) + 6*a*b*d*(B*c*d^3 + 5*A*c*d^2*e + 5*a*B*d*e^2 + 5*a
*A*e^3) + a*(5*a*B*d*e*(3*c*d^2 + 2*a*e^2) + A*(3*c^2*d^4 + 30*a*c*d^2*e^2 + 5*a^2*e^4)))*x^5)/5 + ((5*b^3*d^3
*e*(B*d + 2*A*e) + 3*b^2*d^2*(B*c*d^3 + 5*A*c*d^2*e + 10*a*B*d*e^2 + 10*a*A*e^3) + 3*b*d*(10*a*B*d*e*(c*d^2 +
a*e^2) + A*(c^2*d^4 + 20*a*c*d^2*e^2 + 5*a^2*e^4)) + a*(A*e*(15*c^2*d^4 + 30*a*c*d^2*e^2 + a^2*e^4) + B*(3*c^2
*d^5 + 30*a*c*d^3*e^2 + 5*a^2*d*e^4)))*x^6)/6 + ((10*b^3*d^2*e^2*(B*d + A*e) + 15*b^2*d*e*(B*c*d^3 + 2*A*c*d^2
*e + 2*a*B*d*e^2 + a*A*e^3) + a*B*e*(15*c^2*d^4 + 30*a*c*d^2*e^2 + a^2*e^4) + A*c*d*(c^2*d^4 + 30*a*c*d^2*e^2
+ 15*a^2*e^4) + 3*b*(A*e*(5*c^2*d^4 + 20*a*c*d^2*e^2 + a^2*e^4) + B*(c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)))
*x^7)/7 + ((A*e*(5*c^3*d^4 + 30*c^2*d^2*e*(b*d + a*e) + b^2*e^3*(5*b*d + 3*a*e) + 3*c*e^2*(10*b^2*d^2 + 10*a*b
*d*e + a^2*e^2)) + B*(c^3*d^5 + 15*c^2*d^3*e*(b*d + 2*a*e) + 15*c*d*e^2*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2) + b*
e^3*(10*b^2*d^2 + 15*a*b*d*e + 3*a^2*e^2)))*x^8)/8 + (e*(A*e*(10*c^3*d^3 + b^3*e^3 + 15*c^2*d*e*(2*b*d + a*e)
+ 3*b*c*e^2*(5*b*d + 2*a*e)) + B*(5*c^3*d^4 + 30*c^2*d^2*e*(b*d + a*e) + b^2*e^3*(5*b*d + 3*a*e) + 3*c*e^2*(10
*b^2*d^2 + 10*a*b*d*e + a^2*e^2)))*x^9)/9 + (e^2*(A*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(5*b*d + a*e)) + B*(10
*c^3*d^3 + b^3*e^3 + 15*c^2*d*e*(2*b*d + a*e) + 3*b*c*e^2*(5*b*d + 2*a*e)))*x^10)/10 + (c*e^3*(A*c*e*(5*c*d +
3*b*e) + B*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(5*b*d + a*e)))*x^11)/11 + (c^2*e^4*(5*B*c*d + 3*b*B*e + A*c*e)*x^1
2)/12 + (B*c^3*e^5*x^13)/13

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Maple [B]  time = 0.003, size = 1263, normalized size = 2.3 \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)^5*(c*x^2+b*x+a)^3,x)

[Out]

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

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Maxima [B]  time = 1.00534, size = 1504, normalized size = 2.71 \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)^5*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.95634, size = 3792, normalized size = 6.83 \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)^5*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*c^3*B + 5/12*x^12*e^4*d*c^3*B + 1/4*x^12*e^5*c^2*b*B + 1/12*x^12*e^5*c^3*A + 10/11*x^11*e^3*d^2*
c^3*B + 15/11*x^11*e^4*d*c^2*b*B + 3/11*x^11*e^5*c*b^2*B + 3/11*x^11*e^5*c^2*a*B + 5/11*x^11*e^4*d*c^3*A + 3/1
1*x^11*e^5*c^2*b*A + x^10*e^2*d^3*c^3*B + 3*x^10*e^3*d^2*c^2*b*B + 3/2*x^10*e^4*d*c*b^2*B + 1/10*x^10*e^5*b^3*
B + 3/2*x^10*e^4*d*c^2*a*B + 3/5*x^10*e^5*c*b*a*B + x^10*e^3*d^2*c^3*A + 3/2*x^10*e^4*d*c^2*b*A + 3/10*x^10*e^
5*c*b^2*A + 3/10*x^10*e^5*c^2*a*A + 5/9*x^9*e*d^4*c^3*B + 10/3*x^9*e^2*d^3*c^2*b*B + 10/3*x^9*e^3*d^2*c*b^2*B
+ 5/9*x^9*e^4*d*b^3*B + 10/3*x^9*e^3*d^2*c^2*a*B + 10/3*x^9*e^4*d*c*b*a*B + 1/3*x^9*e^5*b^2*a*B + 1/3*x^9*e^5*
c*a^2*B + 10/9*x^9*e^2*d^3*c^3*A + 10/3*x^9*e^3*d^2*c^2*b*A + 5/3*x^9*e^4*d*c*b^2*A + 1/9*x^9*e^5*b^3*A + 5/3*
x^9*e^4*d*c^2*a*A + 2/3*x^9*e^5*c*b*a*A + 1/8*x^8*d^5*c^3*B + 15/8*x^8*e*d^4*c^2*b*B + 15/4*x^8*e^2*d^3*c*b^2*
B + 5/4*x^8*e^3*d^2*b^3*B + 15/4*x^8*e^2*d^3*c^2*a*B + 15/2*x^8*e^3*d^2*c*b*a*B + 15/8*x^8*e^4*d*b^2*a*B + 15/
8*x^8*e^4*d*c*a^2*B + 3/8*x^8*e^5*b*a^2*B + 5/8*x^8*e*d^4*c^3*A + 15/4*x^8*e^2*d^3*c^2*b*A + 15/4*x^8*e^3*d^2*
c*b^2*A + 5/8*x^8*e^4*d*b^3*A + 15/4*x^8*e^3*d^2*c^2*a*A + 15/4*x^8*e^4*d*c*b*a*A + 3/8*x^8*e^5*b^2*a*A + 3/8*
x^8*e^5*c*a^2*A + 3/7*x^7*d^5*c^2*b*B + 15/7*x^7*e*d^4*c*b^2*B + 10/7*x^7*e^2*d^3*b^3*B + 15/7*x^7*e*d^4*c^2*a
*B + 60/7*x^7*e^2*d^3*c*b*a*B + 30/7*x^7*e^3*d^2*b^2*a*B + 30/7*x^7*e^3*d^2*c*a^2*B + 15/7*x^7*e^4*d*b*a^2*B +
1/7*x^7*e^5*a^3*B + 1/7*x^7*d^5*c^3*A + 15/7*x^7*e*d^4*c^2*b*A + 30/7*x^7*e^2*d^3*c*b^2*A + 10/7*x^7*e^3*d^2*
b^3*A + 30/7*x^7*e^2*d^3*c^2*a*A + 60/7*x^7*e^3*d^2*c*b*a*A + 15/7*x^7*e^4*d*b^2*a*A + 15/7*x^7*e^4*d*c*a^2*A
+ 3/7*x^7*e^5*b*a^2*A + 1/2*x^6*d^5*c*b^2*B + 5/6*x^6*e*d^4*b^3*B + 1/2*x^6*d^5*c^2*a*B + 5*x^6*e*d^4*c*b*a*B
+ 5*x^6*e^2*d^3*b^2*a*B + 5*x^6*e^2*d^3*c*a^2*B + 5*x^6*e^3*d^2*b*a^2*B + 5/6*x^6*e^4*d*a^3*B + 1/2*x^6*d^5*c^
2*b*A + 5/2*x^6*e*d^4*c*b^2*A + 5/3*x^6*e^2*d^3*b^3*A + 5/2*x^6*e*d^4*c^2*a*A + 10*x^6*e^2*d^3*c*b*a*A + 5*x^6
*e^3*d^2*b^2*a*A + 5*x^6*e^3*d^2*c*a^2*A + 5/2*x^6*e^4*d*b*a^2*A + 1/6*x^6*e^5*a^3*A + 1/5*x^5*d^5*b^3*B + 6/5
*x^5*d^5*c*b*a*B + 3*x^5*e*d^4*b^2*a*B + 3*x^5*e*d^4*c*a^2*B + 6*x^5*e^2*d^3*b*a^2*B + 2*x^5*e^3*d^2*a^3*B + 3
/5*x^5*d^5*c*b^2*A + x^5*e*d^4*b^3*A + 3/5*x^5*d^5*c^2*a*A + 6*x^5*e*d^4*c*b*a*A + 6*x^5*e^2*d^3*b^2*a*A + 6*x
^5*e^2*d^3*c*a^2*A + 6*x^5*e^3*d^2*b*a^2*A + x^5*e^4*d*a^3*A + 3/4*x^4*d^5*b^2*a*B + 3/4*x^4*d^5*c*a^2*B + 15/
4*x^4*e*d^4*b*a^2*B + 5/2*x^4*e^2*d^3*a^3*B + 1/4*x^4*d^5*b^3*A + 3/2*x^4*d^5*c*b*a*A + 15/4*x^4*e*d^4*b^2*a*A
+ 15/4*x^4*e*d^4*c*a^2*A + 15/2*x^4*e^2*d^3*b*a^2*A + 5/2*x^4*e^3*d^2*a^3*A + x^3*d^5*b*a^2*B + 5/3*x^3*e*d^4
*a^3*B + x^3*d^5*b^2*a*A + x^3*d^5*c*a^2*A + 5*x^3*e*d^4*b*a^2*A + 10/3*x^3*e^2*d^3*a^3*A + 1/2*x^2*d^5*a^3*B
+ 3/2*x^2*d^5*b*a^2*A + 5/2*x^2*e*d^4*a^3*A + x*d^5*a^3*A

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Sympy [B]  time = 0.247602, size = 1731, normalized size = 3.12 \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)**5*(c*x**2+b*x+a)**3,x)

[Out]

A*a**3*d**5*x + B*c**3*e**5*x**13/13 + x**12*(A*c**3*e**5/12 + B*b*c**2*e**5/4 + 5*B*c**3*d*e**4/12) + x**11*(
3*A*b*c**2*e**5/11 + 5*A*c**3*d*e**4/11 + 3*B*a*c**2*e**5/11 + 3*B*b**2*c*e**5/11 + 15*B*b*c**2*d*e**4/11 + 10
*B*c**3*d**2*e**3/11) + x**10*(3*A*a*c**2*e**5/10 + 3*A*b**2*c*e**5/10 + 3*A*b*c**2*d*e**4/2 + A*c**3*d**2*e**
3 + 3*B*a*b*c*e**5/5 + 3*B*a*c**2*d*e**4/2 + B*b**3*e**5/10 + 3*B*b**2*c*d*e**4/2 + 3*B*b*c**2*d**2*e**3 + B*c
**3*d**3*e**2) + x**9*(2*A*a*b*c*e**5/3 + 5*A*a*c**2*d*e**4/3 + A*b**3*e**5/9 + 5*A*b**2*c*d*e**4/3 + 10*A*b*c
**2*d**2*e**3/3 + 10*A*c**3*d**3*e**2/9 + B*a**2*c*e**5/3 + B*a*b**2*e**5/3 + 10*B*a*b*c*d*e**4/3 + 10*B*a*c**
2*d**2*e**3/3 + 5*B*b**3*d*e**4/9 + 10*B*b**2*c*d**2*e**3/3 + 10*B*b*c**2*d**3*e**2/3 + 5*B*c**3*d**4*e/9) + x
**8*(3*A*a**2*c*e**5/8 + 3*A*a*b**2*e**5/8 + 15*A*a*b*c*d*e**4/4 + 15*A*a*c**2*d**2*e**3/4 + 5*A*b**3*d*e**4/8
+ 15*A*b**2*c*d**2*e**3/4 + 15*A*b*c**2*d**3*e**2/4 + 5*A*c**3*d**4*e/8 + 3*B*a**2*b*e**5/8 + 15*B*a**2*c*d*e
**4/8 + 15*B*a*b**2*d*e**4/8 + 15*B*a*b*c*d**2*e**3/2 + 15*B*a*c**2*d**3*e**2/4 + 5*B*b**3*d**2*e**3/4 + 15*B*
b**2*c*d**3*e**2/4 + 15*B*b*c**2*d**4*e/8 + B*c**3*d**5/8) + x**7*(3*A*a**2*b*e**5/7 + 15*A*a**2*c*d*e**4/7 +
15*A*a*b**2*d*e**4/7 + 60*A*a*b*c*d**2*e**3/7 + 30*A*a*c**2*d**3*e**2/7 + 10*A*b**3*d**2*e**3/7 + 30*A*b**2*c*
d**3*e**2/7 + 15*A*b*c**2*d**4*e/7 + A*c**3*d**5/7 + B*a**3*e**5/7 + 15*B*a**2*b*d*e**4/7 + 30*B*a**2*c*d**2*e
**3/7 + 30*B*a*b**2*d**2*e**3/7 + 60*B*a*b*c*d**3*e**2/7 + 15*B*a*c**2*d**4*e/7 + 10*B*b**3*d**3*e**2/7 + 15*B
*b**2*c*d**4*e/7 + 3*B*b*c**2*d**5/7) + x**6*(A*a**3*e**5/6 + 5*A*a**2*b*d*e**4/2 + 5*A*a**2*c*d**2*e**3 + 5*A
*a*b**2*d**2*e**3 + 10*A*a*b*c*d**3*e**2 + 5*A*a*c**2*d**4*e/2 + 5*A*b**3*d**3*e**2/3 + 5*A*b**2*c*d**4*e/2 +
A*b*c**2*d**5/2 + 5*B*a**3*d*e**4/6 + 5*B*a**2*b*d**2*e**3 + 5*B*a**2*c*d**3*e**2 + 5*B*a*b**2*d**3*e**2 + 5*B
*a*b*c*d**4*e + B*a*c**2*d**5/2 + 5*B*b**3*d**4*e/6 + B*b**2*c*d**5/2) + x**5*(A*a**3*d*e**4 + 6*A*a**2*b*d**2
*e**3 + 6*A*a**2*c*d**3*e**2 + 6*A*a*b**2*d**3*e**2 + 6*A*a*b*c*d**4*e + 3*A*a*c**2*d**5/5 + A*b**3*d**4*e + 3
*A*b**2*c*d**5/5 + 2*B*a**3*d**2*e**3 + 6*B*a**2*b*d**3*e**2 + 3*B*a**2*c*d**4*e + 3*B*a*b**2*d**4*e + 6*B*a*b
*c*d**5/5 + B*b**3*d**5/5) + x**4*(5*A*a**3*d**2*e**3/2 + 15*A*a**2*b*d**3*e**2/2 + 15*A*a**2*c*d**4*e/4 + 15*
A*a*b**2*d**4*e/4 + 3*A*a*b*c*d**5/2 + A*b**3*d**5/4 + 5*B*a**3*d**3*e**2/2 + 15*B*a**2*b*d**4*e/4 + 3*B*a**2*
c*d**5/4 + 3*B*a*b**2*d**5/4) + x**3*(10*A*a**3*d**3*e**2/3 + 5*A*a**2*b*d**4*e + A*a**2*c*d**5 + A*a*b**2*d**
5 + 5*B*a**3*d**4*e/3 + B*a**2*b*d**5) + x**2*(5*A*a**3*d**4*e/2 + 3*A*a**2*b*d**5/2 + B*a**3*d**5/2)

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Giac [B]  time = 1.1143, size = 2164, normalized size = 3.9 \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)^5*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/13*B*c^3*x^13*e^5 + 5/12*B*c^3*d*x^12*e^4 + 10/11*B*c^3*d^2*x^11*e^3 + B*c^3*d^3*x^10*e^2 + 5/9*B*c^3*d^4*x^
9*e + 1/8*B*c^3*d^5*x^8 + 1/4*B*b*c^2*x^12*e^5 + 1/12*A*c^3*x^12*e^5 + 15/11*B*b*c^2*d*x^11*e^4 + 5/11*A*c^3*d
*x^11*e^4 + 3*B*b*c^2*d^2*x^10*e^3 + A*c^3*d^2*x^10*e^3 + 10/3*B*b*c^2*d^3*x^9*e^2 + 10/9*A*c^3*d^3*x^9*e^2 +
15/8*B*b*c^2*d^4*x^8*e + 5/8*A*c^3*d^4*x^8*e + 3/7*B*b*c^2*d^5*x^7 + 1/7*A*c^3*d^5*x^7 + 3/11*B*b^2*c*x^11*e^5
+ 3/11*B*a*c^2*x^11*e^5 + 3/11*A*b*c^2*x^11*e^5 + 3/2*B*b^2*c*d*x^10*e^4 + 3/2*B*a*c^2*d*x^10*e^4 + 3/2*A*b*c
^2*d*x^10*e^4 + 10/3*B*b^2*c*d^2*x^9*e^3 + 10/3*B*a*c^2*d^2*x^9*e^3 + 10/3*A*b*c^2*d^2*x^9*e^3 + 15/4*B*b^2*c*
d^3*x^8*e^2 + 15/4*B*a*c^2*d^3*x^8*e^2 + 15/4*A*b*c^2*d^3*x^8*e^2 + 15/7*B*b^2*c*d^4*x^7*e + 15/7*B*a*c^2*d^4*
x^7*e + 15/7*A*b*c^2*d^4*x^7*e + 1/2*B*b^2*c*d^5*x^6 + 1/2*B*a*c^2*d^5*x^6 + 1/2*A*b*c^2*d^5*x^6 + 1/10*B*b^3*
x^10*e^5 + 3/5*B*a*b*c*x^10*e^5 + 3/10*A*b^2*c*x^10*e^5 + 3/10*A*a*c^2*x^10*e^5 + 5/9*B*b^3*d*x^9*e^4 + 10/3*B
*a*b*c*d*x^9*e^4 + 5/3*A*b^2*c*d*x^9*e^4 + 5/3*A*a*c^2*d*x^9*e^4 + 5/4*B*b^3*d^2*x^8*e^3 + 15/2*B*a*b*c*d^2*x^
8*e^3 + 15/4*A*b^2*c*d^2*x^8*e^3 + 15/4*A*a*c^2*d^2*x^8*e^3 + 10/7*B*b^3*d^3*x^7*e^2 + 60/7*B*a*b*c*d^3*x^7*e^
2 + 30/7*A*b^2*c*d^3*x^7*e^2 + 30/7*A*a*c^2*d^3*x^7*e^2 + 5/6*B*b^3*d^4*x^6*e + 5*B*a*b*c*d^4*x^6*e + 5/2*A*b^
2*c*d^4*x^6*e + 5/2*A*a*c^2*d^4*x^6*e + 1/5*B*b^3*d^5*x^5 + 6/5*B*a*b*c*d^5*x^5 + 3/5*A*b^2*c*d^5*x^5 + 3/5*A*
a*c^2*d^5*x^5 + 1/3*B*a*b^2*x^9*e^5 + 1/9*A*b^3*x^9*e^5 + 1/3*B*a^2*c*x^9*e^5 + 2/3*A*a*b*c*x^9*e^5 + 15/8*B*a
*b^2*d*x^8*e^4 + 5/8*A*b^3*d*x^8*e^4 + 15/8*B*a^2*c*d*x^8*e^4 + 15/4*A*a*b*c*d*x^8*e^4 + 30/7*B*a*b^2*d^2*x^7*
e^3 + 10/7*A*b^3*d^2*x^7*e^3 + 30/7*B*a^2*c*d^2*x^7*e^3 + 60/7*A*a*b*c*d^2*x^7*e^3 + 5*B*a*b^2*d^3*x^6*e^2 + 5
/3*A*b^3*d^3*x^6*e^2 + 5*B*a^2*c*d^3*x^6*e^2 + 10*A*a*b*c*d^3*x^6*e^2 + 3*B*a*b^2*d^4*x^5*e + A*b^3*d^4*x^5*e
+ 3*B*a^2*c*d^4*x^5*e + 6*A*a*b*c*d^4*x^5*e + 3/4*B*a*b^2*d^5*x^4 + 1/4*A*b^3*d^5*x^4 + 3/4*B*a^2*c*d^5*x^4 +
3/2*A*a*b*c*d^5*x^4 + 3/8*B*a^2*b*x^8*e^5 + 3/8*A*a*b^2*x^8*e^5 + 3/8*A*a^2*c*x^8*e^5 + 15/7*B*a^2*b*d*x^7*e^4
+ 15/7*A*a*b^2*d*x^7*e^4 + 15/7*A*a^2*c*d*x^7*e^4 + 5*B*a^2*b*d^2*x^6*e^3 + 5*A*a*b^2*d^2*x^6*e^3 + 5*A*a^2*c
*d^2*x^6*e^3 + 6*B*a^2*b*d^3*x^5*e^2 + 6*A*a*b^2*d^3*x^5*e^2 + 6*A*a^2*c*d^3*x^5*e^2 + 15/4*B*a^2*b*d^4*x^4*e
+ 15/4*A*a*b^2*d^4*x^4*e + 15/4*A*a^2*c*d^4*x^4*e + B*a^2*b*d^5*x^3 + A*a*b^2*d^5*x^3 + A*a^2*c*d^5*x^3 + 1/7*
B*a^3*x^7*e^5 + 3/7*A*a^2*b*x^7*e^5 + 5/6*B*a^3*d*x^6*e^4 + 5/2*A*a^2*b*d*x^6*e^4 + 2*B*a^3*d^2*x^5*e^3 + 6*A*
a^2*b*d^2*x^5*e^3 + 5/2*B*a^3*d^3*x^4*e^2 + 15/2*A*a^2*b*d^3*x^4*e^2 + 5/3*B*a^3*d^4*x^3*e + 5*A*a^2*b*d^4*x^3
*e + 1/2*B*a^3*d^5*x^2 + 3/2*A*a^2*b*d^5*x^2 + 1/6*A*a^3*x^6*e^5 + A*a^3*d*x^5*e^4 + 5/2*A*a^3*d^2*x^4*e^3 + 1
0/3*A*a^3*d^3*x^3*e^2 + 5/2*A*a^3*d^4*x^2*e + A*a^3*d^5*x