3.254 \(\int \frac {(b x+c x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=218 \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac {c^3 x}{e^6} \]

[Out]

c^3*x/e^6-1/5*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^5+3/4*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/(e*x+d)^4-d*(-b*e+c*d)*(b^2
*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^3+1/2*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)/e^7/(e*x+d)^2-3*c*(b^
2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)-3*c^2*(-b*e+2*c*d)*ln(e*x+d)/e^7

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Rubi [A]  time = 0.18, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {c^3 x}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

(c^3*x)/e^6 - (d^3*(c*d - b*e)^3)/(5*e^7*(d + e*x)^5) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(4*e^7*(d + e*x)^4
) - (d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)^3) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*
d*e + b^2*e^2))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - (3*c^2*(2*c*d
- b*e)*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 (b x+c x^2\right )^3}{(d+e x)^6} \, dx &=\int \left (\frac {c^3}{e^6}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^5}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^4}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^3}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^2}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac {c^3 x}{e^6}-\frac {d^3 (c d-b e)^3}{5 e^7 (d+e x)^5}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 242, normalized size = 1.11 \[ -\frac {b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+12 b^2 c e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b c^2 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )}{20 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-1/20*(b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 12*b^2*c*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2
+ 10*d*e^3*x^3 + 5*e^4*x^4) - b*c^2*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^
4) + 2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*
x^6) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

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fricas [B]  time = 1.35, size = 462, normalized size = 2.12 \[ \frac {20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 12 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 20 \, {\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} - 10 \, {\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} - 10 \, {\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} - 5 \, {\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e + {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \, {\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/20*(20*c^3*e^6*x^6 + 100*c^3*d*e^5*x^5 - 174*c^3*d^6 + 137*b*c^2*d^5*e - 12*b^2*c*d^4*e^2 - b^3*d^3*e^3 - 20
*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 - 10*(80*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 +
b^3*e^6)*x^3 - 10*(120*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 - 5*(150*c^3*d^5*e
- 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x - 60*(2*c^3*d^6 - b*c^2*d^5*e + (2*c^3*d*e^5 - b*c^2*
e^6)*x^5 + 5*(2*c^3*d^2*e^4 - b*c^2*d*e^5)*x^4 + 10*(2*c^3*d^3*e^3 - b*c^2*d^2*e^4)*x^3 + 10*(2*c^3*d^4*e^2 -
b*c^2*d^3*e^3)*x^2 + 5*(2*c^3*d^5*e - b*c^2*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x
^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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giac [A]  time = 0.17, size = 251, normalized size = 1.15 \[ c^{3} x e^{\left (-6\right )} - 3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{20 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

c^3*x*e^(-6) - 3*(2*c^3*d - b*c^2*e)*e^(-7)*log(abs(x*e + d)) - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c
*d^4*e^2 + b^3*d^3*e^3 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + b^3*e^6)*x^3 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^
5)*x^2 + 5*(154*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)*e^(-7)/(x*e + d)^5

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maple [A]  time = 0.05, size = 379, normalized size = 1.74 \[ \frac {b^{3} d^{3}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {3 b^{2} c \,d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {3 b \,c^{2} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {c^{3} d^{6}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {3 b^{3} d^{2}}{4 \left (e x +d \right )^{4} e^{4}}+\frac {3 b^{2} c \,d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {15 b \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {3 c^{3} d^{5}}{2 \left (e x +d \right )^{4} e^{7}}+\frac {b^{3} d}{\left (e x +d \right )^{3} e^{4}}-\frac {6 b^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {10 b \,c^{2} d^{3}}{\left (e x +d \right )^{3} e^{6}}-\frac {5 c^{3} d^{4}}{\left (e x +d \right )^{3} e^{7}}-\frac {b^{3}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {6 b^{2} c d}{\left (e x +d \right )^{2} e^{5}}-\frac {15 b \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{6}}+\frac {10 c^{3} d^{3}}{\left (e x +d \right )^{2} e^{7}}-\frac {3 b^{2} c}{\left (e x +d \right ) e^{5}}+\frac {15 b \,c^{2} d}{\left (e x +d \right ) e^{6}}+\frac {3 b \,c^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {15 c^{3} d^{2}}{\left (e x +d \right ) e^{7}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}+\frac {c^{3} x}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^6,x)

[Out]

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

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maxima [A]  time = 1.52, size = 311, normalized size = 1.43 \[ -\frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^
2*c*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 + b^3*e^6)*x^3 + 10*(130*c^3*d^4*e^2 -
110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 5*(154*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^
3 + b^3*d^2*e^4)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7) + c^3
*x/e^6 - 3*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^7

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mupad [B]  time = 0.29, size = 312, normalized size = 1.43 \[ \frac {c^3\,x}{e^6}-\frac {x^4\,\left (3\,b^2\,c\,e^5-15\,b\,c^2\,d\,e^4+15\,c^3\,d^2\,e^3\right )+x^2\,\left (\frac {b^3\,d\,e^4}{2}+6\,b^2\,c\,d^2\,e^3-55\,b\,c^2\,d^3\,e^2+65\,c^3\,d^4\,e\right )+x\,\left (\frac {b^3\,d^2\,e^3}{4}+3\,b^2\,c\,d^3\,e^2-\frac {125\,b\,c^2\,d^4\,e}{4}+\frac {77\,c^3\,d^5}{2}\right )+\frac {b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2-137\,b\,c^2\,d^5\,e+174\,c^3\,d^6}{20\,e}+x^3\,\left (\frac {b^3\,e^5}{2}+6\,b^2\,c\,d\,e^4-45\,b\,c^2\,d^2\,e^3+50\,c^3\,d^3\,e^2\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^3\,d-3\,b\,c^2\,e\right )}{e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x)^6,x)

[Out]

(c^3*x)/e^6 - (x^4*(3*b^2*c*e^5 + 15*c^3*d^2*e^3 - 15*b*c^2*d*e^4) + x^2*((b^3*d*e^4)/2 + 65*c^3*d^4*e - 55*b*
c^2*d^3*e^2 + 6*b^2*c*d^2*e^3) + x*((77*c^3*d^5)/2 + (b^3*d^2*e^3)/4 + 3*b^2*c*d^3*e^2 - (125*b*c^2*d^4*e)/4)
+ (174*c^3*d^6 + b^3*d^3*e^3 + 12*b^2*c*d^4*e^2 - 137*b*c^2*d^5*e)/(20*e) + x^3*((b^3*e^5)/2 + 50*c^3*d^3*e^2
- 45*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4))/(d^5*e^6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^2 + 10*d^
2*e^9*x^3) - (log(d + e*x)*(6*c^3*d - 3*b*c^2*e))/e^7

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sympy [A]  time = 11.62, size = 326, normalized size = 1.50 \[ \frac {c^{3} x}{e^{6}} + \frac {3 c^{2} \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{7}} + \frac {- b^{3} d^{3} e^{3} - 12 b^{2} c d^{4} e^{2} + 137 b c^{2} d^{5} e - 174 c^{3} d^{6} + x^{4} \left (- 60 b^{2} c e^{6} + 300 b c^{2} d e^{5} - 300 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 10 b^{3} e^{6} - 120 b^{2} c d e^{5} + 900 b c^{2} d^{2} e^{4} - 1000 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 10 b^{3} d e^{5} - 120 b^{2} c d^{2} e^{4} + 1100 b c^{2} d^{3} e^{3} - 1300 c^{3} d^{4} e^{2}\right ) + x \left (- 5 b^{3} d^{2} e^{4} - 60 b^{2} c d^{3} e^{3} + 625 b c^{2} d^{4} e^{2} - 770 c^{3} d^{5} e\right )}{20 d^{5} e^{7} + 100 d^{4} e^{8} x + 200 d^{3} e^{9} x^{2} + 200 d^{2} e^{10} x^{3} + 100 d e^{11} x^{4} + 20 e^{12} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**6,x)

[Out]

c**3*x/e**6 + 3*c**2*(b*e - 2*c*d)*log(d + e*x)/e**7 + (-b**3*d**3*e**3 - 12*b**2*c*d**4*e**2 + 137*b*c**2*d**
5*e - 174*c**3*d**6 + x**4*(-60*b**2*c*e**6 + 300*b*c**2*d*e**5 - 300*c**3*d**2*e**4) + x**3*(-10*b**3*e**6 -
120*b**2*c*d*e**5 + 900*b*c**2*d**2*e**4 - 1000*c**3*d**3*e**3) + x**2*(-10*b**3*d*e**5 - 120*b**2*c*d**2*e**4
+ 1100*b*c**2*d**3*e**3 - 1300*c**3*d**4*e**2) + x*(-5*b**3*d**2*e**4 - 60*b**2*c*d**3*e**3 + 625*b*c**2*d**4
*e**2 - 770*c**3*d**5*e))/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e
**11*x**4 + 20*e**12*x**5)

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