3.926 \(\int (d+e x)^m (f+g x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=311 \[ \frac{(d+e x)^{m+4} \left (2 c e (a e g-4 b d g+b e f)+b^2 e^2 g-2 c^2 d (2 e f-5 d g)\right )}{e^6 (m+4)}+\frac{(d+e x)^{m+3} \left (2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))+b e^2 (2 a e g-3 b d g+b e f)+2 c^2 d^2 (3 e f-5 d g)\right )}{e^6 (m+3)}+\frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (4 e f-5 d g)-e (a e g-3 b d g+2 b e f))}{e^6 (m+2)}+\frac{c (d+e x)^{m+5} (2 b e g-5 c d g+c e f)}{e^6 (m+5)}+\frac{c^2 g (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*(d + e*x)^(1 + m))/(e^6*(1 + m)) - ((c*d^2 - b*d*e + a*e^2)*(c*d*(4*e*f
- 5*d*g) - e*(2*b*e*f - 3*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((2*c^2*d^2*(3*e*f - 5*d*g) + b*
e^2*(b*e*f - 3*b*d*g + 2*a*e*g) + 2*c*e*(a*e*(e*f - 3*d*g) - 3*b*d*(e*f - 2*d*g)))*(d + e*x)^(3 + m))/(e^6*(3
+ m)) + ((b^2*e^2*g - 2*c^2*d*(2*e*f - 5*d*g) + 2*c*e*(b*e*f - 4*b*d*g + a*e*g))*(d + e*x)^(4 + m))/(e^6*(4 +
m)) + (c*(c*e*f - 5*c*d*g + 2*b*e*g)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (c^2*g*(d + e*x)^(6 + m))/(e^6*(6 + m)
)

________________________________________________________________________________________

Rubi [A]  time = 0.392187, antiderivative size = 311, 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} \[ \frac{(d+e x)^{m+4} \left (2 c e (a e g-4 b d g+b e f)+b^2 e^2 g-2 c^2 d (2 e f-5 d g)\right )}{e^6 (m+4)}+\frac{(d+e x)^{m+3} \left (2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))+b e^2 (2 a e g-3 b d g+b e f)+2 c^2 d^2 (3 e f-5 d g)\right )}{e^6 (m+3)}+\frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (4 e f-5 d g)-e (a e g-3 b d g+2 b e f))}{e^6 (m+2)}+\frac{c (d+e x)^{m+5} (2 b e g-5 c d g+c e f)}{e^6 (m+5)}+\frac{c^2 g (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^2,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*(d + e*x)^(1 + m))/(e^6*(1 + m)) - ((c*d^2 - b*d*e + a*e^2)*(c*d*(4*e*f
- 5*d*g) - e*(2*b*e*f - 3*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((2*c^2*d^2*(3*e*f - 5*d*g) + b*
e^2*(b*e*f - 3*b*d*g + 2*a*e*g) + 2*c*e*(a*e*(e*f - 3*d*g) - 3*b*d*(e*f - 2*d*g)))*(d + e*x)^(3 + m))/(e^6*(3
+ m)) + ((b^2*e^2*g - 2*c^2*d*(2*e*f - 5*d*g) + 2*c*e*(b*e*f - 4*b*d*g + a*e*g))*(d + e*x)^(4 + m))/(e^6*(4 +
m)) + (c*(c*e*f - 5*c*d*g + 2*b*e*g)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (c^2*g*(d + e*x)^(6 + m))/(e^6*(6 + m)
)

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 (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2 (e f-d g) (d+e x)^m}{e^5}+\frac{\left (c d^2-b d e+a e^2\right ) (-c d (4 e f-5 d g)+e (2 b e f-3 b d g+a e g)) (d+e x)^{1+m}}{e^5}+\frac{\left (2 c^2 d^2 (3 e f-5 d g)+b e^2 (b e f-3 b d g+2 a e g)+2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))\right ) (d+e x)^{2+m}}{e^5}+\frac{\left (b^2 e^2 g-2 c^2 d (2 e f-5 d g)+2 c e (b e f-4 b d g+a e g)\right ) (d+e x)^{3+m}}{e^5}+\frac{c (c e f-5 c d g+2 b e g) (d+e x)^{4+m}}{e^5}+\frac{c^2 g (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 (e f-d g) (d+e x)^{1+m}}{e^6 (1+m)}-\frac{\left (c d^2-b d e+a e^2\right ) (c d (4 e f-5 d g)-e (2 b e f-3 b d g+a e g)) (d+e x)^{2+m}}{e^6 (2+m)}+\frac{\left (2 c^2 d^2 (3 e f-5 d g)+b e^2 (b e f-3 b d g+2 a e g)+2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac{\left (b^2 e^2 g-2 c^2 d (2 e f-5 d g)+2 c e (b e f-4 b d g+a e g)\right ) (d+e x)^{4+m}}{e^6 (4+m)}+\frac{c (c e f-5 c d g+2 b e g) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{c^2 g (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}

Mathematica [B]  time = 1.53982, size = 655, normalized size = 2.11 \[ \frac{(d+e x)^{m+1} \left (\frac{2 \left (\frac{(d+e x) \left (c^2 e^2 \left (4 a^2 e^2 g \left (m^2+8 m+15\right )+2 a b e \left (d g \left (4 m^2+11 m-18\right )+e f \left (2 m^2+19 m+42\right )\right )+b^2 d \left (d g \left (m^2-13 m+6\right )+2 e f \left (m^2+5 m-6\right )\right )\right )-b^2 c e^3 (m+2) (a e g (5 m+21)+b d g (2 m-3)+b e f (m+6))+2 c^3 d e \left (3 b d (d g (m-14)+3 e f (m+6))-2 a e \left (d g \left (m^2-4 m-30\right )+e f \left (2 m^2+19 m+42\right )\right )\right )+b^4 e^4 g \left (m^2+5 m+6\right )+12 c^4 d^3 (5 d g-e f (m+6))\right )}{e^2 (m+2)}+\frac{\left (e (a e-b d)+c d^2\right ) \left (2 c^2 e \left (2 a e \left (d g \left (m^2+m-15\right )+e f \left (m^2+10 m+24\right )\right )-3 b d (d g (m-9)+2 e f (m+6))\right )-b c e^2 (m+1) (2 a e g (2 m+9)+b d g (m-6)+b e f (m+6))+b^3 e^3 g \left (m^2+4 m+3\right )+12 c^3 d^2 (e f (m+6)-5 d g)\right )}{e^2 (m+1)}-(a+x (b+c x)) \left (c e (m+3) x \left (-c e (2 a e g (m+5)+b d g (m-4)+b e f (m+6))+b^2 e^2 g (m+3)+2 c^2 d (e f (m+6)-5 d g)\right )-(3 c d-b e) \left (-c e (2 a e g (m+5)+b d g (m-4)+b e f (m+6))+b^2 e^2 g (m+3)+2 c^2 d (e f (m+6)-5 d g)\right )+c e (m+4) \left (c e f (m+6) (b d-2 a e)+a b e^2 g (m+1)-2 a c d e g m+b d g (2 b e-5 c d)\right )\right )\right )}{c e^2 (m+3) (m+4)}+(a+x (b+c x))^2 (2 b e g+c (-5 d g+e f (m+6)+e g (m+5) x))\right )}{c e^2 (m+5) (m+6)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^2,x]

[Out]

((d + e*x)^(1 + m)*((a + x*(b + c*x))^2*(2*b*e*g + c*(-5*d*g + e*f*(6 + m) + e*g*(5 + m)*x)) + (2*(((c*d^2 + e
*(-(b*d) + a*e))*(b^3*e^3*g*(3 + 4*m + m^2) + 12*c^3*d^2*(-5*d*g + e*f*(6 + m)) - b*c*e^2*(1 + m)*(b*d*g*(-6 +
m) + b*e*f*(6 + m) + 2*a*e*g*(9 + 2*m)) + 2*c^2*e*(-3*b*d*(d*g*(-9 + m) + 2*e*f*(6 + m)) + 2*a*e*(d*g*(-15 +
m + m^2) + e*f*(24 + 10*m + m^2)))))/(e^2*(1 + m)) + ((b^4*e^4*g*(6 + 5*m + m^2) + 12*c^4*d^3*(5*d*g - e*f*(6
+ m)) - b^2*c*e^3*(2 + m)*(b*e*f*(6 + m) + b*d*g*(-3 + 2*m) + a*e*g*(21 + 5*m)) + 2*c^3*d*e*(3*b*d*(d*g*(-14 +
m) + 3*e*f*(6 + m)) - 2*a*e*(d*g*(-30 - 4*m + m^2) + e*f*(42 + 19*m + 2*m^2))) + c^2*e^2*(4*a^2*e^2*g*(15 + 8
*m + m^2) + b^2*d*(d*g*(6 - 13*m + m^2) + 2*e*f*(-6 + 5*m + m^2)) + 2*a*b*e*(e*f*(42 + 19*m + 2*m^2) + d*g*(-1
8 + 11*m + 4*m^2))))*(d + e*x))/(e^2*(2 + m)) - (c*e*(4 + m)*(b*d*(-5*c*d + 2*b*e)*g - 2*a*c*d*e*g*m + a*b*e^2
*g*(1 + m) + c*e*(b*d - 2*a*e)*f*(6 + m)) - (3*c*d - b*e)*(b^2*e^2*g*(3 + m) + 2*c^2*d*(-5*d*g + e*f*(6 + m))
- c*e*(b*d*g*(-4 + m) + 2*a*e*g*(5 + m) + b*e*f*(6 + m))) + c*e*(3 + m)*(b^2*e^2*g*(3 + m) + 2*c^2*d*(-5*d*g +
e*f*(6 + m)) - c*e*(b*d*g*(-4 + m) + 2*a*e*g*(5 + m) + b*e*f*(6 + m)))*x)*(a + x*(b + c*x))))/(c*e^2*(3 + m)*
(4 + m))))/(c*e^2*(5 + m)*(6 + m))

________________________________________________________________________________________

Maple [B]  time = 0.054, size = 2563, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^2,x)

[Out]

-(e*x+d)^(1+m)*(-c^2*e^5*g*m^5*x^5-2*b*c*e^5*g*m^5*x^4-c^2*e^5*f*m^5*x^4-15*c^2*e^5*g*m^4*x^5-2*a*c*e^5*g*m^5*
x^3-b^2*e^5*g*m^5*x^3-2*b*c*e^5*f*m^5*x^3-32*b*c*e^5*g*m^4*x^4+5*c^2*d*e^4*g*m^4*x^4-16*c^2*e^5*f*m^4*x^4-85*c
^2*e^5*g*m^3*x^5-2*a*b*e^5*g*m^5*x^2-2*a*c*e^5*f*m^5*x^2-34*a*c*e^5*g*m^4*x^3-b^2*e^5*f*m^5*x^2-17*b^2*e^5*g*m
^4*x^3+8*b*c*d*e^4*g*m^4*x^3-34*b*c*e^5*f*m^4*x^3-190*b*c*e^5*g*m^3*x^4+4*c^2*d*e^4*f*m^4*x^3+50*c^2*d*e^4*g*m
^3*x^4-95*c^2*e^5*f*m^3*x^4-225*c^2*e^5*g*m^2*x^5-a^2*e^5*g*m^5*x-2*a*b*e^5*f*m^5*x-36*a*b*e^5*g*m^4*x^2+6*a*c
*d*e^4*g*m^4*x^2-36*a*c*e^5*f*m^4*x^2-214*a*c*e^5*g*m^3*x^3+3*b^2*d*e^4*g*m^4*x^2-18*b^2*e^5*f*m^4*x^2-107*b^2
*e^5*g*m^3*x^3+6*b*c*d*e^4*f*m^4*x^2+96*b*c*d*e^4*g*m^3*x^3-214*b*c*e^5*f*m^3*x^3-520*b*c*e^5*g*m^2*x^4-20*c^2
*d^2*e^3*g*m^3*x^3+48*c^2*d*e^4*f*m^3*x^3+175*c^2*d*e^4*g*m^2*x^4-260*c^2*e^5*f*m^2*x^4-274*c^2*e^5*g*m*x^5-a^
2*e^5*f*m^5-19*a^2*e^5*g*m^4*x+4*a*b*d*e^4*g*m^4*x-38*a*b*e^5*f*m^4*x-242*a*b*e^5*g*m^3*x^2+4*a*c*d*e^4*f*m^4*
x+84*a*c*d*e^4*g*m^3*x^2-242*a*c*e^5*f*m^3*x^2-614*a*c*e^5*g*m^2*x^3+2*b^2*d*e^4*f*m^4*x+42*b^2*d*e^4*g*m^3*x^
2-121*b^2*e^5*f*m^3*x^2-307*b^2*e^5*g*m^2*x^3-24*b*c*d^2*e^3*g*m^3*x^2+84*b*c*d*e^4*f*m^3*x^2+376*b*c*d*e^4*g*
m^2*x^3-614*b*c*e^5*f*m^2*x^3-648*b*c*e^5*g*m*x^4-12*c^2*d^2*e^3*f*m^3*x^2-120*c^2*d^2*e^3*g*m^2*x^3+188*c^2*d
*e^4*f*m^2*x^3+250*c^2*d*e^4*g*m*x^4-324*c^2*e^5*f*m*x^4-120*c^2*e^5*g*x^5+a^2*d*e^4*g*m^4-20*a^2*e^5*f*m^4-13
7*a^2*e^5*g*m^3*x+2*a*b*d*e^4*f*m^4+64*a*b*d*e^4*g*m^3*x-274*a*b*e^5*f*m^3*x-744*a*b*e^5*g*m^2*x^2-12*a*c*d^2*
e^3*g*m^3*x+64*a*c*d*e^4*f*m^3*x+390*a*c*d*e^4*g*m^2*x^2-744*a*c*e^5*f*m^2*x^2-792*a*c*e^5*g*m*x^3-6*b^2*d^2*e
^3*g*m^3*x+32*b^2*d*e^4*f*m^3*x+195*b^2*d*e^4*g*m^2*x^2-372*b^2*e^5*f*m^2*x^2-396*b^2*e^5*g*m*x^3-12*b*c*d^2*e
^3*f*m^3*x-216*b*c*d^2*e^3*g*m^2*x^2+390*b*c*d*e^4*f*m^2*x^2+576*b*c*d*e^4*g*m*x^3-792*b*c*e^5*f*m*x^3-288*b*c
*e^5*g*x^4+60*c^2*d^3*e^2*g*m^2*x^2-108*c^2*d^2*e^3*f*m^2*x^2-220*c^2*d^2*e^3*g*m*x^3+288*c^2*d*e^4*f*m*x^3+12
0*c^2*d*e^4*g*x^4-144*c^2*e^5*f*x^4+18*a^2*d*e^4*g*m^3-155*a^2*e^5*f*m^3-461*a^2*e^5*g*m^2*x-4*a*b*d^2*e^3*g*m
^3+36*a*b*d*e^4*f*m^3+356*a*b*d*e^4*g*m^2*x-922*a*b*e^5*f*m^2*x-1016*a*b*e^5*g*m*x^2-4*a*c*d^2*e^3*f*m^3-144*a
*c*d^2*e^3*g*m^2*x+356*a*c*d*e^4*f*m^2*x+672*a*c*d*e^4*g*m*x^2-1016*a*c*e^5*f*m*x^2-360*a*c*e^5*g*x^3-2*b^2*d^
2*e^3*f*m^3-72*b^2*d^2*e^3*g*m^2*x+178*b^2*d*e^4*f*m^2*x+336*b^2*d*e^4*g*m*x^2-508*b^2*e^5*f*m*x^2-180*b^2*e^5
*g*x^3+48*b*c*d^3*e^2*g*m^2*x-144*b*c*d^2*e^3*f*m^2*x-480*b*c*d^2*e^3*g*m*x^2+672*b*c*d*e^4*f*m*x^2+288*b*c*d*
e^4*g*x^3-360*b*c*e^5*f*x^3+24*c^2*d^3*e^2*f*m^2*x+180*c^2*d^3*e^2*g*m*x^2-240*c^2*d^2*e^3*f*m*x^2-120*c^2*d^2
*e^3*g*x^3+144*c^2*d*e^4*f*x^3+119*a^2*d*e^4*g*m^2-580*a^2*e^5*f*m^2-702*a^2*e^5*g*m*x-60*a*b*d^2*e^3*g*m^2+23
8*a*b*d*e^4*f*m^2+776*a*b*d*e^4*g*m*x-1404*a*b*e^5*f*m*x-480*a*b*e^5*g*x^2+12*a*c*d^3*e^2*g*m^2-60*a*c*d^2*e^3
*f*m^2-492*a*c*d^2*e^3*g*m*x+776*a*c*d*e^4*f*m*x+360*a*c*d*e^4*g*x^2-480*a*c*e^5*f*x^2+6*b^2*d^3*e^2*g*m^2-30*
b^2*d^2*e^3*f*m^2-246*b^2*d^2*e^3*g*m*x+388*b^2*d*e^4*f*m*x+180*b^2*d*e^4*g*x^2-240*b^2*e^5*f*x^2+12*b*c*d^3*e
^2*f*m^2+336*b*c*d^3*e^2*g*m*x-492*b*c*d^2*e^3*f*m*x-288*b*c*d^2*e^3*g*x^2+360*b*c*d*e^4*f*x^2-120*c^2*d^4*e*g
*m*x+168*c^2*d^3*e^2*f*m*x+120*c^2*d^3*e^2*g*x^2-144*c^2*d^2*e^3*f*x^2+342*a^2*d*e^4*g*m-1044*a^2*e^5*f*m-360*
a^2*e^5*g*x-296*a*b*d^2*e^3*g*m+684*a*b*d*e^4*f*m+480*a*b*d*e^4*g*x-720*a*b*e^5*f*x+132*a*c*d^3*e^2*g*m-296*a*
c*d^2*e^3*f*m-360*a*c*d^2*e^3*g*x+480*a*c*d*e^4*f*x+66*b^2*d^3*e^2*g*m-148*b^2*d^2*e^3*f*m-180*b^2*d^2*e^3*g*x
+240*b^2*d*e^4*f*x-48*b*c*d^4*e*g*m+132*b*c*d^3*e^2*f*m+288*b*c*d^3*e^2*g*x-360*b*c*d^2*e^3*f*x-24*c^2*d^4*e*f
*m-120*c^2*d^4*e*g*x+144*c^2*d^3*e^2*f*x+360*a^2*d*e^4*g-720*a^2*e^5*f-480*a*b*d^2*e^3*g+720*a*b*d*e^4*f+360*a
*c*d^3*e^2*g-480*a*c*d^2*e^3*f+180*b^2*d^3*e^2*g-240*b^2*d^2*e^3*f-288*b*c*d^4*e*g+360*b*c*d^3*e^2*f+120*c^2*d
^5*g-144*c^2*d^4*e*f)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)

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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((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.95663, size = 5231, normalized size = 16.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

(a^2*d*e^5*f*m^5 + (c^2*e^6*g*m^5 + 15*c^2*e^6*g*m^4 + 85*c^2*e^6*g*m^3 + 225*c^2*e^6*g*m^2 + 274*c^2*e^6*g*m
+ 120*c^2*e^6*g)*x^6 + (144*c^2*e^6*f + 288*b*c*e^6*g + (c^2*e^6*f + (c^2*d*e^5 + 2*b*c*e^6)*g)*m^5 + 2*(8*c^2
*e^6*f + (5*c^2*d*e^5 + 16*b*c*e^6)*g)*m^4 + 5*(19*c^2*e^6*f + (7*c^2*d*e^5 + 38*b*c*e^6)*g)*m^3 + 10*(26*c^2*
e^6*f + (5*c^2*d*e^5 + 52*b*c*e^6)*g)*m^2 + 12*(27*c^2*e^6*f + 2*(c^2*d*e^5 + 27*b*c*e^6)*g)*m)*x^5 - (a^2*d^2
*e^4*g + 2*(a*b*d^2*e^4 - 10*a^2*d*e^5)*f)*m^4 + (360*b*c*e^6*f + 180*(b^2 + 2*a*c)*e^6*g + ((c^2*d*e^5 + 2*b*
c*e^6)*f + (2*b*c*d*e^5 + (b^2 + 2*a*c)*e^6)*g)*m^5 + (2*(6*c^2*d*e^5 + 17*b*c*e^6)*f - (5*c^2*d^2*e^4 - 24*b*
c*d*e^5 - 17*(b^2 + 2*a*c)*e^6)*g)*m^4 + ((47*c^2*d*e^5 + 214*b*c*e^6)*f - (30*c^2*d^2*e^4 - 94*b*c*d*e^5 - 10
7*(b^2 + 2*a*c)*e^6)*g)*m^3 + (2*(36*c^2*d*e^5 + 307*b*c*e^6)*f - (55*c^2*d^2*e^4 - 144*b*c*d*e^5 - 307*(b^2 +
2*a*c)*e^6)*g)*m^2 + 6*(6*(c^2*d*e^5 + 22*b*c*e^6)*f - (5*c^2*d^2*e^4 - 12*b*c*d*e^5 - 66*(b^2 + 2*a*c)*e^6)*
g)*m)*x^4 - ((36*a*b*d^2*e^4 - 155*a^2*d*e^5 - 2*(b^2 + 2*a*c)*d^3*e^3)*f - 2*(2*a*b*d^3*e^3 - 9*a^2*d^2*e^4)*
g)*m^3 + (480*a*b*e^6*g + 240*(b^2 + 2*a*c)*e^6*f + ((2*b*c*d*e^5 + (b^2 + 2*a*c)*e^6)*f + (2*a*b*e^6 + (b^2 +
2*a*c)*d*e^5)*g)*m^5 - 2*((2*c^2*d^2*e^4 - 14*b*c*d*e^5 - 9*(b^2 + 2*a*c)*e^6)*f + (4*b*c*d^2*e^4 - 18*a*b*e^
6 - 7*(b^2 + 2*a*c)*d*e^5)*g)*m^4 - ((36*c^2*d^2*e^4 - 130*b*c*d*e^5 - 121*(b^2 + 2*a*c)*e^6)*f - (20*c^2*d^3*
e^3 - 72*b*c*d^2*e^4 + 242*a*b*e^6 + 65*(b^2 + 2*a*c)*d*e^5)*g)*m^3 - 4*((20*c^2*d^2*e^4 - 56*b*c*d*e^5 - 93*(
b^2 + 2*a*c)*e^6)*f - (15*c^2*d^3*e^3 - 40*b*c*d^2*e^4 + 186*a*b*e^6 + 28*(b^2 + 2*a*c)*d*e^5)*g)*m^2 - 4*((12
*c^2*d^2*e^4 - 30*b*c*d*e^5 - 127*(b^2 + 2*a*c)*e^6)*f - (10*c^2*d^3*e^3 - 24*b*c*d^2*e^4 + 254*a*b*e^6 + 15*(
b^2 + 2*a*c)*d*e^5)*g)*m)*x^3 - (2*(6*b*c*d^4*e^2 + 119*a*b*d^2*e^4 - 290*a^2*d*e^5 - 15*(b^2 + 2*a*c)*d^3*e^3
)*f - (60*a*b*d^3*e^3 - 119*a^2*d^2*e^4 - 6*(b^2 + 2*a*c)*d^4*e^2)*g)*m^2 + (720*a*b*e^6*f + 360*a^2*e^6*g + (
(2*a*b*e^6 + (b^2 + 2*a*c)*d*e^5)*f + (2*a*b*d*e^5 + a^2*e^6)*g)*m^5 - (2*(3*b*c*d^2*e^4 - 19*a*b*e^6 - 8*(b^2
+ 2*a*c)*d*e^5)*f - (32*a*b*d*e^5 + 19*a^2*e^6 - 3*(b^2 + 2*a*c)*d^2*e^4)*g)*m^4 + ((12*c^2*d^3*e^3 - 72*b*c*
d^2*e^4 + 274*a*b*e^6 + 89*(b^2 + 2*a*c)*d*e^5)*f + (24*b*c*d^3*e^3 + 178*a*b*d*e^5 + 137*a^2*e^6 - 36*(b^2 +
2*a*c)*d^2*e^4)*g)*m^3 + (2*(42*c^2*d^3*e^3 - 123*b*c*d^2*e^4 + 461*a*b*e^6 + 97*(b^2 + 2*a*c)*d*e^5)*f - (60*
c^2*d^4*e^2 - 168*b*c*d^3*e^3 - 388*a*b*d*e^5 - 461*a^2*e^6 + 123*(b^2 + 2*a*c)*d^2*e^4)*g)*m^2 + 6*(2*(6*c^2*
d^3*e^3 - 15*b*c*d^2*e^4 + 117*a*b*e^6 + 10*(b^2 + 2*a*c)*d*e^5)*f - (10*c^2*d^4*e^2 - 24*b*c*d^3*e^3 - 40*a*b
*d*e^5 - 117*a^2*e^6 + 15*(b^2 + 2*a*c)*d^2*e^4)*g)*m)*x^2 + 24*(6*c^2*d^5*e - 15*b*c*d^4*e^2 - 30*a*b*d^2*e^4
+ 30*a^2*d*e^5 + 10*(b^2 + 2*a*c)*d^3*e^3)*f - 12*(10*c^2*d^6 - 24*b*c*d^5*e - 40*a*b*d^3*e^3 + 30*a^2*d^2*e^
4 + 15*(b^2 + 2*a*c)*d^4*e^2)*g + 2*(2*(6*c^2*d^5*e - 33*b*c*d^4*e^2 - 171*a*b*d^2*e^4 + 261*a^2*d*e^5 + 37*(b
^2 + 2*a*c)*d^3*e^3)*f + (24*b*c*d^5*e + 148*a*b*d^3*e^3 - 171*a^2*d^2*e^4 - 33*(b^2 + 2*a*c)*d^4*e^2)*g)*m +
(720*a^2*e^6*f + (a^2*d*e^5*g + (2*a*b*d*e^5 + a^2*e^6)*f)*m^5 + 2*((18*a*b*d*e^5 + 10*a^2*e^6 - (b^2 + 2*a*c)
*d^2*e^4)*f - (2*a*b*d^2*e^4 - 9*a^2*d*e^5)*g)*m^4 + ((12*b*c*d^3*e^3 + 238*a*b*d*e^5 + 155*a^2*e^6 - 30*(b^2
+ 2*a*c)*d^2*e^4)*f - (60*a*b*d^2*e^4 - 119*a^2*d*e^5 - 6*(b^2 + 2*a*c)*d^3*e^3)*g)*m^3 - 2*(2*(6*c^2*d^4*e^2
- 33*b*c*d^3*e^3 - 171*a*b*d*e^5 - 145*a^2*e^6 + 37*(b^2 + 2*a*c)*d^2*e^4)*f + (24*b*c*d^4*e^2 + 148*a*b*d^2*e
^4 - 171*a^2*d*e^5 - 33*(b^2 + 2*a*c)*d^3*e^3)*g)*m^2 - 12*((12*c^2*d^4*e^2 - 30*b*c*d^3*e^3 - 60*a*b*d*e^5 -
87*a^2*e^6 + 20*(b^2 + 2*a*c)*d^2*e^4)*f - (10*c^2*d^5*e - 24*b*c*d^4*e^2 - 40*a*b*d^2*e^4 + 30*a^2*d*e^5 + 15
*(b^2 + 2*a*c)*d^3*e^3)*g)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2
+ 1764*e^6*m + 720*e^6)

________________________________________________________________________________________

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((e*x+d)**m*(g*x+f)*(c*x**2+b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.32535, size = 6669, normalized size = 21.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*c^2*g*m^5*x^6*e^6 + (x*e + d)^m*c^2*d*g*m^5*x^5*e^5 + (x*e + d)^m*c^2*f*m^5*x^5*e^6 + 2*(x*e + d)
^m*b*c*g*m^5*x^5*e^6 + 15*(x*e + d)^m*c^2*g*m^4*x^6*e^6 + (x*e + d)^m*c^2*d*f*m^5*x^4*e^5 + 2*(x*e + d)^m*b*c*
d*g*m^5*x^4*e^5 + 10*(x*e + d)^m*c^2*d*g*m^4*x^5*e^5 - 5*(x*e + d)^m*c^2*d^2*g*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c
*f*m^5*x^4*e^6 + (x*e + d)^m*b^2*g*m^5*x^4*e^6 + 2*(x*e + d)^m*a*c*g*m^5*x^4*e^6 + 16*(x*e + d)^m*c^2*f*m^4*x^
5*e^6 + 32*(x*e + d)^m*b*c*g*m^4*x^5*e^6 + 85*(x*e + d)^m*c^2*g*m^3*x^6*e^6 + 2*(x*e + d)^m*b*c*d*f*m^5*x^3*e^
5 + (x*e + d)^m*b^2*d*g*m^5*x^3*e^5 + 2*(x*e + d)^m*a*c*d*g*m^5*x^3*e^5 + 12*(x*e + d)^m*c^2*d*f*m^4*x^4*e^5 +
24*(x*e + d)^m*b*c*d*g*m^4*x^4*e^5 + 35*(x*e + d)^m*c^2*d*g*m^3*x^5*e^5 - 4*(x*e + d)^m*c^2*d^2*f*m^4*x^3*e^4
- 8*(x*e + d)^m*b*c*d^2*g*m^4*x^3*e^4 - 30*(x*e + d)^m*c^2*d^2*g*m^3*x^4*e^4 + 20*(x*e + d)^m*c^2*d^3*g*m^3*x
^3*e^3 + (x*e + d)^m*b^2*f*m^5*x^3*e^6 + 2*(x*e + d)^m*a*c*f*m^5*x^3*e^6 + 2*(x*e + d)^m*a*b*g*m^5*x^3*e^6 + 3
4*(x*e + d)^m*b*c*f*m^4*x^4*e^6 + 17*(x*e + d)^m*b^2*g*m^4*x^4*e^6 + 34*(x*e + d)^m*a*c*g*m^4*x^4*e^6 + 95*(x*
e + d)^m*c^2*f*m^3*x^5*e^6 + 190*(x*e + d)^m*b*c*g*m^3*x^5*e^6 + 225*(x*e + d)^m*c^2*g*m^2*x^6*e^6 + (x*e + d)
^m*b^2*d*f*m^5*x^2*e^5 + 2*(x*e + d)^m*a*c*d*f*m^5*x^2*e^5 + 2*(x*e + d)^m*a*b*d*g*m^5*x^2*e^5 + 28*(x*e + d)^
m*b*c*d*f*m^4*x^3*e^5 + 14*(x*e + d)^m*b^2*d*g*m^4*x^3*e^5 + 28*(x*e + d)^m*a*c*d*g*m^4*x^3*e^5 + 47*(x*e + d)
^m*c^2*d*f*m^3*x^4*e^5 + 94*(x*e + d)^m*b*c*d*g*m^3*x^4*e^5 + 50*(x*e + d)^m*c^2*d*g*m^2*x^5*e^5 - 6*(x*e + d)
^m*b*c*d^2*f*m^4*x^2*e^4 - 3*(x*e + d)^m*b^2*d^2*g*m^4*x^2*e^4 - 6*(x*e + d)^m*a*c*d^2*g*m^4*x^2*e^4 - 36*(x*e
+ d)^m*c^2*d^2*f*m^3*x^3*e^4 - 72*(x*e + d)^m*b*c*d^2*g*m^3*x^3*e^4 - 55*(x*e + d)^m*c^2*d^2*g*m^2*x^4*e^4 +
12*(x*e + d)^m*c^2*d^3*f*m^3*x^2*e^3 + 24*(x*e + d)^m*b*c*d^3*g*m^3*x^2*e^3 + 60*(x*e + d)^m*c^2*d^3*g*m^2*x^3
*e^3 - 60*(x*e + d)^m*c^2*d^4*g*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*f*m^5*x^2*e^6 + (x*e + d)^m*a^2*g*m^5*x^2*e^6
+ 18*(x*e + d)^m*b^2*f*m^4*x^3*e^6 + 36*(x*e + d)^m*a*c*f*m^4*x^3*e^6 + 36*(x*e + d)^m*a*b*g*m^4*x^3*e^6 + 214
*(x*e + d)^m*b*c*f*m^3*x^4*e^6 + 107*(x*e + d)^m*b^2*g*m^3*x^4*e^6 + 214*(x*e + d)^m*a*c*g*m^3*x^4*e^6 + 260*(
x*e + d)^m*c^2*f*m^2*x^5*e^6 + 520*(x*e + d)^m*b*c*g*m^2*x^5*e^6 + 274*(x*e + d)^m*c^2*g*m*x^6*e^6 + 2*(x*e +
d)^m*a*b*d*f*m^5*x*e^5 + (x*e + d)^m*a^2*d*g*m^5*x*e^5 + 16*(x*e + d)^m*b^2*d*f*m^4*x^2*e^5 + 32*(x*e + d)^m*a
*c*d*f*m^4*x^2*e^5 + 32*(x*e + d)^m*a*b*d*g*m^4*x^2*e^5 + 130*(x*e + d)^m*b*c*d*f*m^3*x^3*e^5 + 65*(x*e + d)^m
*b^2*d*g*m^3*x^3*e^5 + 130*(x*e + d)^m*a*c*d*g*m^3*x^3*e^5 + 72*(x*e + d)^m*c^2*d*f*m^2*x^4*e^5 + 144*(x*e + d
)^m*b*c*d*g*m^2*x^4*e^5 + 24*(x*e + d)^m*c^2*d*g*m*x^5*e^5 - 2*(x*e + d)^m*b^2*d^2*f*m^4*x*e^4 - 4*(x*e + d)^m
*a*c*d^2*f*m^4*x*e^4 - 4*(x*e + d)^m*a*b*d^2*g*m^4*x*e^4 - 72*(x*e + d)^m*b*c*d^2*f*m^3*x^2*e^4 - 36*(x*e + d)
^m*b^2*d^2*g*m^3*x^2*e^4 - 72*(x*e + d)^m*a*c*d^2*g*m^3*x^2*e^4 - 80*(x*e + d)^m*c^2*d^2*f*m^2*x^3*e^4 - 160*(
x*e + d)^m*b*c*d^2*g*m^2*x^3*e^4 - 30*(x*e + d)^m*c^2*d^2*g*m*x^4*e^4 + 12*(x*e + d)^m*b*c*d^3*f*m^3*x*e^3 + 6
*(x*e + d)^m*b^2*d^3*g*m^3*x*e^3 + 12*(x*e + d)^m*a*c*d^3*g*m^3*x*e^3 + 84*(x*e + d)^m*c^2*d^3*f*m^2*x^2*e^3 +
168*(x*e + d)^m*b*c*d^3*g*m^2*x^2*e^3 + 40*(x*e + d)^m*c^2*d^3*g*m*x^3*e^3 - 24*(x*e + d)^m*c^2*d^4*f*m^2*x*e
^2 - 48*(x*e + d)^m*b*c*d^4*g*m^2*x*e^2 - 60*(x*e + d)^m*c^2*d^4*g*m*x^2*e^2 + 120*(x*e + d)^m*c^2*d^5*g*m*x*e
+ (x*e + d)^m*a^2*f*m^5*x*e^6 + 38*(x*e + d)^m*a*b*f*m^4*x^2*e^6 + 19*(x*e + d)^m*a^2*g*m^4*x^2*e^6 + 121*(x*
e + d)^m*b^2*f*m^3*x^3*e^6 + 242*(x*e + d)^m*a*c*f*m^3*x^3*e^6 + 242*(x*e + d)^m*a*b*g*m^3*x^3*e^6 + 614*(x*e
+ d)^m*b*c*f*m^2*x^4*e^6 + 307*(x*e + d)^m*b^2*g*m^2*x^4*e^6 + 614*(x*e + d)^m*a*c*g*m^2*x^4*e^6 + 324*(x*e +
d)^m*c^2*f*m*x^5*e^6 + 648*(x*e + d)^m*b*c*g*m*x^5*e^6 + 120*(x*e + d)^m*c^2*g*x^6*e^6 + (x*e + d)^m*a^2*d*f*m
^5*e^5 + 36*(x*e + d)^m*a*b*d*f*m^4*x*e^5 + 18*(x*e + d)^m*a^2*d*g*m^4*x*e^5 + 89*(x*e + d)^m*b^2*d*f*m^3*x^2*
e^5 + 178*(x*e + d)^m*a*c*d*f*m^3*x^2*e^5 + 178*(x*e + d)^m*a*b*d*g*m^3*x^2*e^5 + 224*(x*e + d)^m*b*c*d*f*m^2*
x^3*e^5 + 112*(x*e + d)^m*b^2*d*g*m^2*x^3*e^5 + 224*(x*e + d)^m*a*c*d*g*m^2*x^3*e^5 + 36*(x*e + d)^m*c^2*d*f*m
*x^4*e^5 + 72*(x*e + d)^m*b*c*d*g*m*x^4*e^5 - 2*(x*e + d)^m*a*b*d^2*f*m^4*e^4 - (x*e + d)^m*a^2*d^2*g*m^4*e^4
- 30*(x*e + d)^m*b^2*d^2*f*m^3*x*e^4 - 60*(x*e + d)^m*a*c*d^2*f*m^3*x*e^4 - 60*(x*e + d)^m*a*b*d^2*g*m^3*x*e^4
- 246*(x*e + d)^m*b*c*d^2*f*m^2*x^2*e^4 - 123*(x*e + d)^m*b^2*d^2*g*m^2*x^2*e^4 - 246*(x*e + d)^m*a*c*d^2*g*m
^2*x^2*e^4 - 48*(x*e + d)^m*c^2*d^2*f*m*x^3*e^4 - 96*(x*e + d)^m*b*c*d^2*g*m*x^3*e^4 + 2*(x*e + d)^m*b^2*d^3*f
*m^3*e^3 + 4*(x*e + d)^m*a*c*d^3*f*m^3*e^3 + 4*(x*e + d)^m*a*b*d^3*g*m^3*e^3 + 132*(x*e + d)^m*b*c*d^3*f*m^2*x
*e^3 + 66*(x*e + d)^m*b^2*d^3*g*m^2*x*e^3 + 132*(x*e + d)^m*a*c*d^3*g*m^2*x*e^3 + 72*(x*e + d)^m*c^2*d^3*f*m*x
^2*e^3 + 144*(x*e + d)^m*b*c*d^3*g*m*x^2*e^3 - 12*(x*e + d)^m*b*c*d^4*f*m^2*e^2 - 6*(x*e + d)^m*b^2*d^4*g*m^2*
e^2 - 12*(x*e + d)^m*a*c*d^4*g*m^2*e^2 - 144*(x*e + d)^m*c^2*d^4*f*m*x*e^2 - 288*(x*e + d)^m*b*c*d^4*g*m*x*e^2
+ 24*(x*e + d)^m*c^2*d^5*f*m*e + 48*(x*e + d)^m*b*c*d^5*g*m*e - 120*(x*e + d)^m*c^2*d^6*g + 20*(x*e + d)^m*a^
2*f*m^4*x*e^6 + 274*(x*e + d)^m*a*b*f*m^3*x^2*e^6 + 137*(x*e + d)^m*a^2*g*m^3*x^2*e^6 + 372*(x*e + d)^m*b^2*f*
m^2*x^3*e^6 + 744*(x*e + d)^m*a*c*f*m^2*x^3*e^6 + 744*(x*e + d)^m*a*b*g*m^2*x^3*e^6 + 792*(x*e + d)^m*b*c*f*m*
x^4*e^6 + 396*(x*e + d)^m*b^2*g*m*x^4*e^6 + 792*(x*e + d)^m*a*c*g*m*x^4*e^6 + 144*(x*e + d)^m*c^2*f*x^5*e^6 +
288*(x*e + d)^m*b*c*g*x^5*e^6 + 20*(x*e + d)^m*a^2*d*f*m^4*e^5 + 238*(x*e + d)^m*a*b*d*f*m^3*x*e^5 + 119*(x*e
+ d)^m*a^2*d*g*m^3*x*e^5 + 194*(x*e + d)^m*b^2*d*f*m^2*x^2*e^5 + 388*(x*e + d)^m*a*c*d*f*m^2*x^2*e^5 + 388*(x*
e + d)^m*a*b*d*g*m^2*x^2*e^5 + 120*(x*e + d)^m*b*c*d*f*m*x^3*e^5 + 60*(x*e + d)^m*b^2*d*g*m*x^3*e^5 + 120*(x*e
+ d)^m*a*c*d*g*m*x^3*e^5 - 36*(x*e + d)^m*a*b*d^2*f*m^3*e^4 - 18*(x*e + d)^m*a^2*d^2*g*m^3*e^4 - 148*(x*e + d
)^m*b^2*d^2*f*m^2*x*e^4 - 296*(x*e + d)^m*a*c*d^2*f*m^2*x*e^4 - 296*(x*e + d)^m*a*b*d^2*g*m^2*x*e^4 - 180*(x*e
+ d)^m*b*c*d^2*f*m*x^2*e^4 - 90*(x*e + d)^m*b^2*d^2*g*m*x^2*e^4 - 180*(x*e + d)^m*a*c*d^2*g*m*x^2*e^4 + 30*(x
*e + d)^m*b^2*d^3*f*m^2*e^3 + 60*(x*e + d)^m*a*c*d^3*f*m^2*e^3 + 60*(x*e + d)^m*a*b*d^3*g*m^2*e^3 + 360*(x*e +
d)^m*b*c*d^3*f*m*x*e^3 + 180*(x*e + d)^m*b^2*d^3*g*m*x*e^3 + 360*(x*e + d)^m*a*c*d^3*g*m*x*e^3 - 132*(x*e + d
)^m*b*c*d^4*f*m*e^2 - 66*(x*e + d)^m*b^2*d^4*g*m*e^2 - 132*(x*e + d)^m*a*c*d^4*g*m*e^2 + 144*(x*e + d)^m*c^2*d
^5*f*e + 288*(x*e + d)^m*b*c*d^5*g*e + 155*(x*e + d)^m*a^2*f*m^3*x*e^6 + 922*(x*e + d)^m*a*b*f*m^2*x^2*e^6 + 4
61*(x*e + d)^m*a^2*g*m^2*x^2*e^6 + 508*(x*e + d)^m*b^2*f*m*x^3*e^6 + 1016*(x*e + d)^m*a*c*f*m*x^3*e^6 + 1016*(
x*e + d)^m*a*b*g*m*x^3*e^6 + 360*(x*e + d)^m*b*c*f*x^4*e^6 + 180*(x*e + d)^m*b^2*g*x^4*e^6 + 360*(x*e + d)^m*a
*c*g*x^4*e^6 + 155*(x*e + d)^m*a^2*d*f*m^3*e^5 + 684*(x*e + d)^m*a*b*d*f*m^2*x*e^5 + 342*(x*e + d)^m*a^2*d*g*m
^2*x*e^5 + 120*(x*e + d)^m*b^2*d*f*m*x^2*e^5 + 240*(x*e + d)^m*a*c*d*f*m*x^2*e^5 + 240*(x*e + d)^m*a*b*d*g*m*x
^2*e^5 - 238*(x*e + d)^m*a*b*d^2*f*m^2*e^4 - 119*(x*e + d)^m*a^2*d^2*g*m^2*e^4 - 240*(x*e + d)^m*b^2*d^2*f*m*x
*e^4 - 480*(x*e + d)^m*a*c*d^2*f*m*x*e^4 - 480*(x*e + d)^m*a*b*d^2*g*m*x*e^4 + 148*(x*e + d)^m*b^2*d^3*f*m*e^3
+ 296*(x*e + d)^m*a*c*d^3*f*m*e^3 + 296*(x*e + d)^m*a*b*d^3*g*m*e^3 - 360*(x*e + d)^m*b*c*d^4*f*e^2 - 180*(x*
e + d)^m*b^2*d^4*g*e^2 - 360*(x*e + d)^m*a*c*d^4*g*e^2 + 580*(x*e + d)^m*a^2*f*m^2*x*e^6 + 1404*(x*e + d)^m*a*
b*f*m*x^2*e^6 + 702*(x*e + d)^m*a^2*g*m*x^2*e^6 + 240*(x*e + d)^m*b^2*f*x^3*e^6 + 480*(x*e + d)^m*a*c*f*x^3*e^
6 + 480*(x*e + d)^m*a*b*g*x^3*e^6 + 580*(x*e + d)^m*a^2*d*f*m^2*e^5 + 720*(x*e + d)^m*a*b*d*f*m*x*e^5 + 360*(x
*e + d)^m*a^2*d*g*m*x*e^5 - 684*(x*e + d)^m*a*b*d^2*f*m*e^4 - 342*(x*e + d)^m*a^2*d^2*g*m*e^4 + 240*(x*e + d)^
m*b^2*d^3*f*e^3 + 480*(x*e + d)^m*a*c*d^3*f*e^3 + 480*(x*e + d)^m*a*b*d^3*g*e^3 + 1044*(x*e + d)^m*a^2*f*m*x*e
^6 + 720*(x*e + d)^m*a*b*f*x^2*e^6 + 360*(x*e + d)^m*a^2*g*x^2*e^6 + 1044*(x*e + d)^m*a^2*d*f*m*e^5 - 720*(x*e
+ d)^m*a*b*d^2*f*e^4 - 360*(x*e + d)^m*a^2*d^2*g*e^4 + 720*(x*e + d)^m*a^2*f*x*e^6 + 720*(x*e + d)^m*a^2*d*f*
e^5)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1764*m*e^6 + 720*e^6)