3.2334 \(\int (A+B x) (d+e x)^4 (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=555 \[ -\frac{(d+e x)^8 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{8 e^8}-\frac{3 c (d+e x)^{10} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{10 e^8}-\frac{(d+e x)^9 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{9 e^8}-\frac{3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{6 e^8}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{c^2 (d+e x)^{11} (-A c e-3 b B e+7 B c d)}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8} \]
[Out]
-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*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)^6)/(6*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)^7)/(7*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)^8)/(8*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)^9)/(9*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)^10)/(10*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*e^8)
________________________________________________________________________________________
Rubi [A] time = 1.37815, 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} \[ -\frac{(d+e x)^8 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{8 e^8}-\frac{3 c (d+e x)^{10} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{10 e^8}-\frac{(d+e x)^9 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{9 e^8}-\frac{3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{c^2 (d+e x)^{11} (-A c e-3 b B e+7 B c d)}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^4*(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)^5)/(5*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)^6)/(6*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)^7)/(7*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)^8)/(8*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)^9)/(9*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)^10)/(10*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*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)^4 \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)^4}{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)^5}{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)^6}{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)^7}{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)^8}{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)^9}{e^7}+\frac{c^2 (-7 B c d+3 b B e+A c e) (d+e x)^{10}}{e^7}+\frac{B c^3 (d+e x)^{11}}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 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)^6}{6 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)^7}{7 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)^8}{8 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)^9}{9 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)^{10}}{10 e^8}-\frac{c^2 (7 B c d-3 b B e-A c e) (d+e x)^{11}}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8}\\ \end{align*}
Mathematica [A] time = 0.52126, size = 957, normalized size = 1.72 \[ \frac{1}{12} B c^3 e^4 x^{12}+\frac{1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac{1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )\right ) x^{10}+\frac{1}{9} e \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )+B \left (4 c^3 d^3+6 c^2 e (3 b d+2 a e) d+b^3 e^3+6 b c e^2 (2 b d+a e)\right )\right ) x^9+\frac{1}{8} \left (A e \left (4 c^3 d^3+6 c^2 e (3 b d+2 a e) d+b^3 e^3+6 b c e^2 (2 b d+a e)\right )+B \left (c^3 d^4+6 c^2 e (2 b d+3 a e) d^2+b^2 e^3 (4 b d+3 a e)+3 c e^2 \left (6 b^2 d^2+8 a b e d+a^2 e^2\right )\right )\right ) x^8+\frac{1}{7} \left (2 d e^2 (3 B d+2 A e) b^3+3 e \left (4 B c d^3+6 A c e d^2+4 a B e^2 d+a A e^3\right ) b^2+12 A c d e \left (c d^2+2 a e^2\right ) b+3 B \left (c^2 d^4+12 a c e^2 d^2+a^2 e^4\right ) b+12 a B c d e \left (c d^2+a e^2\right )+A c \left (c^2 d^4+18 a c e^2 d^2+3 a^2 e^4\right )\right ) x^7+\frac{1}{6} \left (2 d^2 e (2 B d+3 A e) b^3+3 d \left (B c d^3+4 A c e d^2+6 a B e^2 d+4 a A e^3\right ) b^2+12 a B d e \left (2 c d^2+a e^2\right ) b+3 A \left (c^2 d^4+12 a c e^2 d^2+a^2 e^4\right ) b+12 a A c d e \left (c d^2+a e^2\right )+a B \left (3 c^2 d^4+18 a c e^2 d^2+a^2 e^4\right )\right ) x^6+\frac{1}{5} \left (b^3 (B d+4 A e) d^3+3 b^2 \left (A c d^2+4 a B e d+6 a A e^2\right ) d^2+6 a b \left (B c d^3+4 A c e d^2+3 a B e^2 d+2 a A e^3\right ) d+a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c e^2 d^2+a^2 e^4\right )\right )\right ) x^5+\frac{1}{4} d \left (3 a B d \left (b^2 d^2+4 a b e d+a \left (c d^2+2 a e^2\right )\right )+A \left (b^3 d^3+12 a b^2 e d^2+6 a b \left (c d^2+3 a e^2\right ) d+4 a^2 e \left (3 c d^2+a e^2\right )\right )\right ) x^4+\frac{1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b e d+a \left (c d^2+2 a e^2\right )\right )\right ) x^3+\frac{1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+a^3 A d^4 x \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x]
[Out]
a^3*A*d^4*x + (a^2*d^3*(3*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a*d^2*(a*B*d*(3*b*d + 4*a*e) + 3*A*(b^2*d^2 + 4*a
*b*d*e + a*(c*d^2 + 2*a*e^2)))*x^3)/3 + (d*(3*a*B*d*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2)) + A*(b^3*d^3 +
12*a*b^2*d^2*e + 4*a^2*e*(3*c*d^2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2)))*x^4)/4 + ((b^3*d^3*(B*d + 4*A*e) + 3
*b^2*d^2*(A*c*d^2 + 4*a*B*d*e + 6*a*A*e^2) + 6*a*b*d*(B*c*d^3 + 4*A*c*d^2*e + 3*a*B*d*e^2 + 2*a*A*e^3) + a*(4*
a*B*d*e*(3*c*d^2 + a*e^2) + A*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4)))*x^5)/5 + ((2*b^3*d^2*e*(2*B*d + 3*A*e)
+ 12*a*A*c*d*e*(c*d^2 + a*e^2) + 12*a*b*B*d*e*(2*c*d^2 + a*e^2) + 3*b^2*d*(B*c*d^3 + 4*A*c*d^2*e + 6*a*B*d*e^2
+ 4*a*A*e^3) + 3*A*b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + a*B*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^6)/
6 + ((2*b^3*d*e^2*(3*B*d + 2*A*e) + 12*a*B*c*d*e*(c*d^2 + a*e^2) + 12*A*b*c*d*e*(c*d^2 + 2*a*e^2) + 3*b^2*e*(4
*B*c*d^3 + 6*A*c*d^2*e + 4*a*B*d*e^2 + a*A*e^3) + 3*b*B*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + A*c*(c^2*d^4 +
18*a*c*d^2*e^2 + 3*a^2*e^4))*x^7)/7 + ((A*e*(4*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d
+ 2*a*e)) + B*(c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3*c*e^2*(6*b^2*d^2 + 8*a*b*d*
e + a^2*e^2)))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e)) + B*(4*c^3*d^3 + b^3*e^3 + 6*b*c
*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e)))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d^2 + b^2
*e^2 + c*e*(4*b*d + a*e)))*x^10)/10 + (c^2*e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12
________________________________________________________________________________________
Maple [A] time = 0.001, size = 1041, normalized size = 1.9 \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)^4*(c*x^2+b*x+a)^3,x)
[Out]
1/12*B*c^3*e^4*x^12+1/11*((A*e^4+4*B*d*e^3)*c^3+3*B*e^4*b*c^2)*x^11+1/10*((4*A*d*e^3+6*B*d^2*e^2)*c^3+3*(A*e^4
+4*B*d*e^3)*b*c^2+B*e^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(4*A*d*e^3+6*B*
d^2*e^2)*b*c^2+(A*e^4+4*B*d*e^3)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*e^4*(4*a*b*c+b*(2*a*c+b^2)))*x^9+1/8*((4*A*d^
3*e+B*d^4)*c^3+3*(6*A*d^2*e^2+4*B*d^3*e)*b*c^2+(4*A*d*e^3+6*B*d^2*e^2)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(A*e^4+4*
B*d*e^3)*(4*a*b*c+b*(2*a*c+b^2))+B*e^4*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*x^8+1/7*(A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b
*c^2+(6*A*d^2*e^2+4*B*d^3*e)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(4*a*b*c+b*(2*a*c+b^2))+(A*
e^4+4*B*d*e^3)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*B*e^4*a^2*b)*x^7+1/6*(3*A*d^4*b*c^2+(4*A*d^3*e+B*d^4)*(a*c^2+2*
b^2*c+c*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^3*e)*(4*a*b*c+b*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(a*(2*a*c+b^2)+2*
b^2*a+c*a^2)+3*(A*e^4+4*B*d*e^3)*a^2*b+B*e^4*a^3)*x^6+1/5*(A*d^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(4*A*d^3*e+B*d^
4)*(4*a*b*c+b*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^3*e)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(4*A*d*e^3+6*B*d^2*e^2)*a^2
*b+(A*e^4+4*B*d*e^3)*a^3)*x^5+1/4*(A*d^4*(4*a*b*c+b*(2*a*c+b^2))+(4*A*d^3*e+B*d^4)*(a*(2*a*c+b^2)+2*b^2*a+c*a^
2)+3*(6*A*d^2*e^2+4*B*d^3*e)*a^2*b+(4*A*d*e^3+6*B*d^2*e^2)*a^3)*x^4+1/3*(A*d^4*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3
*(4*A*d^3*e+B*d^4)*a^2*b+(6*A*d^2*e^2+4*B*d^3*e)*a^3)*x^3+1/2*(3*A*d^4*a^2*b+(4*A*d^3*e+B*d^4)*a^3)*x^2+A*d^4*
a^3*x
________________________________________________________________________________________
Maxima [A] time = 1.05571, size = 1241, normalized size = 2.24 \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)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 + 1/10*(6*B*c^3*d^2*e^2 + 4*(3*B*b*c
^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 + A*c^3)*d^2*e
^2 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^4)*x^9 + A*a^3*d^4*x
+ 1/8*(B*c^3*d^4 + 4*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*a*c^
2 + 3*(2*B*a*b + A*b^2)*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^4)*x^8 + 1/7*((3*B*b*c^2 + A*
c^3)*d^4 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e + 6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^2 + 4*(3
*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^3 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^4)*x^7 + 1/6*(3*(B*b^2*c +
(B*a + A*b)*c^2)*d^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e + 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2
+ 2*A*a*b)*c)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x^6 + 1/5*(A*a^3*e^4
+ (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e + 18*
(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A*a^3*d*e^3 + (3*B*a*b^2 + A
*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^4 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^
4 + 1/3*(6*A*a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^4 + 4*(B*a^3 + 3*A*a^2*b)*d^3*e)*x^3 + 1/2*(4*A*a
^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
________________________________________________________________________________________
Fricas [B] time = 0.901083, size = 3067, normalized size = 5.53 \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)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
1/12*x^12*e^4*c^3*B + 4/11*x^11*e^3*d*c^3*B + 3/11*x^11*e^4*c^2*b*B + 1/11*x^11*e^4*c^3*A + 3/5*x^10*e^2*d^2*c
^3*B + 6/5*x^10*e^3*d*c^2*b*B + 3/10*x^10*e^4*c*b^2*B + 3/10*x^10*e^4*c^2*a*B + 2/5*x^10*e^3*d*c^3*A + 3/10*x^
10*e^4*c^2*b*A + 4/9*x^9*e*d^3*c^3*B + 2*x^9*e^2*d^2*c^2*b*B + 4/3*x^9*e^3*d*c*b^2*B + 1/9*x^9*e^4*b^3*B + 4/3
*x^9*e^3*d*c^2*a*B + 2/3*x^9*e^4*c*b*a*B + 2/3*x^9*e^2*d^2*c^3*A + 4/3*x^9*e^3*d*c^2*b*A + 1/3*x^9*e^4*c*b^2*A
+ 1/3*x^9*e^4*c^2*a*A + 1/8*x^8*d^4*c^3*B + 3/2*x^8*e*d^3*c^2*b*B + 9/4*x^8*e^2*d^2*c*b^2*B + 1/2*x^8*e^3*d*b
^3*B + 9/4*x^8*e^2*d^2*c^2*a*B + 3*x^8*e^3*d*c*b*a*B + 3/8*x^8*e^4*b^2*a*B + 3/8*x^8*e^4*c*a^2*B + 1/2*x^8*e*d
^3*c^3*A + 9/4*x^8*e^2*d^2*c^2*b*A + 3/2*x^8*e^3*d*c*b^2*A + 1/8*x^8*e^4*b^3*A + 3/2*x^8*e^3*d*c^2*a*A + 3/4*x
^8*e^4*c*b*a*A + 3/7*x^7*d^4*c^2*b*B + 12/7*x^7*e*d^3*c*b^2*B + 6/7*x^7*e^2*d^2*b^3*B + 12/7*x^7*e*d^3*c^2*a*B
+ 36/7*x^7*e^2*d^2*c*b*a*B + 12/7*x^7*e^3*d*b^2*a*B + 12/7*x^7*e^3*d*c*a^2*B + 3/7*x^7*e^4*b*a^2*B + 1/7*x^7*
d^4*c^3*A + 12/7*x^7*e*d^3*c^2*b*A + 18/7*x^7*e^2*d^2*c*b^2*A + 4/7*x^7*e^3*d*b^3*A + 18/7*x^7*e^2*d^2*c^2*a*A
+ 24/7*x^7*e^3*d*c*b*a*A + 3/7*x^7*e^4*b^2*a*A + 3/7*x^7*e^4*c*a^2*A + 1/2*x^6*d^4*c*b^2*B + 2/3*x^6*e*d^3*b^
3*B + 1/2*x^6*d^4*c^2*a*B + 4*x^6*e*d^3*c*b*a*B + 3*x^6*e^2*d^2*b^2*a*B + 3*x^6*e^2*d^2*c*a^2*B + 2*x^6*e^3*d*
b*a^2*B + 1/6*x^6*e^4*a^3*B + 1/2*x^6*d^4*c^2*b*A + 2*x^6*e*d^3*c*b^2*A + x^6*e^2*d^2*b^3*A + 2*x^6*e*d^3*c^2*
a*A + 6*x^6*e^2*d^2*c*b*a*A + 2*x^6*e^3*d*b^2*a*A + 2*x^6*e^3*d*c*a^2*A + 1/2*x^6*e^4*b*a^2*A + 1/5*x^5*d^4*b^
3*B + 6/5*x^5*d^4*c*b*a*B + 12/5*x^5*e*d^3*b^2*a*B + 12/5*x^5*e*d^3*c*a^2*B + 18/5*x^5*e^2*d^2*b*a^2*B + 4/5*x
^5*e^3*d*a^3*B + 3/5*x^5*d^4*c*b^2*A + 4/5*x^5*e*d^3*b^3*A + 3/5*x^5*d^4*c^2*a*A + 24/5*x^5*e*d^3*c*b*a*A + 18
/5*x^5*e^2*d^2*b^2*a*A + 18/5*x^5*e^2*d^2*c*a^2*A + 12/5*x^5*e^3*d*b*a^2*A + 1/5*x^5*e^4*a^3*A + 3/4*x^4*d^4*b
^2*a*B + 3/4*x^4*d^4*c*a^2*B + 3*x^4*e*d^3*b*a^2*B + 3/2*x^4*e^2*d^2*a^3*B + 1/4*x^4*d^4*b^3*A + 3/2*x^4*d^4*c
*b*a*A + 3*x^4*e*d^3*b^2*a*A + 3*x^4*e*d^3*c*a^2*A + 9/2*x^4*e^2*d^2*b*a^2*A + x^4*e^3*d*a^3*A + x^3*d^4*b*a^2
*B + 4/3*x^3*e*d^3*a^3*B + x^3*d^4*b^2*a*A + x^3*d^4*c*a^2*A + 4*x^3*e*d^3*b*a^2*A + 2*x^3*e^2*d^2*a^3*A + 1/2
*x^2*d^4*a^3*B + 3/2*x^2*d^4*b*a^2*A + 2*x^2*e*d^3*a^3*A + x*d^4*a^3*A
________________________________________________________________________________________
Sympy [B] time = 0.223935, size = 1401, normalized size = 2.52 \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)**4*(c*x**2+b*x+a)**3,x)
[Out]
A*a**3*d**4*x + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 3*B*b*c**2*e**4/11 + 4*B*c**3*d*e**3/11) + x**1
0*(3*A*b*c**2*e**4/10 + 2*A*c**3*d*e**3/5 + 3*B*a*c**2*e**4/10 + 3*B*b**2*c*e**4/10 + 6*B*b*c**2*d*e**3/5 + 3*
B*c**3*d**2*e**2/5) + x**9*(A*a*c**2*e**4/3 + A*b**2*c*e**4/3 + 4*A*b*c**2*d*e**3/3 + 2*A*c**3*d**2*e**2/3 + 2
*B*a*b*c*e**4/3 + 4*B*a*c**2*d*e**3/3 + B*b**3*e**4/9 + 4*B*b**2*c*d*e**3/3 + 2*B*b*c**2*d**2*e**2 + 4*B*c**3*
d**3*e/9) + x**8*(3*A*a*b*c*e**4/4 + 3*A*a*c**2*d*e**3/2 + A*b**3*e**4/8 + 3*A*b**2*c*d*e**3/2 + 9*A*b*c**2*d*
*2*e**2/4 + A*c**3*d**3*e/2 + 3*B*a**2*c*e**4/8 + 3*B*a*b**2*e**4/8 + 3*B*a*b*c*d*e**3 + 9*B*a*c**2*d**2*e**2/
4 + B*b**3*d*e**3/2 + 9*B*b**2*c*d**2*e**2/4 + 3*B*b*c**2*d**3*e/2 + B*c**3*d**4/8) + x**7*(3*A*a**2*c*e**4/7
+ 3*A*a*b**2*e**4/7 + 24*A*a*b*c*d*e**3/7 + 18*A*a*c**2*d**2*e**2/7 + 4*A*b**3*d*e**3/7 + 18*A*b**2*c*d**2*e**
2/7 + 12*A*b*c**2*d**3*e/7 + A*c**3*d**4/7 + 3*B*a**2*b*e**4/7 + 12*B*a**2*c*d*e**3/7 + 12*B*a*b**2*d*e**3/7 +
36*B*a*b*c*d**2*e**2/7 + 12*B*a*c**2*d**3*e/7 + 6*B*b**3*d**2*e**2/7 + 12*B*b**2*c*d**3*e/7 + 3*B*b*c**2*d**4
/7) + x**6*(A*a**2*b*e**4/2 + 2*A*a**2*c*d*e**3 + 2*A*a*b**2*d*e**3 + 6*A*a*b*c*d**2*e**2 + 2*A*a*c**2*d**3*e
+ A*b**3*d**2*e**2 + 2*A*b**2*c*d**3*e + A*b*c**2*d**4/2 + B*a**3*e**4/6 + 2*B*a**2*b*d*e**3 + 3*B*a**2*c*d**2
*e**2 + 3*B*a*b**2*d**2*e**2 + 4*B*a*b*c*d**3*e + B*a*c**2*d**4/2 + 2*B*b**3*d**3*e/3 + B*b**2*c*d**4/2) + x**
5*(A*a**3*e**4/5 + 12*A*a**2*b*d*e**3/5 + 18*A*a**2*c*d**2*e**2/5 + 18*A*a*b**2*d**2*e**2/5 + 24*A*a*b*c*d**3*
e/5 + 3*A*a*c**2*d**4/5 + 4*A*b**3*d**3*e/5 + 3*A*b**2*c*d**4/5 + 4*B*a**3*d*e**3/5 + 18*B*a**2*b*d**2*e**2/5
+ 12*B*a**2*c*d**3*e/5 + 12*B*a*b**2*d**3*e/5 + 6*B*a*b*c*d**4/5 + B*b**3*d**4/5) + x**4*(A*a**3*d*e**3 + 9*A*
a**2*b*d**2*e**2/2 + 3*A*a**2*c*d**3*e + 3*A*a*b**2*d**3*e + 3*A*a*b*c*d**4/2 + A*b**3*d**4/4 + 3*B*a**3*d**2*
e**2/2 + 3*B*a**2*b*d**3*e + 3*B*a**2*c*d**4/4 + 3*B*a*b**2*d**4/4) + x**3*(2*A*a**3*d**2*e**2 + 4*A*a**2*b*d*
*3*e + A*a**2*c*d**4 + A*a*b**2*d**4 + 4*B*a**3*d**3*e/3 + B*a**2*b*d**4) + x**2*(2*A*a**3*d**3*e + 3*A*a**2*b
*d**4/2 + B*a**3*d**4/2)
________________________________________________________________________________________
Giac [B] time = 1.11946, size = 1773, normalized size = 3.19 \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)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
1/12*B*c^3*x^12*e^4 + 4/11*B*c^3*d*x^11*e^3 + 3/5*B*c^3*d^2*x^10*e^2 + 4/9*B*c^3*d^3*x^9*e + 1/8*B*c^3*d^4*x^8
+ 3/11*B*b*c^2*x^11*e^4 + 1/11*A*c^3*x^11*e^4 + 6/5*B*b*c^2*d*x^10*e^3 + 2/5*A*c^3*d*x^10*e^3 + 2*B*b*c^2*d^2
*x^9*e^2 + 2/3*A*c^3*d^2*x^9*e^2 + 3/2*B*b*c^2*d^3*x^8*e + 1/2*A*c^3*d^3*x^8*e + 3/7*B*b*c^2*d^4*x^7 + 1/7*A*c
^3*d^4*x^7 + 3/10*B*b^2*c*x^10*e^4 + 3/10*B*a*c^2*x^10*e^4 + 3/10*A*b*c^2*x^10*e^4 + 4/3*B*b^2*c*d*x^9*e^3 + 4
/3*B*a*c^2*d*x^9*e^3 + 4/3*A*b*c^2*d*x^9*e^3 + 9/4*B*b^2*c*d^2*x^8*e^2 + 9/4*B*a*c^2*d^2*x^8*e^2 + 9/4*A*b*c^2
*d^2*x^8*e^2 + 12/7*B*b^2*c*d^3*x^7*e + 12/7*B*a*c^2*d^3*x^7*e + 12/7*A*b*c^2*d^3*x^7*e + 1/2*B*b^2*c*d^4*x^6
+ 1/2*B*a*c^2*d^4*x^6 + 1/2*A*b*c^2*d^4*x^6 + 1/9*B*b^3*x^9*e^4 + 2/3*B*a*b*c*x^9*e^4 + 1/3*A*b^2*c*x^9*e^4 +
1/3*A*a*c^2*x^9*e^4 + 1/2*B*b^3*d*x^8*e^3 + 3*B*a*b*c*d*x^8*e^3 + 3/2*A*b^2*c*d*x^8*e^3 + 3/2*A*a*c^2*d*x^8*e^
3 + 6/7*B*b^3*d^2*x^7*e^2 + 36/7*B*a*b*c*d^2*x^7*e^2 + 18/7*A*b^2*c*d^2*x^7*e^2 + 18/7*A*a*c^2*d^2*x^7*e^2 + 2
/3*B*b^3*d^3*x^6*e + 4*B*a*b*c*d^3*x^6*e + 2*A*b^2*c*d^3*x^6*e + 2*A*a*c^2*d^3*x^6*e + 1/5*B*b^3*d^4*x^5 + 6/5
*B*a*b*c*d^4*x^5 + 3/5*A*b^2*c*d^4*x^5 + 3/5*A*a*c^2*d^4*x^5 + 3/8*B*a*b^2*x^8*e^4 + 1/8*A*b^3*x^8*e^4 + 3/8*B
*a^2*c*x^8*e^4 + 3/4*A*a*b*c*x^8*e^4 + 12/7*B*a*b^2*d*x^7*e^3 + 4/7*A*b^3*d*x^7*e^3 + 12/7*B*a^2*c*d*x^7*e^3 +
24/7*A*a*b*c*d*x^7*e^3 + 3*B*a*b^2*d^2*x^6*e^2 + A*b^3*d^2*x^6*e^2 + 3*B*a^2*c*d^2*x^6*e^2 + 6*A*a*b*c*d^2*x^
6*e^2 + 12/5*B*a*b^2*d^3*x^5*e + 4/5*A*b^3*d^3*x^5*e + 12/5*B*a^2*c*d^3*x^5*e + 24/5*A*a*b*c*d^3*x^5*e + 3/4*B
*a*b^2*d^4*x^4 + 1/4*A*b^3*d^4*x^4 + 3/4*B*a^2*c*d^4*x^4 + 3/2*A*a*b*c*d^4*x^4 + 3/7*B*a^2*b*x^7*e^4 + 3/7*A*a
*b^2*x^7*e^4 + 3/7*A*a^2*c*x^7*e^4 + 2*B*a^2*b*d*x^6*e^3 + 2*A*a*b^2*d*x^6*e^3 + 2*A*a^2*c*d*x^6*e^3 + 18/5*B*
a^2*b*d^2*x^5*e^2 + 18/5*A*a*b^2*d^2*x^5*e^2 + 18/5*A*a^2*c*d^2*x^5*e^2 + 3*B*a^2*b*d^3*x^4*e + 3*A*a*b^2*d^3*
x^4*e + 3*A*a^2*c*d^3*x^4*e + B*a^2*b*d^4*x^3 + A*a*b^2*d^4*x^3 + A*a^2*c*d^4*x^3 + 1/6*B*a^3*x^6*e^4 + 1/2*A*
a^2*b*x^6*e^4 + 4/5*B*a^3*d*x^5*e^3 + 12/5*A*a^2*b*d*x^5*e^3 + 3/2*B*a^3*d^2*x^4*e^2 + 9/2*A*a^2*b*d^2*x^4*e^2
+ 4/3*B*a^3*d^3*x^3*e + 4*A*a^2*b*d^3*x^3*e + 1/2*B*a^3*d^4*x^2 + 3/2*A*a^2*b*d^4*x^2 + 1/5*A*a^3*x^5*e^4 + A
*a^3*d*x^4*e^3 + 2*A*a^3*d^2*x^3*e^2 + 2*A*a^3*d^3*x^2*e + A*a^3*d^4*x