### 3.2142 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^7} \, dx$$

Optimal. Leaf size=266 $-\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}$

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(6*e^7*(d + e*x)^6) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^7*(d + e*x)^
5) - (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(4*e^7*(d + e*x)^4) + ((2*c*d - b*e
)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(3*e^7*(d + e*x)^3) - (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d
- a*e)))/(2*e^7*(d + e*x)^2) + (3*c^2*(2*c*d - b*e))/(e^7*(d + e*x)) + (c^3*Log[d + e*x])/e^7

________________________________________________________________________________________

Rubi [A]  time = 0.226392, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $-\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

-(c*d^2 - b*d*e + a*e^2)^3/(6*e^7*(d + e*x)^6) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^7*(d + e*x)^
5) - (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(4*e^7*(d + e*x)^4) + ((2*c*d - b*e
)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(3*e^7*(d + e*x)^3) - (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d
- a*e)))/(2*e^7*(d + e*x)^2) + (3*c^2*(2*c*d - b*e))/(e^7*(d + e*x)) + (c^3*Log[d + e*x])/e^7

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^7}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^5}+\frac{(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 )}{e^6 (d+e x)^4}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^3}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac{c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{\left (c d^2-b d e+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac{(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 )}{3 e^7 (d+e x)^3}-\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^2}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.166499, size = 385, normalized size = 1.45 $\frac{-3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )-e^3 \left (6 a^2 b e^2 (d+6 e x)+10 a^3 e^3+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )-6 c^2 e \left (a e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )+c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - e^3*(10*
a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x
^2 + 20*e^3*x^3)) - 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 +
20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) - 6*c^2*e*(a*e*(d^4 + 6*d^
3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 +
15*d*e^4*x^4 + 6*e^5*x^5)) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)

________________________________________________________________________________________

Maple [B]  time = 0.051, size = 695, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^7,x)

[Out]

-1/6/e/(e*x+d)^6*a^3-1/3/e^4/(e*x+d)^3*b^3+9/2/e^4/(e*x+d)^4*c*a*b*d+4/e^5/(e*x+d)^3*c^2*a*d+6/5/e^7/(e*x+d)^5
*c^3*d^5-3*c^2/e^6/(e*x+d)*b-3/4/e^3/(e*x+d)^4*b^2*a+3/4/e^4/(e*x+d)^4*b^3*d-15/4/e^7/(e*x+d)^4*c^3*d^4+20/3/e
^7/(e*x+d)^3*c^3*d^3+4/e^5/(e*x+d)^3*b^2*c*d-10/e^6/(e*x+d)^3*b*c^2*d^2+15/2*c^2/e^6/(e*x+d)^2*b*d+1/2/e^2/(e*
x+d)^6*b*a^2*d+1/e^4/(e*x+d)^6*d^3*a*b*c-18/5/e^4/(e*x+d)^5*d^2*a*b*c-9/2/e^5/(e*x+d)^4*a*c^2*d^2-3/e^6/(e*x+d
)^5*d^4*b*c^2-1/2/e^5/(e*x+d)^6*a*c^2*d^4-1/2/e^5/(e*x+d)^6*d^4*b^2*c+1/2/e^6/(e*x+d)^6*b*c^2*d^5+6/5/e^3/(e*x
+d)^5*a^2*c*d+6/5/e^3/(e*x+d)^5*a*b^2*d+12/5/e^5/(e*x+d)^5*a*c^2*d^3+12/5/e^5/(e*x+d)^5*d^3*b^2*c-1/2/e^3/(e*x
+d)^6*a^2*c*d^2-1/2/e^3/(e*x+d)^6*a*b^2*d^2-3/4/e^3/(e*x+d)^4*a^2*c-3/2*c^2/e^5/(e*x+d)^2*a-3/2*c/e^5/(e*x+d)^
2*b^2-15/2*c^3/e^7/(e*x+d)^2*d^2+1/6/e^4/(e*x+d)^6*b^3*d^3-1/6/e^7/(e*x+d)^6*c^3*d^6-3/5/e^2/(e*x+d)^5*b*a^2-3
/5/e^4/(e*x+d)^5*b^3*d^2-9/2/e^5/(e*x+d)^4*c*b^2*d^2+15/2/e^6/(e*x+d)^4*d^3*b*c^2-2/e^4/(e*x+d)^3*a*b*c+c^3*ln
(e*x+d)/e^7+6*c^3*d/e^7/(e*x+d)

________________________________________________________________________________________

Maxima [A]  time = 1.15611, size = 633, normalized size = 2.38 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} -{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*
d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 -
(b^2*c + a*c^2)*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)
*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - 3*(a*
b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6*(b^2*c + a*c^2)*d^3*e^3 - (b^3 +
6*a*b*c)*d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^5)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 +
15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + c^3*log(e*x + d)/e^7

________________________________________________________________________________________

Fricas [B]  time = 1.95055, size = 1135, normalized size = 4.27 \begin{align*} \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} -{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*
d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 -
(b^2*c + a*c^2)*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)
*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - 3*(a*
b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6*(b^2*c + a*c^2)*d^3*e^3 - (b^3 +
6*a*b*c)*d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^5)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 + 15*c^3*d^2*e^4*x^4 + 20*c
^3*d^3*e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2
*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

________________________________________________________________________________________

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((c*x**2+b*x+a)**3/(e*x+d)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.13533, size = 574, normalized size = 2.16 \begin{align*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (180 \,{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5} - a c^{2} e^{5}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - 6 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - 3 \, a b^{2} e^{5} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - 6 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - 3 \, a b^{2} d e^{4} - 3 \, a^{2} c d e^{4} - 6 \, a^{2} b e^{5}\right )} x +{\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - 6 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="giac")

[Out]

c^3*e^(-7)*log(abs(x*e + d)) + 1/60*(180*(2*c^3*d*e^4 - b*c^2*e^5)*x^5 + 90*(15*c^3*d^2*e^3 - 5*b*c^2*d*e^4 -
b^2*c*e^5 - a*c^2*e^5)*x^4 + 20*(110*c^3*d^3*e^2 - 30*b*c^2*d^2*e^3 - 6*b^2*c*d*e^4 - 6*a*c^2*d*e^4 - b^3*e^5
- 6*a*b*c*e^5)*x^3 + 15*(125*c^3*d^4*e - 30*b*c^2*d^3*e^2 - 6*b^2*c*d^2*e^3 - 6*a*c^2*d^2*e^3 - b^3*d*e^4 - 6*
a*b*c*d*e^4 - 3*a*b^2*e^5 - 3*a^2*c*e^5)*x^2 + 6*(137*c^3*d^5 - 30*b*c^2*d^4*e - 6*b^2*c*d^3*e^2 - 6*a*c^2*d^3
*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 - 3*a*b^2*d*e^4 - 3*a^2*c*d*e^4 - 6*a^2*b*e^5)*x + (147*c^3*d^6 - 30*b*c^
2*d^5*e - 6*b^2*c*d^4*e^2 - 6*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 - 3*a*b^2*d^2*e^4 - 3*a^2*c*d^2*e^
4 - 6*a^2*b*d*e^5 - 10*a^3*e^6)*e^(-1))*e^(-6)/(x*e + d)^6