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

Optimal. Leaf size=525 \[ \frac{(d+e x)^{m+3} \left (e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )+2 c e \left (a e \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )-b d \left (10 d^2 g^2-12 d e f g+3 e^2 f^2\right )\right )+c^2 d^2 \left (15 d^2 g^2-20 d e f g+6 e^2 f^2\right )\right )}{e^7 (m+3)}+\frac{(d+e x)^{m+5} \left (2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left (15 d^2 g^2-10 d e f g+e^2 f^2\right )\right )}{e^7 (m+5)}+\frac{2 (d+e x)^{m+4} \left (c e \left (2 a e g (e f-2 d g)+b \left (10 d^2 g^2-8 d e f g+e^2 f^2\right )\right )+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left (5 d^2 g^2-5 d e f g+e^2 f^2\right )\right )}{e^7 (m+4)}+\frac{(e f-d g)^2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+1)}-\frac{2 (e f-d g) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^7 (m+2)}+\frac{2 c g (d+e x)^{m+6} (b e g-3 c d g+c e f)}{e^7 (m+6)}+\frac{c^2 g^2 (d+e x)^{m+7}}{e^7 (m+7)} \]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (2*(c*d^2 - b*d*e + a*e^2)*(e*f -
d*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((c^2*d^2*(6*e^2*f
^2 - 20*d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^
2*g^2)) + 2*c*e*(a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2) - b*d*(3*e^2*f^2 - 12*d*e*f*g + 10*d^2*g^2)))*(d + e*x)
^(3 + m))/(e^7*(3 + m)) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) +
c*e*(2*a*e*g*(e*f - 2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((b^2*e
^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^(5 + m))/(e^
7*(5 + m)) + (2*c*g*(c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^2*g^2*(d + e*x)^(7 + m))/(
e^7*(7 + m))

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Rubi [A]  time = 0.61498, antiderivative size = 525, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {947} \[ \frac{(d+e x)^{m+3} \left (e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )+2 c e \left (a e \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )-b d \left (10 d^2 g^2-12 d e f g+3 e^2 f^2\right )\right )+c^2 d^2 \left (15 d^2 g^2-20 d e f g+6 e^2 f^2\right )\right )}{e^7 (m+3)}+\frac{(d+e x)^{m+5} \left (2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left (15 d^2 g^2-10 d e f g+e^2 f^2\right )\right )}{e^7 (m+5)}+\frac{2 (d+e x)^{m+4} \left (c e \left (2 a e g (e f-2 d g)+b \left (10 d^2 g^2-8 d e f g+e^2 f^2\right )\right )+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left (5 d^2 g^2-5 d e f g+e^2 f^2\right )\right )}{e^7 (m+4)}+\frac{(e f-d g)^2 (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+1)}-\frac{2 (e f-d g) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^7 (m+2)}+\frac{2 c g (d+e x)^{m+6} (b e g-3 c d g+c e f)}{e^7 (m+6)}+\frac{c^2 g^2 (d+e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (2*(c*d^2 - b*d*e + a*e^2)*(e*f -
d*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((c^2*d^2*(6*e^2*f
^2 - 20*d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^
2*g^2)) + 2*c*e*(a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2) - b*d*(3*e^2*f^2 - 12*d*e*f*g + 10*d^2*g^2)))*(d + e*x)
^(3 + m))/(e^7*(3 + m)) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) +
c*e*(2*a*e*g*(e*f - 2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((b^2*e
^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^(5 + m))/(e^
7*(5 + m)) + (2*c*g*(c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^2*g^2*(d + e*x)^(7 + m))/(
e^7*(7 + m))

Rule 947

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (E
qQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))

Rubi steps

\begin{align*} \int (d+e x)^m (f+g x)^2 \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)^2 (d+e x)^m}{e^6}+\frac{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)) (d+e x)^{1+m}}{e^6}+\frac{\left (c^2 d^2 \left (6 e^2 f^2-20 d e f g+15 d^2 g^2\right )+e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right )+2 c e \left (a e \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )-b d \left (3 e^2 f^2-12 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{2+m}}{e^6}+\frac{2 \left (b e^2 g (b e f-2 b d g+a e g)-2 c^2 d \left (e^2 f^2-5 d e f g+5 d^2 g^2\right )+c e \left (2 a e g (e f-2 d g)+b \left (e^2 f^2-8 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{3+m}}{e^6}+\frac{\left (b^2 e^2 g^2+2 c e g (2 b e f-5 b d g+a e g)+c^2 \left (e^2 f^2-10 d e f g+15 d^2 g^2\right )\right ) (d+e x)^{4+m}}{e^6}+\frac{2 c g (c e f-3 c d g+b e g) (d+e x)^{5+m}}{e^6}+\frac{c^2 g^2 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2 (d+e x)^{1+m}}{e^7 (1+m)}-\frac{2 \left (c d^2-b d e+a e^2\right ) (e f-d g) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^7 (2+m)}+\frac{\left (c^2 d^2 \left (6 e^2 f^2-20 d e f g+15 d^2 g^2\right )+e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )\right )+2 c e \left (a e \left (e^2 f^2-6 d e f g+6 d^2 g^2\right )-b d \left (3 e^2 f^2-12 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{3+m}}{e^7 (3+m)}+\frac{2 \left (b e^2 g (b e f-2 b d g+a e g)-2 c^2 d \left (e^2 f^2-5 d e f g+5 d^2 g^2\right )+c e \left (2 a e g (e f-2 d g)+b \left (e^2 f^2-8 d e f g+10 d^2 g^2\right )\right )\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac{\left (b^2 e^2 g^2+2 c e g (2 b e f-5 b d g+a e g)+c^2 \left (e^2 f^2-10 d e f g+15 d^2 g^2\right )\right ) (d+e x)^{5+m}}{e^7 (5+m)}+\frac{2 c g (c e f-3 c d g+b e g) (d+e x)^{6+m}}{e^7 (6+m)}+\frac{c^2 g^2 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.847475, size = 492, normalized size = 0.94 \[ \frac{(d+e x)^{m+1} \left (\frac{(d+e x)^2 \left (e^2 \left (a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )\right )+2 c e \left (a e \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )+b d \left (-10 d^2 g^2+12 d e f g-3 e^2 f^2\right )\right )+c^2 d^2 \left (15 d^2 g^2-20 d e f g+6 e^2 f^2\right )\right )}{m+3}+\frac{(d+e x)^4 \left (2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left (15 d^2 g^2-10 d e f g+e^2 f^2\right )\right )}{m+5}+\frac{2 (d+e x)^3 \left (c e \left (2 a e g (e f-2 d g)+b \left (10 d^2 g^2-8 d e f g+e^2 f^2\right )\right )+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left (5 d^2 g^2-5 d e f g+e^2 f^2\right )\right )}{m+4}-\frac{2 (d+e x) (d g-e f) \left (e (a e-b d)+c d^2\right ) (e (a e g-2 b d g+b e f)+c d (3 d g-2 e f))}{m+2}+\frac{(e f-d g)^2 \left (e (a e-b d)+c d^2\right )^2}{m+1}+\frac{2 c g (d+e x)^5 (b e g-3 c d g+c e f)}{m+6}+\frac{c^2 g^2 (d+e x)^6}{m+7}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(((c*d^2 + e*(-(b*d) + a*e))^2*(e*f - d*g)^2)/(1 + m) - (2*(c*d^2 + e*(-(b*d) + a*e))*(-(e*
f) + d*g)*(c*d*(-2*e*f + 3*d*g) + e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x))/(2 + m) + ((c^2*d^2*(6*e^2*f^2 - 20*
d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2))
+ 2*c*e*(b*d*(-3*e^2*f^2 + 12*d*e*f*g - 10*d^2*g^2) + a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2)))*(d + e*x)^2)/(3
+ m) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) + c*e*(2*a*e*g*(e*f -
2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^3)/(4 + m) + ((b^2*e^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b
*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^4)/(5 + m) + (2*c*g*(c*e*f - 3*c*d*g + b*e*
g)*(d + e*x)^5)/(6 + m) + (c^2*g^2*(d + e*x)^6)/(7 + m)))/e^7

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Maple [B]  time = 0.06, size = 5890, normalized size = 11.2 \begin{align*} \text{output 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)^2*(c*x^2+b*x+a)^2,x)

[Out]

result too large to display

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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)^2*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.41609, size = 10647, normalized size = 20.28 \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)^2*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

(a^2*d*e^6*f^2*m^6 + (c^2*e^7*g^2*m^6 + 21*c^2*e^7*g^2*m^5 + 175*c^2*e^7*g^2*m^4 + 735*c^2*e^7*g^2*m^3 + 1624*
c^2*e^7*g^2*m^2 + 1764*c^2*e^7*g^2*m + 720*c^2*e^7*g^2)*x^7 + (1680*c^2*e^7*f*g + 1680*b*c*e^7*g^2 + (2*c^2*e^
7*f*g + (c^2*d*e^6 + 2*b*c*e^7)*g^2)*m^6 + (44*c^2*e^7*f*g + (15*c^2*d*e^6 + 44*b*c*e^7)*g^2)*m^5 + 5*(76*c^2*
e^7*f*g + (17*c^2*d*e^6 + 76*b*c*e^7)*g^2)*m^4 + 5*(328*c^2*e^7*f*g + (45*c^2*d*e^6 + 328*b*c*e^7)*g^2)*m^3 +
2*(1849*c^2*e^7*f*g + (137*c^2*d*e^6 + 1849*b*c*e^7)*g^2)*m^2 + 4*(1019*c^2*e^7*f*g + (30*c^2*d*e^6 + 1019*b*c
*e^7)*g^2)*m)*x^6 - (2*a^2*d^2*e^5*f*g + (2*a*b*d^2*e^5 - 27*a^2*d*e^6)*f^2)*m^5 + (1008*c^2*e^7*f^2 + 4032*b*
c*e^7*f*g + 1008*(b^2 + 2*a*c)*e^7*g^2 + (c^2*e^7*f^2 + 2*(c^2*d*e^6 + 2*b*c*e^7)*f*g + (2*b*c*d*e^6 + (b^2 +
2*a*c)*e^7)*g^2)*m^6 + (23*c^2*e^7*f^2 + 2*(17*c^2*d*e^6 + 46*b*c*e^7)*f*g - (6*c^2*d^2*e^5 - 34*b*c*d*e^6 - 2
3*(b^2 + 2*a*c)*e^7)*g^2)*m^5 + 3*(69*c^2*e^7*f^2 + 2*(35*c^2*d*e^6 + 138*b*c*e^7)*f*g - (20*c^2*d^2*e^5 - 70*
b*c*d*e^6 - 69*(b^2 + 2*a*c)*e^7)*g^2)*m^4 + 5*(185*c^2*e^7*f^2 + 2*(59*c^2*d*e^6 + 370*b*c*e^7)*f*g - (42*c^2
*d^2*e^5 - 118*b*c*d*e^6 - 185*(b^2 + 2*a*c)*e^7)*g^2)*m^3 + 4*(536*c^2*e^7*f^2 + (187*c^2*d*e^6 + 2144*b*c*e^
7)*f*g - (75*c^2*d^2*e^5 - 187*b*c*d*e^6 - 536*(b^2 + 2*a*c)*e^7)*g^2)*m^2 + 12*(201*c^2*e^7*f^2 + 4*(7*c^2*d*
e^6 + 201*b*c*e^7)*f*g - (12*c^2*d^2*e^5 - 28*b*c*d*e^6 - 201*(b^2 + 2*a*c)*e^7)*g^2)*m)*x^5 + (2*a^2*d^3*e^4*
g^2 - (50*a*b*d^2*e^5 - 295*a^2*d*e^6 - 2*(b^2 + 2*a*c)*d^3*e^4)*f^2 + 2*(4*a*b*d^3*e^4 - 25*a^2*d^2*e^5)*f*g)
*m^4 + (2520*b*c*e^7*f^2 + 2520*a*b*e^7*g^2 + 2520*(b^2 + 2*a*c)*e^7*f*g + ((c^2*d*e^6 + 2*b*c*e^7)*f^2 + 2*(2
*b*c*d*e^6 + (b^2 + 2*a*c)*e^7)*f*g + (2*a*b*e^7 + (b^2 + 2*a*c)*d*e^6)*g^2)*m^6 + ((19*c^2*d*e^6 + 48*b*c*e^7
)*f^2 - 2*(5*c^2*d^2*e^5 - 38*b*c*d*e^6 - 24*(b^2 + 2*a*c)*e^7)*f*g - (10*b*c*d^2*e^5 - 48*a*b*e^7 - 19*(b^2 +
2*a*c)*d*e^6)*g^2)*m^5 + ((131*c^2*d*e^6 + 452*b*c*e^7)*f^2 - 2*(65*c^2*d^2*e^5 - 262*b*c*d*e^6 - 226*(b^2 +
2*a*c)*e^7)*f*g + (30*c^2*d^3*e^4 - 130*b*c*d^2*e^5 + 452*a*b*e^7 + 131*(b^2 + 2*a*c)*d*e^6)*g^2)*m^4 + ((401*
c^2*d*e^6 + 2112*b*c*e^7)*f^2 - 2*(265*c^2*d^2*e^5 - 802*b*c*d*e^6 - 1056*(b^2 + 2*a*c)*e^7)*f*g + (180*c^2*d^
3*e^4 - 530*b*c*d^2*e^5 + 2112*a*b*e^7 + 401*(b^2 + 2*a*c)*d*e^6)*g^2)*m^3 + 10*((54*c^2*d*e^6 + 509*b*c*e^7)*
f^2 - (83*c^2*d^2*e^5 - 216*b*c*d*e^6 - 509*(b^2 + 2*a*c)*e^7)*f*g + (33*c^2*d^3*e^4 - 83*b*c*d^2*e^5 + 509*a*
b*e^7 + 54*(b^2 + 2*a*c)*d*e^6)*g^2)*m^2 + 12*(3*(7*c^2*d*e^6 + 164*b*c*e^7)*f^2 - (35*c^2*d^2*e^5 - 84*b*c*d*
e^6 - 492*(b^2 + 2*a*c)*e^7)*f*g + (15*c^2*d^3*e^4 - 35*b*c*d^2*e^5 + 492*a*b*e^7 + 21*(b^2 + 2*a*c)*d*e^6)*g^
2)*m)*x^4 - ((12*b*c*d^4*e^3 + 490*a*b*d^2*e^5 - 1665*a^2*d*e^6 - 44*(b^2 + 2*a*c)*d^3*e^4)*f^2 - 2*(88*a*b*d^
3*e^4 - 245*a^2*d^2*e^5 - 6*(b^2 + 2*a*c)*d^4*e^3)*f*g + 4*(3*a*b*d^4*e^3 - 11*a^2*d^3*e^4)*g^2)*m^3 + (6720*a
*b*e^7*f*g + 1680*a^2*e^7*g^2 + 1680*(b^2 + 2*a*c)*e^7*f^2 + ((2*b*c*d*e^6 + (b^2 + 2*a*c)*e^7)*f^2 + 2*(2*a*b
*e^7 + (b^2 + 2*a*c)*d*e^6)*f*g + (2*a*b*d*e^6 + a^2*e^7)*g^2)*m^6 - ((4*c^2*d^2*e^5 - 42*b*c*d*e^6 - 25*(b^2
+ 2*a*c)*e^7)*f^2 + 2*(8*b*c*d^2*e^5 - 50*a*b*e^7 - 21*(b^2 + 2*a*c)*d*e^6)*f*g - (42*a*b*d*e^6 + 25*a^2*e^7 -
4*(b^2 + 2*a*c)*d^2*e^5)*g^2)*m^5 - ((64*c^2*d^2*e^5 - 326*b*c*d*e^6 - 247*(b^2 + 2*a*c)*e^7)*f^2 - 2*(20*c^2
*d^3*e^4 - 128*b*c*d^2*e^5 + 494*a*b*e^7 + 163*(b^2 + 2*a*c)*d*e^6)*f*g - (40*b*c*d^3*e^4 + 326*a*b*d*e^6 + 24
7*a^2*e^7 - 64*(b^2 + 2*a*c)*d^2*e^5)*g^2)*m^4 - ((332*c^2*d^2*e^5 - 1134*b*c*d*e^6 - 1219*(b^2 + 2*a*c)*e^7)*
f^2 - 2*(200*c^2*d^3*e^4 - 664*b*c*d^2*e^5 + 2438*a*b*e^7 + 567*(b^2 + 2*a*c)*d*e^6)*f*g + (120*c^2*d^4*e^3 -
400*b*c*d^3*e^4 - 1134*a*b*d*e^6 - 1219*a^2*e^7 + 332*(b^2 + 2*a*c)*d^2*e^5)*g^2)*m^3 - 8*((76*c^2*d^2*e^5 - 2
11*b*c*d*e^6 - 389*(b^2 + 2*a*c)*e^7)*f^2 - (115*c^2*d^3*e^4 - 304*b*c*d^2*e^5 + 1556*a*b*e^7 + 211*(b^2 + 2*a
*c)*d*e^6)*f*g + (45*c^2*d^4*e^3 - 115*b*c*d^3*e^4 - 211*a*b*d*e^6 - 389*a^2*e^7 + 76*(b^2 + 2*a*c)*d^2*e^5)*g
^2)*m^2 - 4*((84*c^2*d^2*e^5 - 210*b*c*d*e^6 - 949*(b^2 + 2*a*c)*e^7)*f^2 - 2*(70*c^2*d^3*e^4 - 168*b*c*d^2*e^
5 + 1898*a*b*e^7 + 105*(b^2 + 2*a*c)*d*e^6)*f*g + (60*c^2*d^4*e^3 - 140*b*c*d^3*e^4 - 210*a*b*d*e^6 - 949*a^2*
e^7 + 84*(b^2 + 2*a*c)*d^2*e^5)*g^2)*m)*x^3 + 168*(6*c^2*d^5*e^2 - 15*b*c*d^4*e^3 - 30*a*b*d^2*e^5 + 30*a^2*d*
e^6 + 10*(b^2 + 2*a*c)*d^3*e^4)*f^2 - 168*(10*c^2*d^6*e - 24*b*c*d^5*e^2 - 40*a*b*d^3*e^4 + 30*a^2*d^2*e^5 + 1
5*(b^2 + 2*a*c)*d^4*e^3)*f*g + 24*(30*c^2*d^7 - 70*b*c*d^6*e - 105*a*b*d^4*e^3 + 70*a^2*d^3*e^4 + 42*(b^2 + 2*
a*c)*d^5*e^2)*g^2 + 2*((12*c^2*d^5*e^2 - 108*b*c*d^4*e^3 - 1175*a*b*d^2*e^5 + 2552*a^2*d*e^6 + 179*(b^2 + 2*a*
c)*d^3*e^4)*f^2 + (48*b*c*d^5*e^2 + 716*a*b*d^3*e^4 - 1175*a^2*d^2*e^5 - 108*(b^2 + 2*a*c)*d^4*e^3)*f*g - (108
*a*b*d^4*e^3 - 179*a^2*d^3*e^4 - 12*(b^2 + 2*a*c)*d^5*e^2)*g^2)*m^2 + (5040*a*b*e^7*f^2 + 5040*a^2*e^7*f*g + (
a^2*d*e^6*g^2 + (2*a*b*e^7 + (b^2 + 2*a*c)*d*e^6)*f^2 + 2*(2*a*b*d*e^6 + a^2*e^7)*f*g)*m^6 - ((6*b*c*d^2*e^5 -
52*a*b*e^7 - 23*(b^2 + 2*a*c)*d*e^6)*f^2 - 2*(46*a*b*d*e^6 + 26*a^2*e^7 - 3*(b^2 + 2*a*c)*d^2*e^5)*f*g + (6*a
*b*d^2*e^5 - 23*a^2*d*e^6)*g^2)*m^5 + 3*((4*c^2*d^3*e^4 - 38*b*c*d^2*e^5 + 180*a*b*e^7 + 67*(b^2 + 2*a*c)*d*e^
6)*f^2 + 2*(8*b*c*d^3*e^4 + 134*a*b*d*e^6 + 90*a^2*e^7 - 19*(b^2 + 2*a*c)*d^2*e^5)*f*g - (38*a*b*d^2*e^5 - 67*
a^2*d*e^6 - 4*(b^2 + 2*a*c)*d^3*e^4)*g^2)*m^4 + ((168*c^2*d^3*e^4 - 750*b*c*d^2*e^5 + 2840*a*b*e^7 + 817*(b^2
+ 2*a*c)*d*e^6)*f^2 - 2*(60*c^2*d^4*e^3 - 336*b*c*d^3*e^4 - 1634*a*b*d*e^6 - 1420*a^2*e^7 + 375*(b^2 + 2*a*c)*
d^2*e^5)*f*g - (120*b*c*d^4*e^3 + 750*a*b*d^2*e^5 - 817*a^2*d*e^6 - 168*(b^2 + 2*a*c)*d^3*e^4)*g^2)*m^3 + 2*((
330*c^2*d^3*e^4 - 951*b*c*d^2*e^5 + 3929*a*b*e^7 + 739*(b^2 + 2*a*c)*d*e^6)*f^2 - (480*c^2*d^4*e^3 - 1320*b*c*
d^3*e^4 - 2956*a*b*d*e^6 - 3929*a^2*e^7 + 951*(b^2 + 2*a*c)*d^2*e^5)*f*g + (180*c^2*d^5*e^2 - 480*b*c*d^4*e^3
- 951*a*b*d^2*e^5 + 739*a^2*d*e^6 + 330*(b^2 + 2*a*c)*d^3*e^4)*g^2)*m^2 + 12*((42*c^2*d^3*e^4 - 105*b*c*d^2*e^
5 + 879*a*b*e^7 + 70*(b^2 + 2*a*c)*d*e^6)*f^2 - (70*c^2*d^4*e^3 - 168*b*c*d^3*e^4 - 280*a*b*d*e^6 - 879*a^2*e^
7 + 105*(b^2 + 2*a*c)*d^2*e^5)*f*g + (30*c^2*d^5*e^2 - 70*b*c*d^4*e^3 - 105*a*b*d^2*e^5 + 70*a^2*d*e^6 + 42*(b
^2 + 2*a*c)*d^3*e^4)*g^2)*m)*x^2 + 4*((78*c^2*d^5*e^2 - 321*b*c*d^4*e^3 - 1377*a*b*d^2*e^5 + 2007*a^2*d*e^6 +
319*(b^2 + 2*a*c)*d^3*e^4)*f^2 - (60*c^2*d^6*e - 312*b*c*d^5*e^2 - 1276*a*b*d^3*e^4 + 1377*a^2*d^2*e^5 + 321*(
b^2 + 2*a*c)*d^4*e^3)*f*g - (60*b*c*d^6*e + 321*a*b*d^4*e^3 - 319*a^2*d^3*e^4 - 78*(b^2 + 2*a*c)*d^5*e^2)*g^2)
*m + (5040*a^2*e^7*f^2 + (2*a^2*d*e^6*f*g + (2*a*b*d*e^6 + a^2*e^7)*f^2)*m^6 - (2*a^2*d^2*e^5*g^2 - (50*a*b*d*
e^6 + 27*a^2*e^7 - 2*(b^2 + 2*a*c)*d^2*e^5)*f^2 + 2*(4*a*b*d^2*e^5 - 25*a^2*d*e^6)*f*g)*m^5 + ((12*b*c*d^3*e^4
+ 490*a*b*d*e^6 + 295*a^2*e^7 - 44*(b^2 + 2*a*c)*d^2*e^5)*f^2 - 2*(88*a*b*d^2*e^5 - 245*a^2*d*e^6 - 6*(b^2 +
2*a*c)*d^3*e^4)*f*g + 4*(3*a*b*d^3*e^4 - 11*a^2*d^2*e^5)*g^2)*m^4 - ((24*c^2*d^4*e^3 - 216*b*c*d^3*e^4 - 2350*
a*b*d*e^6 - 1665*a^2*e^7 + 358*(b^2 + 2*a*c)*d^2*e^5)*f^2 + 2*(48*b*c*d^4*e^3 + 716*a*b*d^2*e^5 - 1175*a^2*d*e
^6 - 108*(b^2 + 2*a*c)*d^3*e^4)*f*g - 2*(108*a*b*d^3*e^4 - 179*a^2*d^2*e^5 - 12*(b^2 + 2*a*c)*d^4*e^3)*g^2)*m^
3 - 4*((78*c^2*d^4*e^3 - 321*b*c*d^3*e^4 - 1377*a*b*d*e^6 - 1276*a^2*e^7 + 319*(b^2 + 2*a*c)*d^2*e^5)*f^2 - (6
0*c^2*d^5*e^2 - 312*b*c*d^4*e^3 - 1276*a*b*d^2*e^5 + 1377*a^2*d*e^6 + 321*(b^2 + 2*a*c)*d^3*e^4)*f*g - (60*b*c
*d^5*e^2 + 321*a*b*d^3*e^4 - 319*a^2*d^2*e^5 - 78*(b^2 + 2*a*c)*d^4*e^3)*g^2)*m^2 - 12*((84*c^2*d^4*e^3 - 210*
b*c*d^3*e^4 - 420*a*b*d*e^6 - 669*a^2*e^7 + 140*(b^2 + 2*a*c)*d^2*e^5)*f^2 - 14*(10*c^2*d^5*e^2 - 24*b*c*d^4*e
^3 - 40*a*b*d^2*e^5 + 30*a^2*d*e^6 + 15*(b^2 + 2*a*c)*d^3*e^4)*f*g + 2*(30*c^2*d^6*e - 70*b*c*d^5*e^2 - 105*a*
b*d^3*e^4 + 70*a^2*d^2*e^5 + 42*(b^2 + 2*a*c)*d^4*e^3)*g^2)*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*
m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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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)**2*(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.40195, size = 14160, normalized size = 26.97 \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)^2*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*c^2*g^2*m^6*x^7*e^7 + (x*e + d)^m*c^2*d*g^2*m^6*x^6*e^6 + 2*(x*e + d)^m*c^2*f*g*m^6*x^6*e^7 + 2*(
x*e + d)^m*b*c*g^2*m^6*x^6*e^7 + 21*(x*e + d)^m*c^2*g^2*m^5*x^7*e^7 + 2*(x*e + d)^m*c^2*d*f*g*m^6*x^5*e^6 + 2*
(x*e + d)^m*b*c*d*g^2*m^6*x^5*e^6 + 15*(x*e + d)^m*c^2*d*g^2*m^5*x^6*e^6 - 6*(x*e + d)^m*c^2*d^2*g^2*m^5*x^5*e
^5 + (x*e + d)^m*c^2*f^2*m^6*x^5*e^7 + 4*(x*e + d)^m*b*c*f*g*m^6*x^5*e^7 + (x*e + d)^m*b^2*g^2*m^6*x^5*e^7 + 2
*(x*e + d)^m*a*c*g^2*m^6*x^5*e^7 + 44*(x*e + d)^m*c^2*f*g*m^5*x^6*e^7 + 44*(x*e + d)^m*b*c*g^2*m^5*x^6*e^7 + 1
75*(x*e + d)^m*c^2*g^2*m^4*x^7*e^7 + (x*e + d)^m*c^2*d*f^2*m^6*x^4*e^6 + 4*(x*e + d)^m*b*c*d*f*g*m^6*x^4*e^6 +
(x*e + d)^m*b^2*d*g^2*m^6*x^4*e^6 + 2*(x*e + d)^m*a*c*d*g^2*m^6*x^4*e^6 + 34*(x*e + d)^m*c^2*d*f*g*m^5*x^5*e^
6 + 34*(x*e + d)^m*b*c*d*g^2*m^5*x^5*e^6 + 85*(x*e + d)^m*c^2*d*g^2*m^4*x^6*e^6 - 10*(x*e + d)^m*c^2*d^2*f*g*m
^5*x^4*e^5 - 10*(x*e + d)^m*b*c*d^2*g^2*m^5*x^4*e^5 - 60*(x*e + d)^m*c^2*d^2*g^2*m^4*x^5*e^5 + 30*(x*e + d)^m*
c^2*d^3*g^2*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*f^2*m^6*x^4*e^7 + 2*(x*e + d)^m*b^2*f*g*m^6*x^4*e^7 + 4*(x*e + d)^
m*a*c*f*g*m^6*x^4*e^7 + 2*(x*e + d)^m*a*b*g^2*m^6*x^4*e^7 + 23*(x*e + d)^m*c^2*f^2*m^5*x^5*e^7 + 92*(x*e + d)^
m*b*c*f*g*m^5*x^5*e^7 + 23*(x*e + d)^m*b^2*g^2*m^5*x^5*e^7 + 46*(x*e + d)^m*a*c*g^2*m^5*x^5*e^7 + 380*(x*e + d
)^m*c^2*f*g*m^4*x^6*e^7 + 380*(x*e + d)^m*b*c*g^2*m^4*x^6*e^7 + 735*(x*e + d)^m*c^2*g^2*m^3*x^7*e^7 + 2*(x*e +
d)^m*b*c*d*f^2*m^6*x^3*e^6 + 2*(x*e + d)^m*b^2*d*f*g*m^6*x^3*e^6 + 4*(x*e + d)^m*a*c*d*f*g*m^6*x^3*e^6 + 2*(x
*e + d)^m*a*b*d*g^2*m^6*x^3*e^6 + 19*(x*e + d)^m*c^2*d*f^2*m^5*x^4*e^6 + 76*(x*e + d)^m*b*c*d*f*g*m^5*x^4*e^6
+ 19*(x*e + d)^m*b^2*d*g^2*m^5*x^4*e^6 + 38*(x*e + d)^m*a*c*d*g^2*m^5*x^4*e^6 + 210*(x*e + d)^m*c^2*d*f*g*m^4*
x^5*e^6 + 210*(x*e + d)^m*b*c*d*g^2*m^4*x^5*e^6 + 225*(x*e + d)^m*c^2*d*g^2*m^3*x^6*e^6 - 4*(x*e + d)^m*c^2*d^
2*f^2*m^5*x^3*e^5 - 16*(x*e + d)^m*b*c*d^2*f*g*m^5*x^3*e^5 - 4*(x*e + d)^m*b^2*d^2*g^2*m^5*x^3*e^5 - 8*(x*e +
d)^m*a*c*d^2*g^2*m^5*x^3*e^5 - 130*(x*e + d)^m*c^2*d^2*f*g*m^4*x^4*e^5 - 130*(x*e + d)^m*b*c*d^2*g^2*m^4*x^4*e
^5 - 210*(x*e + d)^m*c^2*d^2*g^2*m^3*x^5*e^5 + 40*(x*e + d)^m*c^2*d^3*f*g*m^4*x^3*e^4 + 40*(x*e + d)^m*b*c*d^3
*g^2*m^4*x^3*e^4 + 180*(x*e + d)^m*c^2*d^3*g^2*m^3*x^4*e^4 - 120*(x*e + d)^m*c^2*d^4*g^2*m^3*x^3*e^3 + (x*e +
d)^m*b^2*f^2*m^6*x^3*e^7 + 2*(x*e + d)^m*a*c*f^2*m^6*x^3*e^7 + 4*(x*e + d)^m*a*b*f*g*m^6*x^3*e^7 + (x*e + d)^m
*a^2*g^2*m^6*x^3*e^7 + 48*(x*e + d)^m*b*c*f^2*m^5*x^4*e^7 + 48*(x*e + d)^m*b^2*f*g*m^5*x^4*e^7 + 96*(x*e + d)^
m*a*c*f*g*m^5*x^4*e^7 + 48*(x*e + d)^m*a*b*g^2*m^5*x^4*e^7 + 207*(x*e + d)^m*c^2*f^2*m^4*x^5*e^7 + 828*(x*e +
d)^m*b*c*f*g*m^4*x^5*e^7 + 207*(x*e + d)^m*b^2*g^2*m^4*x^5*e^7 + 414*(x*e + d)^m*a*c*g^2*m^4*x^5*e^7 + 1640*(x
*e + d)^m*c^2*f*g*m^3*x^6*e^7 + 1640*(x*e + d)^m*b*c*g^2*m^3*x^6*e^7 + 1624*(x*e + d)^m*c^2*g^2*m^2*x^7*e^7 +
(x*e + d)^m*b^2*d*f^2*m^6*x^2*e^6 + 2*(x*e + d)^m*a*c*d*f^2*m^6*x^2*e^6 + 4*(x*e + d)^m*a*b*d*f*g*m^6*x^2*e^6
+ (x*e + d)^m*a^2*d*g^2*m^6*x^2*e^6 + 42*(x*e + d)^m*b*c*d*f^2*m^5*x^3*e^6 + 42*(x*e + d)^m*b^2*d*f*g*m^5*x^3*
e^6 + 84*(x*e + d)^m*a*c*d*f*g*m^5*x^3*e^6 + 42*(x*e + d)^m*a*b*d*g^2*m^5*x^3*e^6 + 131*(x*e + d)^m*c^2*d*f^2*
m^4*x^4*e^6 + 524*(x*e + d)^m*b*c*d*f*g*m^4*x^4*e^6 + 131*(x*e + d)^m*b^2*d*g^2*m^4*x^4*e^6 + 262*(x*e + d)^m*
a*c*d*g^2*m^4*x^4*e^6 + 590*(x*e + d)^m*c^2*d*f*g*m^3*x^5*e^6 + 590*(x*e + d)^m*b*c*d*g^2*m^3*x^5*e^6 + 274*(x
*e + d)^m*c^2*d*g^2*m^2*x^6*e^6 - 6*(x*e + d)^m*b*c*d^2*f^2*m^5*x^2*e^5 - 6*(x*e + d)^m*b^2*d^2*f*g*m^5*x^2*e^
5 - 12*(x*e + d)^m*a*c*d^2*f*g*m^5*x^2*e^5 - 6*(x*e + d)^m*a*b*d^2*g^2*m^5*x^2*e^5 - 64*(x*e + d)^m*c^2*d^2*f^
2*m^4*x^3*e^5 - 256*(x*e + d)^m*b*c*d^2*f*g*m^4*x^3*e^5 - 64*(x*e + d)^m*b^2*d^2*g^2*m^4*x^3*e^5 - 128*(x*e +
d)^m*a*c*d^2*g^2*m^4*x^3*e^5 - 530*(x*e + d)^m*c^2*d^2*f*g*m^3*x^4*e^5 - 530*(x*e + d)^m*b*c*d^2*g^2*m^3*x^4*e
^5 - 300*(x*e + d)^m*c^2*d^2*g^2*m^2*x^5*e^5 + 12*(x*e + d)^m*c^2*d^3*f^2*m^4*x^2*e^4 + 48*(x*e + d)^m*b*c*d^3
*f*g*m^4*x^2*e^4 + 12*(x*e + d)^m*b^2*d^3*g^2*m^4*x^2*e^4 + 24*(x*e + d)^m*a*c*d^3*g^2*m^4*x^2*e^4 + 400*(x*e
+ d)^m*c^2*d^3*f*g*m^3*x^3*e^4 + 400*(x*e + d)^m*b*c*d^3*g^2*m^3*x^3*e^4 + 330*(x*e + d)^m*c^2*d^3*g^2*m^2*x^4
*e^4 - 120*(x*e + d)^m*c^2*d^4*f*g*m^3*x^2*e^3 - 120*(x*e + d)^m*b*c*d^4*g^2*m^3*x^2*e^3 - 360*(x*e + d)^m*c^2
*d^4*g^2*m^2*x^3*e^3 + 360*(x*e + d)^m*c^2*d^5*g^2*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*f^2*m^6*x^2*e^7 + 2*(x*e +
d)^m*a^2*f*g*m^6*x^2*e^7 + 25*(x*e + d)^m*b^2*f^2*m^5*x^3*e^7 + 50*(x*e + d)^m*a*c*f^2*m^5*x^3*e^7 + 100*(x*e
+ d)^m*a*b*f*g*m^5*x^3*e^7 + 25*(x*e + d)^m*a^2*g^2*m^5*x^3*e^7 + 452*(x*e + d)^m*b*c*f^2*m^4*x^4*e^7 + 452*(x
*e + d)^m*b^2*f*g*m^4*x^4*e^7 + 904*(x*e + d)^m*a*c*f*g*m^4*x^4*e^7 + 452*(x*e + d)^m*a*b*g^2*m^4*x^4*e^7 + 92
5*(x*e + d)^m*c^2*f^2*m^3*x^5*e^7 + 3700*(x*e + d)^m*b*c*f*g*m^3*x^5*e^7 + 925*(x*e + d)^m*b^2*g^2*m^3*x^5*e^7
+ 1850*(x*e + d)^m*a*c*g^2*m^3*x^5*e^7 + 3698*(x*e + d)^m*c^2*f*g*m^2*x^6*e^7 + 3698*(x*e + d)^m*b*c*g^2*m^2*
x^6*e^7 + 1764*(x*e + d)^m*c^2*g^2*m*x^7*e^7 + 2*(x*e + d)^m*a*b*d*f^2*m^6*x*e^6 + 2*(x*e + d)^m*a^2*d*f*g*m^6
*x*e^6 + 23*(x*e + d)^m*b^2*d*f^2*m^5*x^2*e^6 + 46*(x*e + d)^m*a*c*d*f^2*m^5*x^2*e^6 + 92*(x*e + d)^m*a*b*d*f*
g*m^5*x^2*e^6 + 23*(x*e + d)^m*a^2*d*g^2*m^5*x^2*e^6 + 326*(x*e + d)^m*b*c*d*f^2*m^4*x^3*e^6 + 326*(x*e + d)^m
*b^2*d*f*g*m^4*x^3*e^6 + 652*(x*e + d)^m*a*c*d*f*g*m^4*x^3*e^6 + 326*(x*e + d)^m*a*b*d*g^2*m^4*x^3*e^6 + 401*(
x*e + d)^m*c^2*d*f^2*m^3*x^4*e^6 + 1604*(x*e + d)^m*b*c*d*f*g*m^3*x^4*e^6 + 401*(x*e + d)^m*b^2*d*g^2*m^3*x^4*
e^6 + 802*(x*e + d)^m*a*c*d*g^2*m^3*x^4*e^6 + 748*(x*e + d)^m*c^2*d*f*g*m^2*x^5*e^6 + 748*(x*e + d)^m*b*c*d*g^
2*m^2*x^5*e^6 + 120*(x*e + d)^m*c^2*d*g^2*m*x^6*e^6 - 2*(x*e + d)^m*b^2*d^2*f^2*m^5*x*e^5 - 4*(x*e + d)^m*a*c*
d^2*f^2*m^5*x*e^5 - 8*(x*e + d)^m*a*b*d^2*f*g*m^5*x*e^5 - 2*(x*e + d)^m*a^2*d^2*g^2*m^5*x*e^5 - 114*(x*e + d)^
m*b*c*d^2*f^2*m^4*x^2*e^5 - 114*(x*e + d)^m*b^2*d^2*f*g*m^4*x^2*e^5 - 228*(x*e + d)^m*a*c*d^2*f*g*m^4*x^2*e^5
- 114*(x*e + d)^m*a*b*d^2*g^2*m^4*x^2*e^5 - 332*(x*e + d)^m*c^2*d^2*f^2*m^3*x^3*e^5 - 1328*(x*e + d)^m*b*c*d^2
*f*g*m^3*x^3*e^5 - 332*(x*e + d)^m*b^2*d^2*g^2*m^3*x^3*e^5 - 664*(x*e + d)^m*a*c*d^2*g^2*m^3*x^3*e^5 - 830*(x*
e + d)^m*c^2*d^2*f*g*m^2*x^4*e^5 - 830*(x*e + d)^m*b*c*d^2*g^2*m^2*x^4*e^5 - 144*(x*e + d)^m*c^2*d^2*g^2*m*x^5
*e^5 + 12*(x*e + d)^m*b*c*d^3*f^2*m^4*x*e^4 + 12*(x*e + d)^m*b^2*d^3*f*g*m^4*x*e^4 + 24*(x*e + d)^m*a*c*d^3*f*
g*m^4*x*e^4 + 12*(x*e + d)^m*a*b*d^3*g^2*m^4*x*e^4 + 168*(x*e + d)^m*c^2*d^3*f^2*m^3*x^2*e^4 + 672*(x*e + d)^m
*b*c*d^3*f*g*m^3*x^2*e^4 + 168*(x*e + d)^m*b^2*d^3*g^2*m^3*x^2*e^4 + 336*(x*e + d)^m*a*c*d^3*g^2*m^3*x^2*e^4 +
920*(x*e + d)^m*c^2*d^3*f*g*m^2*x^3*e^4 + 920*(x*e + d)^m*b*c*d^3*g^2*m^2*x^3*e^4 + 180*(x*e + d)^m*c^2*d^3*g
^2*m*x^4*e^4 - 24*(x*e + d)^m*c^2*d^4*f^2*m^3*x*e^3 - 96*(x*e + d)^m*b*c*d^4*f*g*m^3*x*e^3 - 24*(x*e + d)^m*b^
2*d^4*g^2*m^3*x*e^3 - 48*(x*e + d)^m*a*c*d^4*g^2*m^3*x*e^3 - 960*(x*e + d)^m*c^2*d^4*f*g*m^2*x^2*e^3 - 960*(x*
e + d)^m*b*c*d^4*g^2*m^2*x^2*e^3 - 240*(x*e + d)^m*c^2*d^4*g^2*m*x^3*e^3 + 240*(x*e + d)^m*c^2*d^5*f*g*m^2*x*e
^2 + 240*(x*e + d)^m*b*c*d^5*g^2*m^2*x*e^2 + 360*(x*e + d)^m*c^2*d^5*g^2*m*x^2*e^2 - 720*(x*e + d)^m*c^2*d^6*g
^2*m*x*e + (x*e + d)^m*a^2*f^2*m^6*x*e^7 + 52*(x*e + d)^m*a*b*f^2*m^5*x^2*e^7 + 52*(x*e + d)^m*a^2*f*g*m^5*x^2
*e^7 + 247*(x*e + d)^m*b^2*f^2*m^4*x^3*e^7 + 494*(x*e + d)^m*a*c*f^2*m^4*x^3*e^7 + 988*(x*e + d)^m*a*b*f*g*m^4
*x^3*e^7 + 247*(x*e + d)^m*a^2*g^2*m^4*x^3*e^7 + 2112*(x*e + d)^m*b*c*f^2*m^3*x^4*e^7 + 2112*(x*e + d)^m*b^2*f
*g*m^3*x^4*e^7 + 4224*(x*e + d)^m*a*c*f*g*m^3*x^4*e^7 + 2112*(x*e + d)^m*a*b*g^2*m^3*x^4*e^7 + 2144*(x*e + d)^
m*c^2*f^2*m^2*x^5*e^7 + 8576*(x*e + d)^m*b*c*f*g*m^2*x^5*e^7 + 2144*(x*e + d)^m*b^2*g^2*m^2*x^5*e^7 + 4288*(x*
e + d)^m*a*c*g^2*m^2*x^5*e^7 + 4076*(x*e + d)^m*c^2*f*g*m*x^6*e^7 + 4076*(x*e + d)^m*b*c*g^2*m*x^6*e^7 + 720*(
x*e + d)^m*c^2*g^2*x^7*e^7 + (x*e + d)^m*a^2*d*f^2*m^6*e^6 + 50*(x*e + d)^m*a*b*d*f^2*m^5*x*e^6 + 50*(x*e + d)
^m*a^2*d*f*g*m^5*x*e^6 + 201*(x*e + d)^m*b^2*d*f^2*m^4*x^2*e^6 + 402*(x*e + d)^m*a*c*d*f^2*m^4*x^2*e^6 + 804*(
x*e + d)^m*a*b*d*f*g*m^4*x^2*e^6 + 201*(x*e + d)^m*a^2*d*g^2*m^4*x^2*e^6 + 1134*(x*e + d)^m*b*c*d*f^2*m^3*x^3*
e^6 + 1134*(x*e + d)^m*b^2*d*f*g*m^3*x^3*e^6 + 2268*(x*e + d)^m*a*c*d*f*g*m^3*x^3*e^6 + 1134*(x*e + d)^m*a*b*d
*g^2*m^3*x^3*e^6 + 540*(x*e + d)^m*c^2*d*f^2*m^2*x^4*e^6 + 2160*(x*e + d)^m*b*c*d*f*g*m^2*x^4*e^6 + 540*(x*e +
d)^m*b^2*d*g^2*m^2*x^4*e^6 + 1080*(x*e + d)^m*a*c*d*g^2*m^2*x^4*e^6 + 336*(x*e + d)^m*c^2*d*f*g*m*x^5*e^6 + 3
36*(x*e + d)^m*b*c*d*g^2*m*x^5*e^6 - 2*(x*e + d)^m*a*b*d^2*f^2*m^5*e^5 - 2*(x*e + d)^m*a^2*d^2*f*g*m^5*e^5 - 4
4*(x*e + d)^m*b^2*d^2*f^2*m^4*x*e^5 - 88*(x*e + d)^m*a*c*d^2*f^2*m^4*x*e^5 - 176*(x*e + d)^m*a*b*d^2*f*g*m^4*x
*e^5 - 44*(x*e + d)^m*a^2*d^2*g^2*m^4*x*e^5 - 750*(x*e + d)^m*b*c*d^2*f^2*m^3*x^2*e^5 - 750*(x*e + d)^m*b^2*d^
2*f*g*m^3*x^2*e^5 - 1500*(x*e + d)^m*a*c*d^2*f*g*m^3*x^2*e^5 - 750*(x*e + d)^m*a*b*d^2*g^2*m^3*x^2*e^5 - 608*(
x*e + d)^m*c^2*d^2*f^2*m^2*x^3*e^5 - 2432*(x*e + d)^m*b*c*d^2*f*g*m^2*x^3*e^5 - 608*(x*e + d)^m*b^2*d^2*g^2*m^
2*x^3*e^5 - 1216*(x*e + d)^m*a*c*d^2*g^2*m^2*x^3*e^5 - 420*(x*e + d)^m*c^2*d^2*f*g*m*x^4*e^5 - 420*(x*e + d)^m
*b*c*d^2*g^2*m*x^4*e^5 + 2*(x*e + d)^m*b^2*d^3*f^2*m^4*e^4 + 4*(x*e + d)^m*a*c*d^3*f^2*m^4*e^4 + 8*(x*e + d)^m
*a*b*d^3*f*g*m^4*e^4 + 2*(x*e + d)^m*a^2*d^3*g^2*m^4*e^4 + 216*(x*e + d)^m*b*c*d^3*f^2*m^3*x*e^4 + 216*(x*e +
d)^m*b^2*d^3*f*g*m^3*x*e^4 + 432*(x*e + d)^m*a*c*d^3*f*g*m^3*x*e^4 + 216*(x*e + d)^m*a*b*d^3*g^2*m^3*x*e^4 + 6
60*(x*e + d)^m*c^2*d^3*f^2*m^2*x^2*e^4 + 2640*(x*e + d)^m*b*c*d^3*f*g*m^2*x^2*e^4 + 660*(x*e + d)^m*b^2*d^3*g^
2*m^2*x^2*e^4 + 1320*(x*e + d)^m*a*c*d^3*g^2*m^2*x^2*e^4 + 560*(x*e + d)^m*c^2*d^3*f*g*m*x^3*e^4 + 560*(x*e +
d)^m*b*c*d^3*g^2*m*x^3*e^4 - 12*(x*e + d)^m*b*c*d^4*f^2*m^3*e^3 - 12*(x*e + d)^m*b^2*d^4*f*g*m^3*e^3 - 24*(x*e
+ d)^m*a*c*d^4*f*g*m^3*e^3 - 12*(x*e + d)^m*a*b*d^4*g^2*m^3*e^3 - 312*(x*e + d)^m*c^2*d^4*f^2*m^2*x*e^3 - 124
8*(x*e + d)^m*b*c*d^4*f*g*m^2*x*e^3 - 312*(x*e + d)^m*b^2*d^4*g^2*m^2*x*e^3 - 624*(x*e + d)^m*a*c*d^4*g^2*m^2*
x*e^3 - 840*(x*e + d)^m*c^2*d^4*f*g*m*x^2*e^3 - 840*(x*e + d)^m*b*c*d^4*g^2*m*x^2*e^3 + 24*(x*e + d)^m*c^2*d^5
*f^2*m^2*e^2 + 96*(x*e + d)^m*b*c*d^5*f*g*m^2*e^2 + 24*(x*e + d)^m*b^2*d^5*g^2*m^2*e^2 + 48*(x*e + d)^m*a*c*d^
5*g^2*m^2*e^2 + 1680*(x*e + d)^m*c^2*d^5*f*g*m*x*e^2 + 1680*(x*e + d)^m*b*c*d^5*g^2*m*x*e^2 - 240*(x*e + d)^m*
c^2*d^6*f*g*m*e - 240*(x*e + d)^m*b*c*d^6*g^2*m*e + 720*(x*e + d)^m*c^2*d^7*g^2 + 27*(x*e + d)^m*a^2*f^2*m^5*x
*e^7 + 540*(x*e + d)^m*a*b*f^2*m^4*x^2*e^7 + 540*(x*e + d)^m*a^2*f*g*m^4*x^2*e^7 + 1219*(x*e + d)^m*b^2*f^2*m^
3*x^3*e^7 + 2438*(x*e + d)^m*a*c*f^2*m^3*x^3*e^7 + 4876*(x*e + d)^m*a*b*f*g*m^3*x^3*e^7 + 1219*(x*e + d)^m*a^2
*g^2*m^3*x^3*e^7 + 5090*(x*e + d)^m*b*c*f^2*m^2*x^4*e^7 + 5090*(x*e + d)^m*b^2*f*g*m^2*x^4*e^7 + 10180*(x*e +
d)^m*a*c*f*g*m^2*x^4*e^7 + 5090*(x*e + d)^m*a*b*g^2*m^2*x^4*e^7 + 2412*(x*e + d)^m*c^2*f^2*m*x^5*e^7 + 9648*(x
*e + d)^m*b*c*f*g*m*x^5*e^7 + 2412*(x*e + d)^m*b^2*g^2*m*x^5*e^7 + 4824*(x*e + d)^m*a*c*g^2*m*x^5*e^7 + 1680*(
x*e + d)^m*c^2*f*g*x^6*e^7 + 1680*(x*e + d)^m*b*c*g^2*x^6*e^7 + 27*(x*e + d)^m*a^2*d*f^2*m^5*e^6 + 490*(x*e +
d)^m*a*b*d*f^2*m^4*x*e^6 + 490*(x*e + d)^m*a^2*d*f*g*m^4*x*e^6 + 817*(x*e + d)^m*b^2*d*f^2*m^3*x^2*e^6 + 1634*
(x*e + d)^m*a*c*d*f^2*m^3*x^2*e^6 + 3268*(x*e + d)^m*a*b*d*f*g*m^3*x^2*e^6 + 817*(x*e + d)^m*a^2*d*g^2*m^3*x^2
*e^6 + 1688*(x*e + d)^m*b*c*d*f^2*m^2*x^3*e^6 + 1688*(x*e + d)^m*b^2*d*f*g*m^2*x^3*e^6 + 3376*(x*e + d)^m*a*c*
d*f*g*m^2*x^3*e^6 + 1688*(x*e + d)^m*a*b*d*g^2*m^2*x^3*e^6 + 252*(x*e + d)^m*c^2*d*f^2*m*x^4*e^6 + 1008*(x*e +
d)^m*b*c*d*f*g*m*x^4*e^6 + 252*(x*e + d)^m*b^2*d*g^2*m*x^4*e^6 + 504*(x*e + d)^m*a*c*d*g^2*m*x^4*e^6 - 50*(x*
e + d)^m*a*b*d^2*f^2*m^4*e^5 - 50*(x*e + d)^m*a^2*d^2*f*g*m^4*e^5 - 358*(x*e + d)^m*b^2*d^2*f^2*m^3*x*e^5 - 71
6*(x*e + d)^m*a*c*d^2*f^2*m^3*x*e^5 - 1432*(x*e + d)^m*a*b*d^2*f*g*m^3*x*e^5 - 358*(x*e + d)^m*a^2*d^2*g^2*m^3
*x*e^5 - 1902*(x*e + d)^m*b*c*d^2*f^2*m^2*x^2*e^5 - 1902*(x*e + d)^m*b^2*d^2*f*g*m^2*x^2*e^5 - 3804*(x*e + d)^
m*a*c*d^2*f*g*m^2*x^2*e^5 - 1902*(x*e + d)^m*a*b*d^2*g^2*m^2*x^2*e^5 - 336*(x*e + d)^m*c^2*d^2*f^2*m*x^3*e^5 -
1344*(x*e + d)^m*b*c*d^2*f*g*m*x^3*e^5 - 336*(x*e + d)^m*b^2*d^2*g^2*m*x^3*e^5 - 672*(x*e + d)^m*a*c*d^2*g^2*
m*x^3*e^5 + 44*(x*e + d)^m*b^2*d^3*f^2*m^3*e^4 + 88*(x*e + d)^m*a*c*d^3*f^2*m^3*e^4 + 176*(x*e + d)^m*a*b*d^3*
f*g*m^3*e^4 + 44*(x*e + d)^m*a^2*d^3*g^2*m^3*e^4 + 1284*(x*e + d)^m*b*c*d^3*f^2*m^2*x*e^4 + 1284*(x*e + d)^m*b
^2*d^3*f*g*m^2*x*e^4 + 2568*(x*e + d)^m*a*c*d^3*f*g*m^2*x*e^4 + 1284*(x*e + d)^m*a*b*d^3*g^2*m^2*x*e^4 + 504*(
x*e + d)^m*c^2*d^3*f^2*m*x^2*e^4 + 2016*(x*e + d)^m*b*c*d^3*f*g*m*x^2*e^4 + 504*(x*e + d)^m*b^2*d^3*g^2*m*x^2*
e^4 + 1008*(x*e + d)^m*a*c*d^3*g^2*m*x^2*e^4 - 216*(x*e + d)^m*b*c*d^4*f^2*m^2*e^3 - 216*(x*e + d)^m*b^2*d^4*f
*g*m^2*e^3 - 432*(x*e + d)^m*a*c*d^4*f*g*m^2*e^3 - 216*(x*e + d)^m*a*b*d^4*g^2*m^2*e^3 - 1008*(x*e + d)^m*c^2*
d^4*f^2*m*x*e^3 - 4032*(x*e + d)^m*b*c*d^4*f*g*m*x*e^3 - 1008*(x*e + d)^m*b^2*d^4*g^2*m*x*e^3 - 2016*(x*e + d)
^m*a*c*d^4*g^2*m*x*e^3 + 312*(x*e + d)^m*c^2*d^5*f^2*m*e^2 + 1248*(x*e + d)^m*b*c*d^5*f*g*m*e^2 + 312*(x*e + d
)^m*b^2*d^5*g^2*m*e^2 + 624*(x*e + d)^m*a*c*d^5*g^2*m*e^2 - 1680*(x*e + d)^m*c^2*d^6*f*g*e - 1680*(x*e + d)^m*
b*c*d^6*g^2*e + 295*(x*e + d)^m*a^2*f^2*m^4*x*e^7 + 2840*(x*e + d)^m*a*b*f^2*m^3*x^2*e^7 + 2840*(x*e + d)^m*a^
2*f*g*m^3*x^2*e^7 + 3112*(x*e + d)^m*b^2*f^2*m^2*x^3*e^7 + 6224*(x*e + d)^m*a*c*f^2*m^2*x^3*e^7 + 12448*(x*e +
d)^m*a*b*f*g*m^2*x^3*e^7 + 3112*(x*e + d)^m*a^2*g^2*m^2*x^3*e^7 + 5904*(x*e + d)^m*b*c*f^2*m*x^4*e^7 + 5904*(
x*e + d)^m*b^2*f*g*m*x^4*e^7 + 11808*(x*e + d)^m*a*c*f*g*m*x^4*e^7 + 5904*(x*e + d)^m*a*b*g^2*m*x^4*e^7 + 1008
*(x*e + d)^m*c^2*f^2*x^5*e^7 + 4032*(x*e + d)^m*b*c*f*g*x^5*e^7 + 1008*(x*e + d)^m*b^2*g^2*x^5*e^7 + 2016*(x*e
+ d)^m*a*c*g^2*x^5*e^7 + 295*(x*e + d)^m*a^2*d*f^2*m^4*e^6 + 2350*(x*e + d)^m*a*b*d*f^2*m^3*x*e^6 + 2350*(x*e
+ d)^m*a^2*d*f*g*m^3*x*e^6 + 1478*(x*e + d)^m*b^2*d*f^2*m^2*x^2*e^6 + 2956*(x*e + d)^m*a*c*d*f^2*m^2*x^2*e^6
+ 5912*(x*e + d)^m*a*b*d*f*g*m^2*x^2*e^6 + 1478*(x*e + d)^m*a^2*d*g^2*m^2*x^2*e^6 + 840*(x*e + d)^m*b*c*d*f^2*
m*x^3*e^6 + 840*(x*e + d)^m*b^2*d*f*g*m*x^3*e^6 + 1680*(x*e + d)^m*a*c*d*f*g*m*x^3*e^6 + 840*(x*e + d)^m*a*b*d
*g^2*m*x^3*e^6 - 490*(x*e + d)^m*a*b*d^2*f^2*m^3*e^5 - 490*(x*e + d)^m*a^2*d^2*f*g*m^3*e^5 - 1276*(x*e + d)^m*
b^2*d^2*f^2*m^2*x*e^5 - 2552*(x*e + d)^m*a*c*d^2*f^2*m^2*x*e^5 - 5104*(x*e + d)^m*a*b*d^2*f*g*m^2*x*e^5 - 1276
*(x*e + d)^m*a^2*d^2*g^2*m^2*x*e^5 - 1260*(x*e + d)^m*b*c*d^2*f^2*m*x^2*e^5 - 1260*(x*e + d)^m*b^2*d^2*f*g*m*x
^2*e^5 - 2520*(x*e + d)^m*a*c*d^2*f*g*m*x^2*e^5 - 1260*(x*e + d)^m*a*b*d^2*g^2*m*x^2*e^5 + 358*(x*e + d)^m*b^2
*d^3*f^2*m^2*e^4 + 716*(x*e + d)^m*a*c*d^3*f^2*m^2*e^4 + 1432*(x*e + d)^m*a*b*d^3*f*g*m^2*e^4 + 358*(x*e + d)^
m*a^2*d^3*g^2*m^2*e^4 + 2520*(x*e + d)^m*b*c*d^3*f^2*m*x*e^4 + 2520*(x*e + d)^m*b^2*d^3*f*g*m*x*e^4 + 5040*(x*
e + d)^m*a*c*d^3*f*g*m*x*e^4 + 2520*(x*e + d)^m*a*b*d^3*g^2*m*x*e^4 - 1284*(x*e + d)^m*b*c*d^4*f^2*m*e^3 - 128
4*(x*e + d)^m*b^2*d^4*f*g*m*e^3 - 2568*(x*e + d)^m*a*c*d^4*f*g*m*e^3 - 1284*(x*e + d)^m*a*b*d^4*g^2*m*e^3 + 10
08*(x*e + d)^m*c^2*d^5*f^2*e^2 + 4032*(x*e + d)^m*b*c*d^5*f*g*e^2 + 1008*(x*e + d)^m*b^2*d^5*g^2*e^2 + 2016*(x
*e + d)^m*a*c*d^5*g^2*e^2 + 1665*(x*e + d)^m*a^2*f^2*m^3*x*e^7 + 7858*(x*e + d)^m*a*b*f^2*m^2*x^2*e^7 + 7858*(
x*e + d)^m*a^2*f*g*m^2*x^2*e^7 + 3796*(x*e + d)^m*b^2*f^2*m*x^3*e^7 + 7592*(x*e + d)^m*a*c*f^2*m*x^3*e^7 + 151
84*(x*e + d)^m*a*b*f*g*m*x^3*e^7 + 3796*(x*e + d)^m*a^2*g^2*m*x^3*e^7 + 2520*(x*e + d)^m*b*c*f^2*x^4*e^7 + 252
0*(x*e + d)^m*b^2*f*g*x^4*e^7 + 5040*(x*e + d)^m*a*c*f*g*x^4*e^7 + 2520*(x*e + d)^m*a*b*g^2*x^4*e^7 + 1665*(x*
e + d)^m*a^2*d*f^2*m^3*e^6 + 5508*(x*e + d)^m*a*b*d*f^2*m^2*x*e^6 + 5508*(x*e + d)^m*a^2*d*f*g*m^2*x*e^6 + 840
*(x*e + d)^m*b^2*d*f^2*m*x^2*e^6 + 1680*(x*e + d)^m*a*c*d*f^2*m*x^2*e^6 + 3360*(x*e + d)^m*a*b*d*f*g*m*x^2*e^6
+ 840*(x*e + d)^m*a^2*d*g^2*m*x^2*e^6 - 2350*(x*e + d)^m*a*b*d^2*f^2*m^2*e^5 - 2350*(x*e + d)^m*a^2*d^2*f*g*m
^2*e^5 - 1680*(x*e + d)^m*b^2*d^2*f^2*m*x*e^5 - 3360*(x*e + d)^m*a*c*d^2*f^2*m*x*e^5 - 6720*(x*e + d)^m*a*b*d^
2*f*g*m*x*e^5 - 1680*(x*e + d)^m*a^2*d^2*g^2*m*x*e^5 + 1276*(x*e + d)^m*b^2*d^3*f^2*m*e^4 + 2552*(x*e + d)^m*a
*c*d^3*f^2*m*e^4 + 5104*(x*e + d)^m*a*b*d^3*f*g*m*e^4 + 1276*(x*e + d)^m*a^2*d^3*g^2*m*e^4 - 2520*(x*e + d)^m*
b*c*d^4*f^2*e^3 - 2520*(x*e + d)^m*b^2*d^4*f*g*e^3 - 5040*(x*e + d)^m*a*c*d^4*f*g*e^3 - 2520*(x*e + d)^m*a*b*d
^4*g^2*e^3 + 5104*(x*e + d)^m*a^2*f^2*m^2*x*e^7 + 10548*(x*e + d)^m*a*b*f^2*m*x^2*e^7 + 10548*(x*e + d)^m*a^2*
f*g*m*x^2*e^7 + 1680*(x*e + d)^m*b^2*f^2*x^3*e^7 + 3360*(x*e + d)^m*a*c*f^2*x^3*e^7 + 6720*(x*e + d)^m*a*b*f*g
*x^3*e^7 + 1680*(x*e + d)^m*a^2*g^2*x^3*e^7 + 5104*(x*e + d)^m*a^2*d*f^2*m^2*e^6 + 5040*(x*e + d)^m*a*b*d*f^2*
m*x*e^6 + 5040*(x*e + d)^m*a^2*d*f*g*m*x*e^6 - 5508*(x*e + d)^m*a*b*d^2*f^2*m*e^5 - 5508*(x*e + d)^m*a^2*d^2*f
*g*m*e^5 + 1680*(x*e + d)^m*b^2*d^3*f^2*e^4 + 3360*(x*e + d)^m*a*c*d^3*f^2*e^4 + 6720*(x*e + d)^m*a*b*d^3*f*g*
e^4 + 1680*(x*e + d)^m*a^2*d^3*g^2*e^4 + 8028*(x*e + d)^m*a^2*f^2*m*x*e^7 + 5040*(x*e + d)^m*a*b*f^2*x^2*e^7 +
5040*(x*e + d)^m*a^2*f*g*x^2*e^7 + 8028*(x*e + d)^m*a^2*d*f^2*m*e^6 - 5040*(x*e + d)^m*a*b*d^2*f^2*e^5 - 5040
*(x*e + d)^m*a^2*d^2*f*g*e^5 + 5040*(x*e + d)^m*a^2*f^2*x*e^7 + 5040*(x*e + d)^m*a^2*d*f^2*e^6)/(m^7*e^7 + 28*
m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e^7 + 13068*m*e^7 + 5040*e^7)