3.1487 \(\int (A+B x) (d+e x)^m (a+c x^2)^3 \, dx\)
Optimal. Leaf size=372 \[ -\frac{c (d+e x)^{m+4} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (m+4)}-\frac{c^2 (d+e x)^{m+5} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (m+6)}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) (d+e x)^{m+1}}{e^8 (m+1)}+\frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+2} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (m+2)}-\frac{3 c \left (a e^2+c d^2\right ) (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (m+3)}-\frac{c^3 (7 B d-A e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]
[Out]
-(((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e
+ a*B*e^2)*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a
*A*e^3)*(d + e*x)^(3 + m))/(e^8*(3 + m)) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2
+ 3*a^2*e^4))*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(
d + e*x)^(5 + m))/(e^8*(5 + m)) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(6 + m))/(e^8*(6 + m)) -
(c^3*(7*B*d - A*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))
________________________________________________________________________________________
Rubi [A] time = 0.250634, antiderivative size = 372, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{772} \[ -\frac{c (d+e x)^{m+4} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (m+4)}-\frac{c^2 (d+e x)^{m+5} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (m+5)}+\frac{3 c^2 (d+e x)^{m+6} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (m+6)}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) (d+e x)^{m+1}}{e^8 (m+1)}+\frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+2} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (m+2)}-\frac{3 c \left (a e^2+c d^2\right ) (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (m+3)}-\frac{c^3 (7 B d-A e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(a + c*x^2)^3,x]
[Out]
-(((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e
+ a*B*e^2)*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a
*A*e^3)*(d + e*x)^(3 + m))/(e^8*(3 + m)) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2
+ 3*a^2*e^4))*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(
d + e*x)^(5 + m))/(e^8*(5 + m)) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(6 + m))/(e^8*(6 + m)) -
(c^3*(7*B*d - A*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))
Rule 772
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3 (d+e x)^m}{e^7}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{1+m}}{e^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{2+m}}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^{3+m}}{e^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^{4+m}}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{5+m}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{6+m}}{e^7}+\frac{B c^3 (d+e x)^{7+m}}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^8 (2+m)}-\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3+m}}{e^8 (3+m)}-\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}-\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{5+m}}{e^8 (5+m)}+\frac{3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac{c^3 (7 B d-A e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac{B c^3 (d+e x)^{8+m}}{e^8 (8+m)}\\ \end{align*}
Mathematica [B] time = 1.63859, size = 773, normalized size = 2.08 \[ \frac{(d+e x)^{m+1} \left (B (m+1) (d+e x) \left (6 (m+7) \left (a e^2+c d^2\right ) \left (4 (m+5) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )-6 c d (m+2) (d+e x) \left (4 (m+6) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )-4 c d (m+3) (d+e x) \left (a e^2 \left (m^2+11 m+30\right )+c \left (2 d^2-2 d e (m+4) x+e^2 \left (m^2+9 m+20\right ) x^2\right )\right )+e^4 (m+3) (m+4) (m+5) (m+6) \left (a+c x^2\right )^2\right )+e^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \left (a+c x^2\right )^3\right )-(m+8) (B d-A e) \left (6 (m+6) \left (a e^2+c d^2\right ) \left (4 (m+4) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )-4 c d (m+1) (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )+e^4 (m+1) (m+2) (m+3) (m+4) \left (a+c x^2\right )^2\right )-6 c d (m+1) (d+e x) \left (4 (m+5) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )+e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) \left (a+c x^2\right )^3\right )\right )}{e^8 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(a + c*x^2)^3,x]
[Out]
((d + e*x)^(1 + m)*(-((B*d - A*e)*(8 + m)*(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(a + c*x^2)^3 +
6*(c*d^2 + a*e^2)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(4 + m)*(a*e
^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) - 4*c*d*(1 + m)*(d + e*x)*(a*e^2*(
12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2))) - 6*c*d*(1 + m)*(d + e*x)*(e^4*(2 +
m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*
e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(
3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))))) + B*(1 + m)*(d + e*x)*(e^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7
+ m)*(a + c*x^2)^3 + 6*(c*d^2 + a*e^2)*(7 + m)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2
+ a*e^2)*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 +
m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))) - 6*c*d*(2 +
m)*(d + e*x)*(e^4*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(6 + m)*(a*e^2*(20 + 9*m +
m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)) - 4*c*d*(3 + m)*(d + e*x)*(a*e^2*(30 + 11*m +
m^2) + c*(2*d^2 - 2*d*e*(4 + m)*x + e^2*(20 + 9*m + m^2)*x^2))))))/(e^8*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m
)*(6 + m)*(7 + m)*(8 + m))
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Maple [B] time = 0.02, size = 3176, normalized size = 8.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^m*(c*x^2+a)^3,x)
[Out]
(e*x+d)^(1+m)*(B*c^3*e^7*m^7*x^7+A*c^3*e^7*m^7*x^6+28*B*c^3*e^7*m^6*x^7+29*A*c^3*e^7*m^6*x^6+3*B*a*c^2*e^7*m^7
*x^5-7*B*c^3*d*e^6*m^6*x^6+322*B*c^3*e^7*m^5*x^7+3*A*a*c^2*e^7*m^7*x^4-6*A*c^3*d*e^6*m^6*x^5+343*A*c^3*e^7*m^5
*x^6+90*B*a*c^2*e^7*m^6*x^5-147*B*c^3*d*e^6*m^5*x^6+1960*B*c^3*e^7*m^4*x^7+93*A*a*c^2*e^7*m^6*x^4-138*A*c^3*d*
e^6*m^5*x^5+2135*A*c^3*e^7*m^4*x^6+3*B*a^2*c*e^7*m^7*x^3-15*B*a*c^2*d*e^6*m^6*x^4+1098*B*a*c^2*e^7*m^5*x^5+42*
B*c^3*d^2*e^5*m^5*x^5-1225*B*c^3*d*e^6*m^4*x^6+6769*B*c^3*e^7*m^3*x^7+3*A*a^2*c*e^7*m^7*x^2-12*A*a*c^2*d*e^6*m
^6*x^3+1173*A*a*c^2*e^7*m^5*x^4+30*A*c^3*d^2*e^5*m^5*x^4-1230*A*c^3*d*e^6*m^4*x^5+7504*A*c^3*e^7*m^3*x^6+96*B*
a^2*c*e^7*m^6*x^3-375*B*a*c^2*d*e^6*m^5*x^4+7020*B*a*c^2*e^7*m^4*x^5+630*B*c^3*d^2*e^5*m^4*x^5-5145*B*c^3*d*e^
6*m^3*x^6+13132*B*c^3*e^7*m^2*x^7+99*A*a^2*c*e^7*m^6*x^2-324*A*a*c^2*d*e^6*m^5*x^3+7743*A*a*c^2*e^7*m^4*x^4+54
0*A*c^3*d^2*e^5*m^4*x^4-5430*A*c^3*d*e^6*m^3*x^5+14756*A*c^3*e^7*m^2*x^6+B*a^3*e^7*m^7*x-9*B*a^2*c*d*e^6*m^6*x
^2+1254*B*a^2*c*e^7*m^5*x^3+60*B*a*c^2*d^2*e^5*m^5*x^3-3615*B*a*c^2*d*e^6*m^4*x^4+25227*B*a*c^2*e^7*m^3*x^5-21
0*B*c^3*d^3*e^4*m^4*x^4+3570*B*c^3*d^2*e^5*m^3*x^5-11368*B*c^3*d*e^6*m^2*x^6+13068*B*c^3*e^7*m*x^7+A*a^3*e^7*m
^7-6*A*a^2*c*d*e^6*m^6*x+1341*A*a^2*c*e^7*m^5*x^2+36*A*a*c^2*d^2*e^5*m^5*x^2-3396*A*a*c^2*d*e^6*m^4*x^3+28632*
A*a*c^2*e^7*m^3*x^4-120*A*c^3*d^3*e^4*m^4*x^3+3450*A*c^3*d^2*e^5*m^3*x^4-12444*A*c^3*d*e^6*m^2*x^5+14832*A*c^3
*e^7*m*x^6+34*B*a^3*e^7*m^6*x-261*B*a^2*c*d*e^6*m^5*x^2+8592*B*a^2*c*e^7*m^4*x^3+1260*B*a*c^2*d^2*e^5*m^4*x^3-
17025*B*a*c^2*d*e^6*m^3*x^4+50490*B*a*c^2*e^7*m^2*x^5-2100*B*c^3*d^3*e^4*m^3*x^4+9450*B*c^3*d^2*e^5*m^2*x^5-12
348*B*c^3*d*e^6*m*x^6+5040*B*c^3*e^7*x^7+35*A*a^3*e^7*m^6-186*A*a^2*c*d*e^6*m^5*x+9585*A*a^2*c*e^7*m^4*x^2+864
*A*a*c^2*d^2*e^5*m^4*x^2-17388*A*a*c^2*d*e^6*m^3*x^3+58692*A*a*c^2*e^7*m^2*x^4-1680*A*c^3*d^3*e^4*m^3*x^3+9900
*A*c^3*d^2*e^5*m^2*x^4-13872*A*c^3*d*e^6*m*x^5+5760*A*c^3*e^7*x^6-B*a^3*d*e^6*m^6+478*B*a^3*e^7*m^5*x+18*B*a^2
*c*d^2*e^5*m^5*x-2979*B*a^2*c*d*e^6*m^4*x^2+32979*B*a^2*c*e^7*m^3*x^3-180*B*a*c^2*d^3*e^4*m^4*x^2+9420*B*a*c^2
*d^2*e^5*m^3*x^3-41010*B*a*c^2*d*e^6*m^2*x^4+51432*B*a*c^2*e^7*m*x^5+840*B*c^3*d^4*e^3*m^3*x^3-7350*B*c^3*d^3*
e^4*m^2*x^4+11508*B*c^3*d^2*e^5*m*x^5-5040*B*c^3*d*e^6*x^6+511*A*a^3*e^7*m^5+6*A*a^2*c*d^2*e^5*m^5-2310*A*a^2*
c*d*e^6*m^4*x+38592*A*a^2*c*e^7*m^3*x^2-72*A*a*c^2*d^3*e^4*m^4*x+7596*A*a*c^2*d^2*e^5*m^3*x^2-44976*A*a*c^2*d*
e^6*m^2*x^3+60912*A*a*c^2*e^7*m*x^4+360*A*c^3*d^4*e^3*m^3*x^2-7080*A*c^3*d^3*e^4*m^2*x^3+12720*A*c^3*d^2*e^5*m
*x^4-5760*A*c^3*d*e^6*x^5-33*B*a^3*d*e^6*m^5+3580*B*a^3*e^7*m^4*x+486*B*a^2*c*d^2*e^5*m^4*x-16839*B*a^2*c*d*e^
6*m^3*x^2+69936*B*a^2*c*e^7*m^2*x^3-3240*B*a*c^2*d^3*e^4*m^3*x^2+30420*B*a*c^2*d^2*e^5*m^2*x^3-47400*B*a*c^2*d
*e^6*m*x^4+20160*B*a*c^2*e^7*x^5+5040*B*c^3*d^4*e^3*m^2*x^3-10500*B*c^3*d^3*e^4*m*x^4+5040*B*c^3*d^2*e^5*x^5+4
025*A*a^3*e^7*m^4+180*A*a^2*c*d^2*e^5*m^4-14550*A*a^2*c*d*e^6*m^3*x+86076*A*a^2*c*e^7*m^2*x^2-1584*A*a*c^2*d^3
*e^4*m^3*x+29376*A*a*c^2*d^2*e^5*m^2*x^2-54864*A*a*c^2*d*e^6*m*x^3+24192*A*a*c^2*e^7*x^4+3960*A*c^3*d^4*e^3*m^
2*x^2-11280*A*c^3*d^3*e^4*m*x^3+5760*A*c^3*d^2*e^5*x^4-445*B*a^3*d*e^6*m^4+15289*B*a^3*e^7*m^3*x-18*B*a^2*c*d^
3*e^4*m^4+4986*B*a^2*c*d^2*e^5*m^3*x-48420*B*a^2*c*d*e^6*m^2*x^2+74628*B*a^2*c*e^7*m*x^3+360*B*a*c^2*d^4*e^3*m
^3*x-18540*B*a*c^2*d^3*e^4*m^2*x^2+42360*B*a*c^2*d^2*e^5*m*x^3-20160*B*a*c^2*d*e^6*x^4-2520*B*c^3*d^5*e^2*m^2*
x^2+9240*B*c^3*d^4*e^3*m*x^3-5040*B*c^3*d^3*e^4*x^4+18424*A*a^3*e^7*m^3+2130*A*a^2*c*d^2*e^5*m^3-48084*A*a^2*c
*d*e^6*m^2*x+96144*A*a^2*c*e^7*m*x^2+72*A*a*c^2*d^4*e^3*m^3-12024*A*a*c^2*d^3*e^4*m^2*x+46800*A*a*c^2*d^2*e^5*
m*x^2-24192*A*a*c^2*d*e^6*x^3-720*A*c^3*d^5*e^2*m^2*x+9360*A*c^3*d^4*e^3*m*x^2-5760*A*c^3*d^3*e^4*x^3-3135*B*a
^3*d*e^6*m^3+36706*B*a^3*e^7*m^2*x-468*B*a^2*c*d^3*e^4*m^3+23706*B*a^2*c*d^2*e^5*m^2*x-64548*B*a^2*c*d*e^6*m*x
^2+30240*B*a^2*c*e^7*x^3+5760*B*a*c^2*d^4*e^3*m^2*x-35640*B*a*c^2*d^3*e^4*m*x^2+20160*B*a*c^2*d^2*e^5*x^3-7560
*B*c^3*d^5*e^2*m*x^2+5040*B*c^3*d^4*e^3*x^3+48860*A*a^3*e^7*m^2+12420*A*a^2*c*d^2*e^5*m^2-75984*A*a^2*c*d*e^6*
m*x+40320*A*a^2*c*e^7*x^2+1512*A*a*c^2*d^4*e^3*m^2-34704*A*a*c^2*d^3*e^4*m*x+24192*A*a*c^2*d^2*e^5*x^2-6480*A*
c^3*d^5*e^2*m*x+5760*A*c^3*d^4*e^3*x^2-12154*B*a^3*d*e^6*m^2+44712*B*a^3*e^7*m*x-4518*B*a^2*c*d^3*e^4*m^2+4942
8*B*a^2*c*d^2*e^5*m*x-30240*B*a^2*c*d*e^6*x^2-360*B*a*c^2*d^5*e^2*m^2+25560*B*a*c^2*d^4*e^3*m*x-20160*B*a*c^2*
d^3*e^4*x^2+5040*B*c^3*d^6*e*m*x-5040*B*c^3*d^5*e^2*x^2+69264*A*a^3*e^7*m+35664*A*a^2*c*d^2*e^5*m-40320*A*a^2*
c*d*e^6*x+10512*A*a*c^2*d^4*e^3*m-24192*A*a*c^2*d^3*e^4*x+720*A*c^3*d^6*e*m-5760*A*c^3*d^5*e^2*x-24552*B*a^3*d
*e^6*m+20160*B*a^3*e^7*x-19188*B*a^2*c*d^3*e^4*m+30240*B*a^2*c*d^2*e^5*x-5400*B*a*c^2*d^5*e^2*m+20160*B*a*c^2*
d^4*e^3*x+5040*B*c^3*d^6*e*x+40320*A*a^3*e^7+40320*A*a^2*c*d^2*e^5+24192*A*a*c^2*d^4*e^3+5760*A*c^3*d^6*e-2016
0*B*a^3*d*e^6-30240*B*a^2*c*d^3*e^4-20160*B*a*c^2*d^5*e^2-5040*B*c^3*d^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+224
49*m^4+67284*m^3+118124*m^2+109584*m+40320)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 2.24973, size = 7097, normalized size = 19.08 \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)^m*(c*x^2+a)^3,x, algorithm="fricas")
[Out]
(A*a^3*d*e^7*m^7 - 5040*B*c^3*d^8 + 5760*A*c^3*d^7*e - 20160*B*a*c^2*d^6*e^2 + 24192*A*a*c^2*d^5*e^3 - 30240*B
*a^2*c*d^4*e^4 + 40320*A*a^2*c*d^3*e^5 - 20160*B*a^3*d^2*e^6 + 40320*A*a^3*d*e^7 + (B*c^3*e^8*m^7 + 28*B*c^3*e
^8*m^6 + 322*B*c^3*e^8*m^5 + 1960*B*c^3*e^8*m^4 + 6769*B*c^3*e^8*m^3 + 13132*B*c^3*e^8*m^2 + 13068*B*c^3*e^8*m
+ 5040*B*c^3*e^8)*x^8 + (5760*A*c^3*e^8 + (B*c^3*d*e^7 + A*c^3*e^8)*m^7 + (21*B*c^3*d*e^7 + 29*A*c^3*e^8)*m^6
+ 7*(25*B*c^3*d*e^7 + 49*A*c^3*e^8)*m^5 + 35*(21*B*c^3*d*e^7 + 61*A*c^3*e^8)*m^4 + 56*(29*B*c^3*d*e^7 + 134*A
*c^3*e^8)*m^3 + 28*(63*B*c^3*d*e^7 + 527*A*c^3*e^8)*m^2 + 144*(5*B*c^3*d*e^7 + 103*A*c^3*e^8)*m)*x^7 - (B*a^3*
d^2*e^6 - 35*A*a^3*d*e^7)*m^6 + (20160*B*a*c^2*e^8 + (A*c^3*d*e^7 + 3*B*a*c^2*e^8)*m^7 - (7*B*c^3*d^2*e^6 - 23
*A*c^3*d*e^7 - 90*B*a*c^2*e^8)*m^6 - (105*B*c^3*d^2*e^6 - 205*A*c^3*d*e^7 - 1098*B*a*c^2*e^8)*m^5 - 5*(119*B*c
^3*d^2*e^6 - 181*A*c^3*d*e^7 - 1404*B*a*c^2*e^8)*m^4 - (1575*B*c^3*d^2*e^6 - 2074*A*c^3*d*e^7 - 25227*B*a*c^2*
e^8)*m^3 - 2*(959*B*c^3*d^2*e^6 - 1156*A*c^3*d*e^7 - 25245*B*a*c^2*e^8)*m^2 - 24*(35*B*c^3*d^2*e^6 - 40*A*c^3*
d*e^7 - 2143*B*a*c^2*e^8)*m)*x^6 + (6*A*a^2*c*d^3*e^5 - 33*B*a^3*d^2*e^6 + 511*A*a^3*d*e^7)*m^5 + 3*(8064*A*a*
c^2*e^8 + (B*a*c^2*d*e^7 + A*a*c^2*e^8)*m^7 - (2*A*c^3*d^2*e^6 - 25*B*a*c^2*d*e^7 - 31*A*a*c^2*e^8)*m^6 + (14*
B*c^3*d^3*e^5 - 36*A*c^3*d^2*e^6 + 241*B*a*c^2*d*e^7 + 391*A*a*c^2*e^8)*m^5 + (140*B*c^3*d^3*e^5 - 230*A*c^3*d
^2*e^6 + 1135*B*a*c^2*d*e^7 + 2581*A*a*c^2*e^8)*m^4 + 2*(245*B*c^3*d^3*e^5 - 330*A*c^3*d^2*e^6 + 1367*B*a*c^2*
d*e^7 + 4772*A*a*c^2*e^8)*m^3 + 4*(175*B*c^3*d^3*e^5 - 212*A*c^3*d^2*e^6 + 790*B*a*c^2*d*e^7 + 4891*A*a*c^2*e^
8)*m^2 + 48*(7*B*c^3*d^3*e^5 - 8*A*c^3*d^2*e^6 + 28*B*a*c^2*d*e^7 + 423*A*a*c^2*e^8)*m)*x^5 - (18*B*a^2*c*d^4*
e^4 - 180*A*a^2*c*d^3*e^5 + 445*B*a^3*d^2*e^6 - 4025*A*a^3*d*e^7)*m^4 + 3*(10080*B*a^2*c*e^8 + (A*a*c^2*d*e^7
+ B*a^2*c*e^8)*m^7 - (5*B*a*c^2*d^2*e^6 - 27*A*a*c^2*d*e^7 - 32*B*a^2*c*e^8)*m^6 + (10*A*c^3*d^3*e^5 - 105*B*a
*c^2*d^2*e^6 + 283*A*a*c^2*d*e^7 + 418*B*a^2*c*e^8)*m^5 - (70*B*c^3*d^4*e^4 - 140*A*c^3*d^3*e^5 + 785*B*a*c^2*
d^2*e^6 - 1449*A*a*c^2*d*e^7 - 2864*B*a^2*c*e^8)*m^4 - (420*B*c^3*d^4*e^4 - 590*A*c^3*d^3*e^5 + 2535*B*a*c^2*d
^2*e^6 - 3748*A*a*c^2*d*e^7 - 10993*B*a^2*c*e^8)*m^3 - 2*(385*B*c^3*d^4*e^4 - 470*A*c^3*d^3*e^5 + 1765*B*a*c^2
*d^2*e^6 - 2286*A*a*c^2*d*e^7 - 11656*B*a^2*c*e^8)*m^2 - 12*(35*B*c^3*d^4*e^4 - 40*A*c^3*d^3*e^5 + 140*B*a*c^2
*d^2*e^6 - 168*A*a*c^2*d*e^7 - 2073*B*a^2*c*e^8)*m)*x^4 + (72*A*a*c^2*d^5*e^3 - 468*B*a^2*c*d^4*e^4 + 2130*A*a
^2*c*d^3*e^5 - 3135*B*a^3*d^2*e^6 + 18424*A*a^3*d*e^7)*m^3 + 3*(13440*A*a^2*c*e^8 + (B*a^2*c*d*e^7 + A*a^2*c*e
^8)*m^7 - (4*A*a*c^2*d^2*e^6 - 29*B*a^2*c*d*e^7 - 33*A*a^2*c*e^8)*m^6 + (20*B*a*c^2*d^3*e^5 - 96*A*a*c^2*d^2*e
^6 + 331*B*a^2*c*d*e^7 + 447*A*a^2*c*e^8)*m^5 - (40*A*c^3*d^4*e^4 - 360*B*a*c^2*d^3*e^5 + 844*A*a*c^2*d^2*e^6
- 1871*B*a^2*c*d*e^7 - 3195*A*a^2*c*e^8)*m^4 + 4*(70*B*c^3*d^5*e^3 - 110*A*c^3*d^4*e^4 + 515*B*a*c^2*d^3*e^5 -
816*A*a*c^2*d^2*e^6 + 1345*B*a^2*c*d*e^7 + 3216*A*a^2*c*e^8)*m^3 + 4*(210*B*c^3*d^5*e^3 - 260*A*c^3*d^4*e^4 +
990*B*a*c^2*d^3*e^5 - 1300*A*a*c^2*d^2*e^6 + 1793*B*a^2*c*d*e^7 + 7173*A*a^2*c*e^8)*m^2 + 16*(35*B*c^3*d^5*e^
3 - 40*A*c^3*d^4*e^4 + 140*B*a*c^2*d^3*e^5 - 168*A*a*c^2*d^2*e^6 + 210*B*a^2*c*d*e^7 + 2003*A*a^2*c*e^8)*m)*x^
3 - 2*(180*B*a*c^2*d^6*e^2 - 756*A*a*c^2*d^5*e^3 + 2259*B*a^2*c*d^4*e^4 - 6210*A*a^2*c*d^3*e^5 + 6077*B*a^3*d^
2*e^6 - 24430*A*a^3*d*e^7)*m^2 + (20160*B*a^3*e^8 + (3*A*a^2*c*d*e^7 + B*a^3*e^8)*m^7 - (9*B*a^2*c*d^2*e^6 - 9
3*A*a^2*c*d*e^7 - 34*B*a^3*e^8)*m^6 + (36*A*a*c^2*d^3*e^5 - 243*B*a^2*c*d^2*e^6 + 1155*A*a^2*c*d*e^7 + 478*B*a
^3*e^8)*m^5 - (180*B*a*c^2*d^4*e^4 - 792*A*a*c^2*d^3*e^5 + 2493*B*a^2*c*d^2*e^6 - 7275*A*a^2*c*d*e^7 - 3580*B*
a^3*e^8)*m^4 + (360*A*c^3*d^5*e^3 - 2880*B*a*c^2*d^4*e^4 + 6012*A*a*c^2*d^3*e^5 - 11853*B*a^2*c*d^2*e^6 + 2404
2*A*a^2*c*d*e^7 + 15289*B*a^3*e^8)*m^3 - 2*(1260*B*c^3*d^6*e^2 - 1620*A*c^3*d^5*e^3 + 6390*B*a*c^2*d^4*e^4 - 8
676*A*a*c^2*d^3*e^5 + 12357*B*a^2*c*d^2*e^6 - 18996*A*a^2*c*d*e^7 - 18353*B*a^3*e^8)*m^2 - 72*(35*B*c^3*d^6*e^
2 - 40*A*c^3*d^5*e^3 + 140*B*a*c^2*d^4*e^4 - 168*A*a*c^2*d^3*e^5 + 210*B*a^2*c*d^2*e^6 - 280*A*a^2*c*d*e^7 - 6
21*B*a^3*e^8)*m)*x^2 + 12*(60*A*c^3*d^7*e - 450*B*a*c^2*d^6*e^2 + 876*A*a*c^2*d^5*e^3 - 1599*B*a^2*c*d^4*e^4 +
2972*A*a^2*c*d^3*e^5 - 2046*B*a^3*d^2*e^6 + 5772*A*a^3*d*e^7)*m + (40320*A*a^3*e^8 + (B*a^3*d*e^7 + A*a^3*e^8
)*m^7 - (6*A*a^2*c*d^2*e^6 - 33*B*a^3*d*e^7 - 35*A*a^3*e^8)*m^6 + (18*B*a^2*c*d^3*e^5 - 180*A*a^2*c*d^2*e^6 +
445*B*a^3*d*e^7 + 511*A*a^3*e^8)*m^5 - (72*A*a*c^2*d^4*e^4 - 468*B*a^2*c*d^3*e^5 + 2130*A*a^2*c*d^2*e^6 - 3135
*B*a^3*d*e^7 - 4025*A*a^3*e^8)*m^4 + 2*(180*B*a*c^2*d^5*e^3 - 756*A*a*c^2*d^4*e^4 + 2259*B*a^2*c*d^3*e^5 - 621
0*A*a^2*c*d^2*e^6 + 6077*B*a^3*d*e^7 + 9212*A*a^3*e^8)*m^3 - 4*(180*A*c^3*d^6*e^2 - 1350*B*a*c^2*d^5*e^3 + 262
8*A*a*c^2*d^4*e^4 - 4797*B*a^2*c*d^3*e^5 + 8916*A*a^2*c*d^2*e^6 - 6138*B*a^3*d*e^7 - 12215*A*a^3*e^8)*m^2 + 14
4*(35*B*c^3*d^7*e - 40*A*c^3*d^6*e^2 + 140*B*a*c^2*d^5*e^3 - 168*A*a*c^2*d^4*e^4 + 210*B*a^2*c*d^3*e^5 - 280*A
*a^2*c*d^2*e^6 + 140*B*a^3*d*e^7 + 481*A*a^3*e^8)*m)*x)*(e*x + d)^m/(e^8*m^8 + 36*e^8*m^7 + 546*e^8*m^6 + 4536
*e^8*m^5 + 22449*e^8*m^4 + 67284*e^8*m^3 + 118124*e^8*m^2 + 109584*e^8*m + 40320*e^8)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.26841, size = 7506, normalized size = 20.18 \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)^m*(c*x^2+a)^3,x, algorithm="giac")
[Out]
((x*e + d)^m*B*c^3*m^7*x^8*e^8 + (x*e + d)^m*B*c^3*d*m^7*x^7*e^7 + (x*e + d)^m*A*c^3*m^7*x^7*e^8 + 28*(x*e + d
)^m*B*c^3*m^6*x^8*e^8 + (x*e + d)^m*A*c^3*d*m^7*x^6*e^7 + 21*(x*e + d)^m*B*c^3*d*m^6*x^7*e^7 - 7*(x*e + d)^m*B
*c^3*d^2*m^6*x^6*e^6 + 3*(x*e + d)^m*B*a*c^2*m^7*x^6*e^8 + 29*(x*e + d)^m*A*c^3*m^6*x^7*e^8 + 322*(x*e + d)^m*
B*c^3*m^5*x^8*e^8 + 3*(x*e + d)^m*B*a*c^2*d*m^7*x^5*e^7 + 23*(x*e + d)^m*A*c^3*d*m^6*x^6*e^7 + 175*(x*e + d)^m
*B*c^3*d*m^5*x^7*e^7 - 6*(x*e + d)^m*A*c^3*d^2*m^6*x^5*e^6 - 105*(x*e + d)^m*B*c^3*d^2*m^5*x^6*e^6 + 42*(x*e +
d)^m*B*c^3*d^3*m^5*x^5*e^5 + 3*(x*e + d)^m*A*a*c^2*m^7*x^5*e^8 + 90*(x*e + d)^m*B*a*c^2*m^6*x^6*e^8 + 343*(x*
e + d)^m*A*c^3*m^5*x^7*e^8 + 1960*(x*e + d)^m*B*c^3*m^4*x^8*e^8 + 3*(x*e + d)^m*A*a*c^2*d*m^7*x^4*e^7 + 75*(x*
e + d)^m*B*a*c^2*d*m^6*x^5*e^7 + 205*(x*e + d)^m*A*c^3*d*m^5*x^6*e^7 + 735*(x*e + d)^m*B*c^3*d*m^4*x^7*e^7 - 1
5*(x*e + d)^m*B*a*c^2*d^2*m^6*x^4*e^6 - 108*(x*e + d)^m*A*c^3*d^2*m^5*x^5*e^6 - 595*(x*e + d)^m*B*c^3*d^2*m^4*
x^6*e^6 + 30*(x*e + d)^m*A*c^3*d^3*m^5*x^4*e^5 + 420*(x*e + d)^m*B*c^3*d^3*m^4*x^5*e^5 - 210*(x*e + d)^m*B*c^3
*d^4*m^4*x^4*e^4 + 3*(x*e + d)^m*B*a^2*c*m^7*x^4*e^8 + 93*(x*e + d)^m*A*a*c^2*m^6*x^5*e^8 + 1098*(x*e + d)^m*B
*a*c^2*m^5*x^6*e^8 + 2135*(x*e + d)^m*A*c^3*m^4*x^7*e^8 + 6769*(x*e + d)^m*B*c^3*m^3*x^8*e^8 + 3*(x*e + d)^m*B
*a^2*c*d*m^7*x^3*e^7 + 81*(x*e + d)^m*A*a*c^2*d*m^6*x^4*e^7 + 723*(x*e + d)^m*B*a*c^2*d*m^5*x^5*e^7 + 905*(x*e
+ d)^m*A*c^3*d*m^4*x^6*e^7 + 1624*(x*e + d)^m*B*c^3*d*m^3*x^7*e^7 - 12*(x*e + d)^m*A*a*c^2*d^2*m^6*x^3*e^6 -
315*(x*e + d)^m*B*a*c^2*d^2*m^5*x^4*e^6 - 690*(x*e + d)^m*A*c^3*d^2*m^4*x^5*e^6 - 1575*(x*e + d)^m*B*c^3*d^2*m
^3*x^6*e^6 + 60*(x*e + d)^m*B*a*c^2*d^3*m^5*x^3*e^5 + 420*(x*e + d)^m*A*c^3*d^3*m^4*x^4*e^5 + 1470*(x*e + d)^m
*B*c^3*d^3*m^3*x^5*e^5 - 120*(x*e + d)^m*A*c^3*d^4*m^4*x^3*e^4 - 1260*(x*e + d)^m*B*c^3*d^4*m^3*x^4*e^4 + 840*
(x*e + d)^m*B*c^3*d^5*m^3*x^3*e^3 + 3*(x*e + d)^m*A*a^2*c*m^7*x^3*e^8 + 96*(x*e + d)^m*B*a^2*c*m^6*x^4*e^8 + 1
173*(x*e + d)^m*A*a*c^2*m^5*x^5*e^8 + 7020*(x*e + d)^m*B*a*c^2*m^4*x^6*e^8 + 7504*(x*e + d)^m*A*c^3*m^3*x^7*e^
8 + 13132*(x*e + d)^m*B*c^3*m^2*x^8*e^8 + 3*(x*e + d)^m*A*a^2*c*d*m^7*x^2*e^7 + 87*(x*e + d)^m*B*a^2*c*d*m^6*x
^3*e^7 + 849*(x*e + d)^m*A*a*c^2*d*m^5*x^4*e^7 + 3405*(x*e + d)^m*B*a*c^2*d*m^4*x^5*e^7 + 2074*(x*e + d)^m*A*c
^3*d*m^3*x^6*e^7 + 1764*(x*e + d)^m*B*c^3*d*m^2*x^7*e^7 - 9*(x*e + d)^m*B*a^2*c*d^2*m^6*x^2*e^6 - 288*(x*e + d
)^m*A*a*c^2*d^2*m^5*x^3*e^6 - 2355*(x*e + d)^m*B*a*c^2*d^2*m^4*x^4*e^6 - 1980*(x*e + d)^m*A*c^3*d^2*m^3*x^5*e^
6 - 1918*(x*e + d)^m*B*c^3*d^2*m^2*x^6*e^6 + 36*(x*e + d)^m*A*a*c^2*d^3*m^5*x^2*e^5 + 1080*(x*e + d)^m*B*a*c^2
*d^3*m^4*x^3*e^5 + 1770*(x*e + d)^m*A*c^3*d^3*m^3*x^4*e^5 + 2100*(x*e + d)^m*B*c^3*d^3*m^2*x^5*e^5 - 180*(x*e
+ d)^m*B*a*c^2*d^4*m^4*x^2*e^4 - 1320*(x*e + d)^m*A*c^3*d^4*m^3*x^3*e^4 - 2310*(x*e + d)^m*B*c^3*d^4*m^2*x^4*e
^4 + 360*(x*e + d)^m*A*c^3*d^5*m^3*x^2*e^3 + 2520*(x*e + d)^m*B*c^3*d^5*m^2*x^3*e^3 - 2520*(x*e + d)^m*B*c^3*d
^6*m^2*x^2*e^2 + (x*e + d)^m*B*a^3*m^7*x^2*e^8 + 99*(x*e + d)^m*A*a^2*c*m^6*x^3*e^8 + 1254*(x*e + d)^m*B*a^2*c
*m^5*x^4*e^8 + 7743*(x*e + d)^m*A*a*c^2*m^4*x^5*e^8 + 25227*(x*e + d)^m*B*a*c^2*m^3*x^6*e^8 + 14756*(x*e + d)^
m*A*c^3*m^2*x^7*e^8 + 13068*(x*e + d)^m*B*c^3*m*x^8*e^8 + (x*e + d)^m*B*a^3*d*m^7*x*e^7 + 93*(x*e + d)^m*A*a^2
*c*d*m^6*x^2*e^7 + 993*(x*e + d)^m*B*a^2*c*d*m^5*x^3*e^7 + 4347*(x*e + d)^m*A*a*c^2*d*m^4*x^4*e^7 + 8202*(x*e
+ d)^m*B*a*c^2*d*m^3*x^5*e^7 + 2312*(x*e + d)^m*A*c^3*d*m^2*x^6*e^7 + 720*(x*e + d)^m*B*c^3*d*m*x^7*e^7 - 6*(x
*e + d)^m*A*a^2*c*d^2*m^6*x*e^6 - 243*(x*e + d)^m*B*a^2*c*d^2*m^5*x^2*e^6 - 2532*(x*e + d)^m*A*a*c^2*d^2*m^4*x
^3*e^6 - 7605*(x*e + d)^m*B*a*c^2*d^2*m^3*x^4*e^6 - 2544*(x*e + d)^m*A*c^3*d^2*m^2*x^5*e^6 - 840*(x*e + d)^m*B
*c^3*d^2*m*x^6*e^6 + 18*(x*e + d)^m*B*a^2*c*d^3*m^5*x*e^5 + 792*(x*e + d)^m*A*a*c^2*d^3*m^4*x^2*e^5 + 6180*(x*
e + d)^m*B*a*c^2*d^3*m^3*x^3*e^5 + 2820*(x*e + d)^m*A*c^3*d^3*m^2*x^4*e^5 + 1008*(x*e + d)^m*B*c^3*d^3*m*x^5*e
^5 - 72*(x*e + d)^m*A*a*c^2*d^4*m^4*x*e^4 - 2880*(x*e + d)^m*B*a*c^2*d^4*m^3*x^2*e^4 - 3120*(x*e + d)^m*A*c^3*
d^4*m^2*x^3*e^4 - 1260*(x*e + d)^m*B*c^3*d^4*m*x^4*e^4 + 360*(x*e + d)^m*B*a*c^2*d^5*m^3*x*e^3 + 3240*(x*e + d
)^m*A*c^3*d^5*m^2*x^2*e^3 + 1680*(x*e + d)^m*B*c^3*d^5*m*x^3*e^3 - 720*(x*e + d)^m*A*c^3*d^6*m^2*x*e^2 - 2520*
(x*e + d)^m*B*c^3*d^6*m*x^2*e^2 + 5040*(x*e + d)^m*B*c^3*d^7*m*x*e + (x*e + d)^m*A*a^3*m^7*x*e^8 + 34*(x*e + d
)^m*B*a^3*m^6*x^2*e^8 + 1341*(x*e + d)^m*A*a^2*c*m^5*x^3*e^8 + 8592*(x*e + d)^m*B*a^2*c*m^4*x^4*e^8 + 28632*(x
*e + d)^m*A*a*c^2*m^3*x^5*e^8 + 50490*(x*e + d)^m*B*a*c^2*m^2*x^6*e^8 + 14832*(x*e + d)^m*A*c^3*m*x^7*e^8 + 50
40*(x*e + d)^m*B*c^3*x^8*e^8 + (x*e + d)^m*A*a^3*d*m^7*e^7 + 33*(x*e + d)^m*B*a^3*d*m^6*x*e^7 + 1155*(x*e + d)
^m*A*a^2*c*d*m^5*x^2*e^7 + 5613*(x*e + d)^m*B*a^2*c*d*m^4*x^3*e^7 + 11244*(x*e + d)^m*A*a*c^2*d*m^3*x^4*e^7 +
9480*(x*e + d)^m*B*a*c^2*d*m^2*x^5*e^7 + 960*(x*e + d)^m*A*c^3*d*m*x^6*e^7 - (x*e + d)^m*B*a^3*d^2*m^6*e^6 - 1
80*(x*e + d)^m*A*a^2*c*d^2*m^5*x*e^6 - 2493*(x*e + d)^m*B*a^2*c*d^2*m^4*x^2*e^6 - 9792*(x*e + d)^m*A*a*c^2*d^2
*m^3*x^3*e^6 - 10590*(x*e + d)^m*B*a*c^2*d^2*m^2*x^4*e^6 - 1152*(x*e + d)^m*A*c^3*d^2*m*x^5*e^6 + 6*(x*e + d)^
m*A*a^2*c*d^3*m^5*e^5 + 468*(x*e + d)^m*B*a^2*c*d^3*m^4*x*e^5 + 6012*(x*e + d)^m*A*a*c^2*d^3*m^3*x^2*e^5 + 118
80*(x*e + d)^m*B*a*c^2*d^3*m^2*x^3*e^5 + 1440*(x*e + d)^m*A*c^3*d^3*m*x^4*e^5 - 18*(x*e + d)^m*B*a^2*c*d^4*m^4
*e^4 - 1512*(x*e + d)^m*A*a*c^2*d^4*m^3*x*e^4 - 12780*(x*e + d)^m*B*a*c^2*d^4*m^2*x^2*e^4 - 1920*(x*e + d)^m*A
*c^3*d^4*m*x^3*e^4 + 72*(x*e + d)^m*A*a*c^2*d^5*m^3*e^3 + 5400*(x*e + d)^m*B*a*c^2*d^5*m^2*x*e^3 + 2880*(x*e +
d)^m*A*c^3*d^5*m*x^2*e^3 - 360*(x*e + d)^m*B*a*c^2*d^6*m^2*e^2 - 5760*(x*e + d)^m*A*c^3*d^6*m*x*e^2 + 720*(x*
e + d)^m*A*c^3*d^7*m*e - 5040*(x*e + d)^m*B*c^3*d^8 + 35*(x*e + d)^m*A*a^3*m^6*x*e^8 + 478*(x*e + d)^m*B*a^3*m
^5*x^2*e^8 + 9585*(x*e + d)^m*A*a^2*c*m^4*x^3*e^8 + 32979*(x*e + d)^m*B*a^2*c*m^3*x^4*e^8 + 58692*(x*e + d)^m*
A*a*c^2*m^2*x^5*e^8 + 51432*(x*e + d)^m*B*a*c^2*m*x^6*e^8 + 5760*(x*e + d)^m*A*c^3*x^7*e^8 + 35*(x*e + d)^m*A*
a^3*d*m^6*e^7 + 445*(x*e + d)^m*B*a^3*d*m^5*x*e^7 + 7275*(x*e + d)^m*A*a^2*c*d*m^4*x^2*e^7 + 16140*(x*e + d)^m
*B*a^2*c*d*m^3*x^3*e^7 + 13716*(x*e + d)^m*A*a*c^2*d*m^2*x^4*e^7 + 4032*(x*e + d)^m*B*a*c^2*d*m*x^5*e^7 - 33*(
x*e + d)^m*B*a^3*d^2*m^5*e^6 - 2130*(x*e + d)^m*A*a^2*c*d^2*m^4*x*e^6 - 11853*(x*e + d)^m*B*a^2*c*d^2*m^3*x^2*
e^6 - 15600*(x*e + d)^m*A*a*c^2*d^2*m^2*x^3*e^6 - 5040*(x*e + d)^m*B*a*c^2*d^2*m*x^4*e^6 + 180*(x*e + d)^m*A*a
^2*c*d^3*m^4*e^5 + 4518*(x*e + d)^m*B*a^2*c*d^3*m^3*x*e^5 + 17352*(x*e + d)^m*A*a*c^2*d^3*m^2*x^2*e^5 + 6720*(
x*e + d)^m*B*a*c^2*d^3*m*x^3*e^5 - 468*(x*e + d)^m*B*a^2*c*d^4*m^3*e^4 - 10512*(x*e + d)^m*A*a*c^2*d^4*m^2*x*e
^4 - 10080*(x*e + d)^m*B*a*c^2*d^4*m*x^2*e^4 + 1512*(x*e + d)^m*A*a*c^2*d^5*m^2*e^3 + 20160*(x*e + d)^m*B*a*c^
2*d^5*m*x*e^3 - 5400*(x*e + d)^m*B*a*c^2*d^6*m*e^2 + 5760*(x*e + d)^m*A*c^3*d^7*e + 511*(x*e + d)^m*A*a^3*m^5*
x*e^8 + 3580*(x*e + d)^m*B*a^3*m^4*x^2*e^8 + 38592*(x*e + d)^m*A*a^2*c*m^3*x^3*e^8 + 69936*(x*e + d)^m*B*a^2*c
*m^2*x^4*e^8 + 60912*(x*e + d)^m*A*a*c^2*m*x^5*e^8 + 20160*(x*e + d)^m*B*a*c^2*x^6*e^8 + 511*(x*e + d)^m*A*a^3
*d*m^5*e^7 + 3135*(x*e + d)^m*B*a^3*d*m^4*x*e^7 + 24042*(x*e + d)^m*A*a^2*c*d*m^3*x^2*e^7 + 21516*(x*e + d)^m*
B*a^2*c*d*m^2*x^3*e^7 + 6048*(x*e + d)^m*A*a*c^2*d*m*x^4*e^7 - 445*(x*e + d)^m*B*a^3*d^2*m^4*e^6 - 12420*(x*e
+ d)^m*A*a^2*c*d^2*m^3*x*e^6 - 24714*(x*e + d)^m*B*a^2*c*d^2*m^2*x^2*e^6 - 8064*(x*e + d)^m*A*a*c^2*d^2*m*x^3*
e^6 + 2130*(x*e + d)^m*A*a^2*c*d^3*m^3*e^5 + 19188*(x*e + d)^m*B*a^2*c*d^3*m^2*x*e^5 + 12096*(x*e + d)^m*A*a*c
^2*d^3*m*x^2*e^5 - 4518*(x*e + d)^m*B*a^2*c*d^4*m^2*e^4 - 24192*(x*e + d)^m*A*a*c^2*d^4*m*x*e^4 + 10512*(x*e +
d)^m*A*a*c^2*d^5*m*e^3 - 20160*(x*e + d)^m*B*a*c^2*d^6*e^2 + 4025*(x*e + d)^m*A*a^3*m^4*x*e^8 + 15289*(x*e +
d)^m*B*a^3*m^3*x^2*e^8 + 86076*(x*e + d)^m*A*a^2*c*m^2*x^3*e^8 + 74628*(x*e + d)^m*B*a^2*c*m*x^4*e^8 + 24192*(
x*e + d)^m*A*a*c^2*x^5*e^8 + 4025*(x*e + d)^m*A*a^3*d*m^4*e^7 + 12154*(x*e + d)^m*B*a^3*d*m^3*x*e^7 + 37992*(x
*e + d)^m*A*a^2*c*d*m^2*x^2*e^7 + 10080*(x*e + d)^m*B*a^2*c*d*m*x^3*e^7 - 3135*(x*e + d)^m*B*a^3*d^2*m^3*e^6 -
35664*(x*e + d)^m*A*a^2*c*d^2*m^2*x*e^6 - 15120*(x*e + d)^m*B*a^2*c*d^2*m*x^2*e^6 + 12420*(x*e + d)^m*A*a^2*c
*d^3*m^2*e^5 + 30240*(x*e + d)^m*B*a^2*c*d^3*m*x*e^5 - 19188*(x*e + d)^m*B*a^2*c*d^4*m*e^4 + 24192*(x*e + d)^m
*A*a*c^2*d^5*e^3 + 18424*(x*e + d)^m*A*a^3*m^3*x*e^8 + 36706*(x*e + d)^m*B*a^3*m^2*x^2*e^8 + 96144*(x*e + d)^m
*A*a^2*c*m*x^3*e^8 + 30240*(x*e + d)^m*B*a^2*c*x^4*e^8 + 18424*(x*e + d)^m*A*a^3*d*m^3*e^7 + 24552*(x*e + d)^m
*B*a^3*d*m^2*x*e^7 + 20160*(x*e + d)^m*A*a^2*c*d*m*x^2*e^7 - 12154*(x*e + d)^m*B*a^3*d^2*m^2*e^6 - 40320*(x*e
+ d)^m*A*a^2*c*d^2*m*x*e^6 + 35664*(x*e + d)^m*A*a^2*c*d^3*m*e^5 - 30240*(x*e + d)^m*B*a^2*c*d^4*e^4 + 48860*(
x*e + d)^m*A*a^3*m^2*x*e^8 + 44712*(x*e + d)^m*B*a^3*m*x^2*e^8 + 40320*(x*e + d)^m*A*a^2*c*x^3*e^8 + 48860*(x*
e + d)^m*A*a^3*d*m^2*e^7 + 20160*(x*e + d)^m*B*a^3*d*m*x*e^7 - 24552*(x*e + d)^m*B*a^3*d^2*m*e^6 + 40320*(x*e
+ d)^m*A*a^2*c*d^3*e^5 + 69264*(x*e + d)^m*A*a^3*m*x*e^8 + 20160*(x*e + d)^m*B*a^3*x^2*e^8 + 69264*(x*e + d)^m
*A*a^3*d*m*e^7 - 20160*(x*e + d)^m*B*a^3*d^2*e^6 + 40320*(x*e + d)^m*A*a^3*x*e^8 + 40320*(x*e + d)^m*A*a^3*d*e
^7)/(m^8*e^8 + 36*m^7*e^8 + 546*m^6*e^8 + 4536*m^5*e^8 + 22449*m^4*e^8 + 67284*m^3*e^8 + 118124*m^2*e^8 + 1095
84*m*e^8 + 40320*e^8)