3.2335 \(\int (A+B x) (d+e x)^3 (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=555 \[ -\frac{(d+e x)^7 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{7 e^8}-\frac{c (d+e x)^9 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{3 e^8}-\frac{(d+e x)^8 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{8 e^8}-\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{5 e^8}-\frac{(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8}-\frac{c^2 (d+e x)^{10} (-A c e-3 b B e+7 B c d)}{10 e^8}+\frac{B c^3 (d+e x)^{11}}{11 e^8} \]
[Out]
-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^4)/(4*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)^5)/(5*e^8) - ((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)^6)/(2*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)^7)/(7*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)^8)/(8*e^8) - (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
)^9)/(3*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^10)/(10*e^8) + (B*c^3*(d + e*x)^11)/(11*e^8)
________________________________________________________________________________________
Rubi [A] time = 1.10997, 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} \[ -\frac{(d+e x)^7 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{7 e^8}-\frac{c (d+e x)^9 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{3 e^8}-\frac{(d+e x)^8 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{8 e^8}-\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{5 e^8}-\frac{(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8}-\frac{c^2 (d+e x)^{10} (-A c e-3 b B e+7 B c d)}{10 e^8}+\frac{B c^3 (d+e x)^{11}}{11 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^3*(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)^4)/(4*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)^5)/(5*e^8) - ((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)^6)/(2*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)^7)/(7*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)^8)/(8*e^8) - (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
)^9)/(3*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^10)/(10*e^8) + (B*c^3*(d + e*x)^11)/(11*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)^3 \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)^3}{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)^4}{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)^5}{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)^6}{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)^7}{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)^8}{e^7}+\frac{c^2 (-7 B c d+3 b B e+A c e) (d+e x)^9}{e^7}+\frac{B c^3 (d+e x)^{10}}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{4 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)^5}{5 e^8}-\frac{\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)^6}{2 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)^7}{7 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)^8}{8 e^8}-\frac{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)^9}{3 e^8}-\frac{c^2 (7 B c d-3 b B e-A c e) (d+e x)^{10}}{10 e^8}+\frac{B c^3 (d+e x)^{11}}{11 e^8}\\ \end{align*}
Mathematica [A] time = 0.388154, size = 715, normalized size = 1.29 \[ \frac{1}{4} x^4 \left (A \left (a^2 e \left (a e^2+9 c d^2\right )+9 a b^2 d^2 e+3 a b d \left (3 a e^2+2 c d^2\right )+b^3 d^3\right )+3 a B d \left (3 a b d e+a \left (a e^2+c d^2\right )+b^2 d^2\right )\right )+\frac{1}{2} a^2 d^2 x^2 (3 A (a e+b d)+a B d)+a^3 A d^3 x+\frac{1}{3} c e x^9 \left (B \left (c e (a e+3 b d)+b^2 e^2+c^2 d^2\right )+A c e (b e+c d)\right )+\frac{1}{8} x^8 \left (3 A c e \left (c e (a e+3 b d)+b^2 e^2+c^2 d^2\right )+B \left (9 c^2 d e (a e+b d)+3 b c e^2 (2 a e+3 b d)+b^3 e^3+c^3 d^3\right )\right )+\frac{1}{7} x^7 \left (3 b^2 e \left (a B e^2+3 A c d e+3 B c d^2\right )+3 b c \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )+c \left (A c d \left (9 a e^2+c d^2\right )+3 a B e \left (a e^2+3 c d^2\right )\right )+b^3 e^2 (A e+3 B d)\right )+\frac{1}{2} x^6 \left (b^2 \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+A b c d \left (6 a e^2+c d^2\right )+a c \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+a b B e \left (a e^2+6 c d^2\right )+b^3 d e (A e+B d)\right )+\frac{1}{5} x^5 \left (3 b^2 d \left (3 a A e^2+3 a B d e+A c d^2\right )+3 a b \left (a A e^3+3 a B d e^2+6 A c d^2 e+2 B c d^3\right )+a \left (3 A c d \left (3 a e^2+c d^2\right )+a B e \left (a e^2+9 c d^2\right )\right )+b^3 d^2 (3 A e+B d)\right )+a d x^3 \left (A \left (3 a b d e+a \left (a e^2+c d^2\right )+b^2 d^2\right )+a B d (a e+b d)\right )+\frac{1}{10} c^2 e^2 x^{10} (A c e+3 B (b e+c d))+\frac{1}{11} B c^3 e^3 x^{11} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^3*(a + b*x + c*x^2)^3,x]
[Out]
a^3*A*d^3*x + (a^2*d^2*(a*B*d + 3*A*(b*d + a*e))*x^2)/2 + a*d*(a*B*d*(b*d + a*e) + A*(b^2*d^2 + 3*a*b*d*e + a*
(c*d^2 + a*e^2)))*x^3 + ((3*a*B*d*(b^2*d^2 + 3*a*b*d*e + a*(c*d^2 + a*e^2)) + A*(b^3*d^3 + 9*a*b^2*d^2*e + a^2
*e*(9*c*d^2 + a*e^2) + 3*a*b*d*(2*c*d^2 + 3*a*e^2)))*x^4)/4 + ((b^3*d^2*(B*d + 3*A*e) + 3*b^2*d*(A*c*d^2 + 3*a
*B*d*e + 3*a*A*e^2) + 3*a*b*(2*B*c*d^3 + 6*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3) + a*(a*B*e*(9*c*d^2 + a*e^2) + 3
*A*c*d*(c*d^2 + 3*a*e^2)))*x^5)/5 + ((b^3*d*e*(B*d + A*e) + a*b*B*e*(6*c*d^2 + a*e^2) + A*b*c*d*(c*d^2 + 6*a*e
^2) + b^2*(B*c*d^3 + 3*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3) + a*c*(B*c*d^3 + 3*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3
))*x^6)/2 + ((b^3*e^2*(3*B*d + A*e) + 3*b^2*e*(3*B*c*d^2 + 3*A*c*d*e + a*B*e^2) + 3*b*c*(B*c*d^3 + 3*A*c*d^2*e
+ 6*a*B*d*e^2 + 2*a*A*e^3) + c*(3*a*B*e*(3*c*d^2 + a*e^2) + A*c*d*(c*d^2 + 9*a*e^2)))*x^7)/7 + ((3*A*c*e*(c^2
*d^2 + b^2*e^2 + c*e*(3*b*d + a*e)) + B*(c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(b*d + a*e) + 3*b*c*e^2*(3*b*d + 2*a*e)
))*x^8)/8 + (c*e*(A*c*e*(c*d + b*e) + B*(c^2*d^2 + b^2*e^2 + c*e*(3*b*d + a*e)))*x^9)/3 + (c^2*e^2*(A*c*e + 3*
B*(c*d + b*e))*x^10)/10 + (B*c^3*e^3*x^11)/11
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Maple [A] time = 0., size = 819, normalized size = 1.5 \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)^3*(c*x^2+b*x+a)^3,x)
[Out]
1/11*B*c^3*e^3*x^11+1/10*((A*e^3+3*B*d*e^2)*c^3+3*B*e^3*b*c^2)*x^10+1/9*((3*A*d*e^2+3*B*d^2*e)*c^3+3*(A*e^3+3*
B*d*e^2)*b*c^2+B*e^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^9+1/8*((3*A*d^2*e+B*d^3)*c^3+3*(3*A*d*e^2+3*B*d^2*e)*b*c
^2+(A*e^3+3*B*d*e^2)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*e^3*(4*a*b*c+b*(2*a*c+b^2)))*x^8+1/7*(A*c^3*d^3+3*(3*A*d^
2*e+B*d^3)*b*c^2+(3*A*d*e^2+3*B*d^2*e)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(A*e^3+3*B*d*e^2)*(4*a*b*c+b*(2*a*c+b^2))
+B*e^3*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*x^7+1/6*(3*A*d^3*b*c^2+(3*A*d^2*e+B*d^3)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(
3*A*d*e^2+3*B*d^2*e)*(4*a*b*c+b*(2*a*c+b^2))+(A*e^3+3*B*d*e^2)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*B*e^3*a^2*b)*x^
6+1/5*(A*d^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(3*A*d^2*e+B*d^3)*(4*a*b*c+b*(2*a*c+b^2))+(3*A*d*e^2+3*B*d^2*e)*(a*
(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(A*e^3+3*B*d*e^2)*a^2*b+B*e^3*a^3)*x^5+1/4*(A*d^3*(4*a*b*c+b*(2*a*c+b^2))+(3*A*d^
2*e+B*d^3)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(3*A*d*e^2+3*B*d^2*e)*a^2*b+(A*e^3+3*B*d*e^2)*a^3)*x^4+1/3*(A*d^3*(
a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(3*A*d^2*e+B*d^3)*a^2*b+(3*A*d*e^2+3*B*d^2*e)*a^3)*x^3+1/2*(3*A*d^3*a^2*b+(3*A*
d^2*e+B*d^3)*a^3)*x^2+A*d^3*a^3*x
________________________________________________________________________________________
Maxima [A] time = 1.02479, size = 963, normalized size = 1.74 \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)^3*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
1/11*B*c^3*e^3*x^11 + 1/10*(3*B*c^3*d*e^2 + (3*B*b*c^2 + A*c^3)*e^3)*x^10 + 1/3*(B*c^3*d^2*e + (3*B*b*c^2 + A*
c^3)*d*e^2 + (B*b^2*c + (B*a + A*b)*c^2)*e^3)*x^9 + 1/8*(B*c^3*d^3 + 3*(3*B*b*c^2 + A*c^3)*d^2*e + 9*(B*b^2*c
+ (B*a + A*b)*c^2)*d*e^2 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^3)*x^8 + A*a^3*d^3*x + 1/7*((3*B*b*c^
2 + A*c^3)*d^3 + 9*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e + 3*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^2 + (
3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^3)*x^7 + 1/2*((B*b^2*c + (B*a + A*b)*c^2)*d^3 + (B*b^3 + 3*A*a*c^
2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^2 + (B*a^2*b + A*a*b^2 + A*
a^2*c)*e^3)*x^6 + 1/5*((B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A
*a*b)*c)*d^2*e + 9*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^2 + (B*a^3 + 3*A*a^2*b)*e^3)*x^5 + 1/4*(A*a^3*e^3 + (3*B*
a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3 + 9*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e + 3*(B*a^3 + 3*A*a^2*b)*d*e
^2)*x^4 + (A*a^3*d*e^2 + (B*a^2*b + A*a*b^2 + A*a^2*c)*d^3 + (B*a^3 + 3*A*a^2*b)*d^2*e)*x^3 + 1/2*(3*A*a^3*d^2
*e + (B*a^3 + 3*A*a^2*b)*d^3)*x^2
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Fricas [A] time = 0.86437, size = 2383, normalized size = 4.29 \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)^3*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
1/11*x^11*e^3*c^3*B + 3/10*x^10*e^2*d*c^3*B + 3/10*x^10*e^3*c^2*b*B + 1/10*x^10*e^3*c^3*A + 1/3*x^9*e*d^2*c^3*
B + x^9*e^2*d*c^2*b*B + 1/3*x^9*e^3*c*b^2*B + 1/3*x^9*e^3*c^2*a*B + 1/3*x^9*e^2*d*c^3*A + 1/3*x^9*e^3*c^2*b*A
+ 1/8*x^8*d^3*c^3*B + 9/8*x^8*e*d^2*c^2*b*B + 9/8*x^8*e^2*d*c*b^2*B + 1/8*x^8*e^3*b^3*B + 9/8*x^8*e^2*d*c^2*a*
B + 3/4*x^8*e^3*c*b*a*B + 3/8*x^8*e*d^2*c^3*A + 9/8*x^8*e^2*d*c^2*b*A + 3/8*x^8*e^3*c*b^2*A + 3/8*x^8*e^3*c^2*
a*A + 3/7*x^7*d^3*c^2*b*B + 9/7*x^7*e*d^2*c*b^2*B + 3/7*x^7*e^2*d*b^3*B + 9/7*x^7*e*d^2*c^2*a*B + 18/7*x^7*e^2
*d*c*b*a*B + 3/7*x^7*e^3*b^2*a*B + 3/7*x^7*e^3*c*a^2*B + 1/7*x^7*d^3*c^3*A + 9/7*x^7*e*d^2*c^2*b*A + 9/7*x^7*e
^2*d*c*b^2*A + 1/7*x^7*e^3*b^3*A + 9/7*x^7*e^2*d*c^2*a*A + 6/7*x^7*e^3*c*b*a*A + 1/2*x^6*d^3*c*b^2*B + 1/2*x^6
*e*d^2*b^3*B + 1/2*x^6*d^3*c^2*a*B + 3*x^6*e*d^2*c*b*a*B + 3/2*x^6*e^2*d*b^2*a*B + 3/2*x^6*e^2*d*c*a^2*B + 1/2
*x^6*e^3*b*a^2*B + 1/2*x^6*d^3*c^2*b*A + 3/2*x^6*e*d^2*c*b^2*A + 1/2*x^6*e^2*d*b^3*A + 3/2*x^6*e*d^2*c^2*a*A +
3*x^6*e^2*d*c*b*a*A + 1/2*x^6*e^3*b^2*a*A + 1/2*x^6*e^3*c*a^2*A + 1/5*x^5*d^3*b^3*B + 6/5*x^5*d^3*c*b*a*B + 9
/5*x^5*e*d^2*b^2*a*B + 9/5*x^5*e*d^2*c*a^2*B + 9/5*x^5*e^2*d*b*a^2*B + 1/5*x^5*e^3*a^3*B + 3/5*x^5*d^3*c*b^2*A
+ 3/5*x^5*e*d^2*b^3*A + 3/5*x^5*d^3*c^2*a*A + 18/5*x^5*e*d^2*c*b*a*A + 9/5*x^5*e^2*d*b^2*a*A + 9/5*x^5*e^2*d*
c*a^2*A + 3/5*x^5*e^3*b*a^2*A + 3/4*x^4*d^3*b^2*a*B + 3/4*x^4*d^3*c*a^2*B + 9/4*x^4*e*d^2*b*a^2*B + 3/4*x^4*e^
2*d*a^3*B + 1/4*x^4*d^3*b^3*A + 3/2*x^4*d^3*c*b*a*A + 9/4*x^4*e*d^2*b^2*a*A + 9/4*x^4*e*d^2*c*a^2*A + 9/4*x^4*
e^2*d*b*a^2*A + 1/4*x^4*e^3*a^3*A + x^3*d^3*b*a^2*B + x^3*e*d^2*a^3*B + x^3*d^3*b^2*a*A + x^3*d^3*c*a^2*A + 3*
x^3*e*d^2*b*a^2*A + x^3*e^2*d*a^3*A + 1/2*x^2*d^3*a^3*B + 3/2*x^2*d^3*b*a^2*A + 3/2*x^2*e*d^2*a^3*A + x*d^3*a^
3*A
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Sympy [B] time = 0.189384, size = 1080, normalized size = 1.95 \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)**3*(c*x**2+b*x+a)**3,x)
[Out]
A*a**3*d**3*x + B*c**3*e**3*x**11/11 + x**10*(A*c**3*e**3/10 + 3*B*b*c**2*e**3/10 + 3*B*c**3*d*e**2/10) + x**9
*(A*b*c**2*e**3/3 + A*c**3*d*e**2/3 + B*a*c**2*e**3/3 + B*b**2*c*e**3/3 + B*b*c**2*d*e**2 + B*c**3*d**2*e/3) +
x**8*(3*A*a*c**2*e**3/8 + 3*A*b**2*c*e**3/8 + 9*A*b*c**2*d*e**2/8 + 3*A*c**3*d**2*e/8 + 3*B*a*b*c*e**3/4 + 9*
B*a*c**2*d*e**2/8 + B*b**3*e**3/8 + 9*B*b**2*c*d*e**2/8 + 9*B*b*c**2*d**2*e/8 + B*c**3*d**3/8) + x**7*(6*A*a*b
*c*e**3/7 + 9*A*a*c**2*d*e**2/7 + A*b**3*e**3/7 + 9*A*b**2*c*d*e**2/7 + 9*A*b*c**2*d**2*e/7 + A*c**3*d**3/7 +
3*B*a**2*c*e**3/7 + 3*B*a*b**2*e**3/7 + 18*B*a*b*c*d*e**2/7 + 9*B*a*c**2*d**2*e/7 + 3*B*b**3*d*e**2/7 + 9*B*b*
*2*c*d**2*e/7 + 3*B*b*c**2*d**3/7) + x**6*(A*a**2*c*e**3/2 + A*a*b**2*e**3/2 + 3*A*a*b*c*d*e**2 + 3*A*a*c**2*d
**2*e/2 + A*b**3*d*e**2/2 + 3*A*b**2*c*d**2*e/2 + A*b*c**2*d**3/2 + B*a**2*b*e**3/2 + 3*B*a**2*c*d*e**2/2 + 3*
B*a*b**2*d*e**2/2 + 3*B*a*b*c*d**2*e + B*a*c**2*d**3/2 + B*b**3*d**2*e/2 + B*b**2*c*d**3/2) + x**5*(3*A*a**2*b
*e**3/5 + 9*A*a**2*c*d*e**2/5 + 9*A*a*b**2*d*e**2/5 + 18*A*a*b*c*d**2*e/5 + 3*A*a*c**2*d**3/5 + 3*A*b**3*d**2*
e/5 + 3*A*b**2*c*d**3/5 + B*a**3*e**3/5 + 9*B*a**2*b*d*e**2/5 + 9*B*a**2*c*d**2*e/5 + 9*B*a*b**2*d**2*e/5 + 6*
B*a*b*c*d**3/5 + B*b**3*d**3/5) + x**4*(A*a**3*e**3/4 + 9*A*a**2*b*d*e**2/4 + 9*A*a**2*c*d**2*e/4 + 9*A*a*b**2
*d**2*e/4 + 3*A*a*b*c*d**3/2 + A*b**3*d**3/4 + 3*B*a**3*d*e**2/4 + 9*B*a**2*b*d**2*e/4 + 3*B*a**2*c*d**3/4 + 3
*B*a*b**2*d**3/4) + x**3*(A*a**3*d*e**2 + 3*A*a**2*b*d**2*e + A*a**2*c*d**3 + A*a*b**2*d**3 + B*a**3*d**2*e +
B*a**2*b*d**3) + x**2*(3*A*a**3*d**2*e/2 + 3*A*a**2*b*d**3/2 + B*a**3*d**3/2)
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Giac [A] time = 1.10685, size = 1377, normalized size = 2.48 \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)^3*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
1/11*B*c^3*x^11*e^3 + 3/10*B*c^3*d*x^10*e^2 + 1/3*B*c^3*d^2*x^9*e + 1/8*B*c^3*d^3*x^8 + 3/10*B*b*c^2*x^10*e^3
+ 1/10*A*c^3*x^10*e^3 + B*b*c^2*d*x^9*e^2 + 1/3*A*c^3*d*x^9*e^2 + 9/8*B*b*c^2*d^2*x^8*e + 3/8*A*c^3*d^2*x^8*e
+ 3/7*B*b*c^2*d^3*x^7 + 1/7*A*c^3*d^3*x^7 + 1/3*B*b^2*c*x^9*e^3 + 1/3*B*a*c^2*x^9*e^3 + 1/3*A*b*c^2*x^9*e^3 +
9/8*B*b^2*c*d*x^8*e^2 + 9/8*B*a*c^2*d*x^8*e^2 + 9/8*A*b*c^2*d*x^8*e^2 + 9/7*B*b^2*c*d^2*x^7*e + 9/7*B*a*c^2*d^
2*x^7*e + 9/7*A*b*c^2*d^2*x^7*e + 1/2*B*b^2*c*d^3*x^6 + 1/2*B*a*c^2*d^3*x^6 + 1/2*A*b*c^2*d^3*x^6 + 1/8*B*b^3*
x^8*e^3 + 3/4*B*a*b*c*x^8*e^3 + 3/8*A*b^2*c*x^8*e^3 + 3/8*A*a*c^2*x^8*e^3 + 3/7*B*b^3*d*x^7*e^2 + 18/7*B*a*b*c
*d*x^7*e^2 + 9/7*A*b^2*c*d*x^7*e^2 + 9/7*A*a*c^2*d*x^7*e^2 + 1/2*B*b^3*d^2*x^6*e + 3*B*a*b*c*d^2*x^6*e + 3/2*A
*b^2*c*d^2*x^6*e + 3/2*A*a*c^2*d^2*x^6*e + 1/5*B*b^3*d^3*x^5 + 6/5*B*a*b*c*d^3*x^5 + 3/5*A*b^2*c*d^3*x^5 + 3/5
*A*a*c^2*d^3*x^5 + 3/7*B*a*b^2*x^7*e^3 + 1/7*A*b^3*x^7*e^3 + 3/7*B*a^2*c*x^7*e^3 + 6/7*A*a*b*c*x^7*e^3 + 3/2*B
*a*b^2*d*x^6*e^2 + 1/2*A*b^3*d*x^6*e^2 + 3/2*B*a^2*c*d*x^6*e^2 + 3*A*a*b*c*d*x^6*e^2 + 9/5*B*a*b^2*d^2*x^5*e +
3/5*A*b^3*d^2*x^5*e + 9/5*B*a^2*c*d^2*x^5*e + 18/5*A*a*b*c*d^2*x^5*e + 3/4*B*a*b^2*d^3*x^4 + 1/4*A*b^3*d^3*x^
4 + 3/4*B*a^2*c*d^3*x^4 + 3/2*A*a*b*c*d^3*x^4 + 1/2*B*a^2*b*x^6*e^3 + 1/2*A*a*b^2*x^6*e^3 + 1/2*A*a^2*c*x^6*e^
3 + 9/5*B*a^2*b*d*x^5*e^2 + 9/5*A*a*b^2*d*x^5*e^2 + 9/5*A*a^2*c*d*x^5*e^2 + 9/4*B*a^2*b*d^2*x^4*e + 9/4*A*a*b^
2*d^2*x^4*e + 9/4*A*a^2*c*d^2*x^4*e + B*a^2*b*d^3*x^3 + A*a*b^2*d^3*x^3 + A*a^2*c*d^3*x^3 + 1/5*B*a^3*x^5*e^3
+ 3/5*A*a^2*b*x^5*e^3 + 3/4*B*a^3*d*x^4*e^2 + 9/4*A*a^2*b*d*x^4*e^2 + B*a^3*d^2*x^3*e + 3*A*a^2*b*d^2*x^3*e +
1/2*B*a^3*d^3*x^2 + 3/2*A*a^2*b*d^3*x^2 + 1/4*A*a^3*x^4*e^3 + A*a^3*d*x^3*e^2 + 3/2*A*a^3*d^2*x^2*e + A*a^3*d^
3*x