\(\int \frac {A+B x^2+C x^4+D x^6}{x^6 (d+e x^2)^2} \, dx\) [176]

Optimal result

Integrand size = 30, antiderivative size = 151 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {A}{5 d^2 x^5}-\frac {B d-2 A e}{3 d^3 x^3}-\frac {C d^2-e (2 B d-3 A e)}{d^4 x}+\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x}{2 d^4 \left (d+e x^2\right )}+\frac {\left (d^3 D-3 C d^2 e+5 B d e^2-7 A e^3\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{9/2} \sqrt {e}} \] Output:

-1/5*A/d^2/x^5-1/3*(-2*A*e+B*d)/d^3/x^3-(C*d^2-e*(-3*A*e+2*B*d))/d^4/x+1/2 
*(-A*e^3+B*d*e^2-C*d^2*e+D*d^3)*x/d^4/(e*x^2+d)+1/2*(-7*A*e^3+5*B*d*e^2-3* 
C*d^2*e+D*d^3)*arctan(e^(1/2)*x/d^(1/2))/d^(9/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=-\frac {A}{5 d^2 x^5}+\frac {-B d+2 A e}{3 d^3 x^3}+\frac {-C d^2+2 B d e-3 A e^2}{d^4 x}+\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x}{2 d^4 \left (d+e x^2\right )}+\frac {\left (d^3 D-3 C d^2 e+5 B d e^2-7 A e^3\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{9/2} \sqrt {e}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(d + e*x^2)^2),x]
 

Output:

-1/5*A/(d^2*x^5) + (-(B*d) + 2*A*e)/(3*d^3*x^3) + (-(C*d^2) + 2*B*d*e - 3* 
A*e^2)/(d^4*x) + ((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x)/(2*d^4*(d + e*x^2 
)) + ((d^3*D - 3*C*d^2*e + 5*B*d*e^2 - 7*A*e^3)*ArcTan[(Sqrt[e]*x)/Sqrt[d] 
])/(2*d^(9/2)*Sqrt[e])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2336, 25, 2333, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {x \left (-A e^3+B d e^2-C d^2 e+d^3 D\right )}{2 d^4 \left (d+e x^2\right )}-\frac {\int -\frac {\frac {\left (D d^3-C e d^2+B e^2 d-A e^3\right ) x^6}{d^3}+2 \left (C-\frac {e (B d-A e)}{d^2}\right ) x^4+2 \left (B-\frac {A e}{d}\right ) x^2+2 A}{x^6 \left (e x^2+d\right )}dx}{2 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {\left (D d^3-C e d^2+B e^2 d-A e^3\right ) x^6}{d^3}+2 \left (C-\frac {e (B d-A e)}{d^2}\right ) x^4+2 \left (B-\frac {A e}{d}\right ) x^2+2 A}{x^6 \left (e x^2+d\right )}dx}{2 d}+\frac {x \left (-A e^3+B d e^2-C d^2 e+d^3 D\right )}{2 d^4 \left (d+e x^2\right )}\)

\(\Big \downarrow \) 2333

\(\displaystyle \frac {\int \left (\frac {2 A}{d x^6}+\frac {D d^3-3 C e d^2+5 B e^2 d-7 A e^3}{d^3 \left (e x^2+d\right )}+\frac {2 \left (C d^2-e (2 B d-3 A e)\right )}{d^3 x^2}+\frac {2 (B d-2 A e)}{d^2 x^4}\right )dx}{2 d}+\frac {x \left (-A e^3+B d e^2-C d^2 e+d^3 D\right )}{2 d^4 \left (d+e x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-7 A e^3+5 B d e^2-3 C d^2 e+d^3 D\right )}{d^{7/2} \sqrt {e}}-\frac {2 \left (C d^2-e (2 B d-3 A e)\right )}{d^3 x}-\frac {2 (B d-2 A e)}{3 d^2 x^3}-\frac {2 A}{5 d x^5}}{2 d}+\frac {x \left (-A e^3+B d e^2-C d^2 e+d^3 D\right )}{2 d^4 \left (d+e x^2\right )}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(d + e*x^2)^2),x]
 

Output:

((d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x)/(2*d^4*(d + e*x^2)) + ((-2*A)/(5*d 
*x^5) - (2*(B*d - 2*A*e))/(3*d^2*x^3) - (2*(C*d^2 - e*(2*B*d - 3*A*e)))/(d 
^3*x) + ((d^3*D - 3*C*d^2*e + 5*B*d*e^2 - 7*A*e^3)*ArcTan[(Sqrt[e]*x)/Sqrt 
[d]])/(d^(7/2)*Sqrt[e]))/(2*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\frac {\left (\frac {1}{2} A \,e^{3}-\frac {1}{2} B d \,e^{2}+\frac {1}{2} C e \,d^{2}-\frac {1}{2} D d^{3}\right ) x}{e \,x^{2}+d}+\frac {\left (7 A \,e^{3}-5 B d \,e^{2}+3 C e \,d^{2}-D d^{3}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}}{d^{4}}-\frac {A}{5 d^{2} x^{5}}-\frac {-2 A e +B d}{3 d^{3} x^{3}}-\frac {3 A \,e^{2}-2 d B e +d^{2} C}{d^{4} x}\)	\(139\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/d^4*((1/2*A*e^3-1/2*B*d*e^2+1/2*C*e*d^2-1/2*D*d^3)*x/(e*x^2+d)+1/2*(7*A 
*e^3-5*B*d*e^2+3*C*d^2*e-D*d^3)/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-1/5*A 
/d^2/x^5-1/3*(-2*A*e+B*d)/d^3/x^3-(3*A*e^2-2*B*d*e+C*d^2)/d^4/x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.86 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=\left [\frac {30 \, {\left (D d^{4} e - 3 \, C d^{3} e^{2} + 5 \, B d^{2} e^{3} - 7 \, A d e^{4}\right )} x^{6} - 12 \, A d^{4} e - 20 \, {\left (3 \, C d^{4} e - 5 \, B d^{3} e^{2} + 7 \, A d^{2} e^{3}\right )} x^{4} - 4 \, {\left (5 \, B d^{4} e - 7 \, A d^{3} e^{2}\right )} x^{2} + 15 \, {\left ({\left (D d^{3} e - 3 \, C d^{2} e^{2} + 5 \, B d e^{3} - 7 \, A e^{4}\right )} x^{7} + {\left (D d^{4} - 3 \, C d^{3} e + 5 \, B d^{2} e^{2} - 7 \, A d e^{3}\right )} x^{5}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{60 \, {\left (d^{5} e^{2} x^{7} + d^{6} e x^{5}\right )}}, \frac {15 \, {\left (D d^{4} e - 3 \, C d^{3} e^{2} + 5 \, B d^{2} e^{3} - 7 \, A d e^{4}\right )} x^{6} - 6 \, A d^{4} e - 10 \, {\left (3 \, C d^{4} e - 5 \, B d^{3} e^{2} + 7 \, A d^{2} e^{3}\right )} x^{4} - 2 \, {\left (5 \, B d^{4} e - 7 \, A d^{3} e^{2}\right )} x^{2} + 15 \, {\left ({\left (D d^{3} e - 3 \, C d^{2} e^{2} + 5 \, B d e^{3} - 7 \, A e^{4}\right )} x^{7} + {\left (D d^{4} - 3 \, C d^{3} e + 5 \, B d^{2} e^{2} - 7 \, A d e^{3}\right )} x^{5}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right )}{30 \, {\left (d^{5} e^{2} x^{7} + d^{6} e x^{5}\right )}}\right ] \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^2,x, algorithm="fricas")
 

Output:

[1/60*(30*(D*d^4*e - 3*C*d^3*e^2 + 5*B*d^2*e^3 - 7*A*d*e^4)*x^6 - 12*A*d^4 
*e - 20*(3*C*d^4*e - 5*B*d^3*e^2 + 7*A*d^2*e^3)*x^4 - 4*(5*B*d^4*e - 7*A*d 
^3*e^2)*x^2 + 15*((D*d^3*e - 3*C*d^2*e^2 + 5*B*d*e^3 - 7*A*e^4)*x^7 + (D*d 
^4 - 3*C*d^3*e + 5*B*d^2*e^2 - 7*A*d*e^3)*x^5)*sqrt(-d*e)*log((e*x^2 + 2*s 
qrt(-d*e)*x - d)/(e*x^2 + d)))/(d^5*e^2*x^7 + d^6*e*x^5), 1/30*(15*(D*d^4* 
e - 3*C*d^3*e^2 + 5*B*d^2*e^3 - 7*A*d*e^4)*x^6 - 6*A*d^4*e - 10*(3*C*d^4*e 
 - 5*B*d^3*e^2 + 7*A*d^2*e^3)*x^4 - 2*(5*B*d^4*e - 7*A*d^3*e^2)*x^2 + 15*( 
(D*d^3*e - 3*C*d^2*e^2 + 5*B*d*e^3 - 7*A*e^4)*x^7 + (D*d^4 - 3*C*d^3*e + 5 
*B*d^2*e^2 - 7*A*d*e^3)*x^5)*sqrt(d*e)*arctan(sqrt(d*e)*x/d))/(d^5*e^2*x^7 
 + d^6*e*x^5)]
 

Sympy [A] (verification not implemented)

Time = 11.95 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{d^{9} e}} \left (- 7 A e^{3} + 5 B d e^{2} - 3 C d^{2} e + D d^{3}\right ) \log {\left (- d^{5} \sqrt {- \frac {1}{d^{9} e}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{9} e}} \left (- 7 A e^{3} + 5 B d e^{2} - 3 C d^{2} e + D d^{3}\right ) \log {\left (d^{5} \sqrt {- \frac {1}{d^{9} e}} + x \right )}}{4} + \frac {- 6 A d^{3} + x^{6} \left (- 105 A e^{3} + 75 B d e^{2} - 45 C d^{2} e + 15 D d^{3}\right ) + x^{4} \left (- 70 A d e^{2} + 50 B d^{2} e - 30 C d^{3}\right ) + x^{2} \cdot \left (14 A d^{2} e - 10 B d^{3}\right )}{30 d^{5} x^{5} + 30 d^{4} e x^{7}} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**6/(e*x**2+d)**2,x)
 

Output:

-sqrt(-1/(d**9*e))*(-7*A*e**3 + 5*B*d*e**2 - 3*C*d**2*e + D*d**3)*log(-d** 
5*sqrt(-1/(d**9*e)) + x)/4 + sqrt(-1/(d**9*e))*(-7*A*e**3 + 5*B*d*e**2 - 3 
*C*d**2*e + D*d**3)*log(d**5*sqrt(-1/(d**9*e)) + x)/4 + (-6*A*d**3 + x**6* 
(-105*A*e**3 + 75*B*d*e**2 - 45*C*d**2*e + 15*D*d**3) + x**4*(-70*A*d*e**2 
 + 50*B*d**2*e - 30*C*d**3) + x**2*(14*A*d**2*e - 10*B*d**3))/(30*d**5*x** 
5 + 30*d**4*e*x**7)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=\frac {{\left (D d^{3} - 3 \, C d^{2} e + 5 \, B d e^{2} - 7 \, A e^{3}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{4}} + \frac {D d^{3} x - C d^{2} e x + B d e^{2} x - A e^{3} x}{2 \, {\left (e x^{2} + d\right )} d^{4}} - \frac {15 \, C d^{2} x^{4} - 30 \, B d e x^{4} + 45 \, A e^{2} x^{4} + 5 \, B d^{2} x^{2} - 10 \, A d e x^{2} + 3 \, A d^{2}}{15 \, d^{4} x^{5}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^2,x, algorithm="giac")
 

Output:

1/2*(D*d^3 - 3*C*d^2*e + 5*B*d*e^2 - 7*A*e^3)*arctan(e*x/sqrt(d*e))/(sqrt( 
d*e)*d^4) + 1/2*(D*d^3*x - C*d^2*e*x + B*d*e^2*x - A*e^3*x)/((e*x^2 + d)*d 
^4) - 1/15*(15*C*d^2*x^4 - 30*B*d*e*x^4 + 45*A*e^2*x^4 + 5*B*d^2*x^2 - 10* 
A*d*e*x^2 + 3*A*d^2)/(d^4*x^5)
 

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=\frac {\frac {5\,B\,e^2\,x^4}{2\,d^3}-\frac {B}{3\,d}+\frac {5\,B\,e\,x^2}{3\,d^2}}{e\,x^5+d\,x^3}-\frac {\frac {A}{5\,d}-\frac {7\,A\,e\,x^2}{15\,d^2}+\frac {7\,A\,e^2\,x^4}{3\,d^3}+\frac {7\,A\,e^3\,x^6}{2\,d^4}}{e\,x^7+d\,x^5}-\frac {\frac {C}{d}+\frac {3\,C\,e\,x^2}{2\,d^2}}{e\,x^3+d\,x}+\frac {x\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ \frac {3}{2};\ -\frac {e\,x^2}{d}\right )}{d^2}-\frac {7\,A\,e^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{2\,d^{9/2}}+\frac {5\,B\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{2\,d^{7/2}}-\frac {3\,C\,\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{2\,d^{5/2}} \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^6*(d + e*x^2)^2),x)
 

Output:

((5*B*e^2*x^4)/(2*d^3) - B/(3*d) + (5*B*e*x^2)/(3*d^2))/(d*x^3 + e*x^5) - 
(A/(5*d) - (7*A*e*x^2)/(15*d^2) + (7*A*e^2*x^4)/(3*d^3) + (7*A*e^3*x^6)/(2 
*d^4))/(d*x^5 + e*x^7) - (C/d + (3*C*e*x^2)/(2*d^2))/(d*x + e*x^3) + (x*D* 
hypergeom([1/2, 2], 3/2, -(e*x^2)/d))/d^2 - (7*A*e^(5/2)*atan((e^(1/2)*x)/ 
d^(1/2)))/(2*d^(9/2)) + (5*B*e^(3/2)*atan((e^(1/2)*x)/d^(1/2)))/(2*d^(7/2) 
) - (3*C*e^(1/2)*atan((e^(1/2)*x)/d^(1/2)))/(2*d^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.22 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^2} \, dx=\frac {-105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d \,e^{3} x^{5}-105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{4} x^{7}+75 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{2} e^{2} x^{5}+75 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b d \,e^{3} x^{7}-45 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c \,d^{3} e \,x^{5}-45 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) c \,d^{2} e^{2} x^{7}+15 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d^{5} x^{5}+15 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d^{4} e \,x^{7}-6 a \,d^{4} e +14 a \,d^{3} e^{2} x^{2}-70 a \,d^{2} e^{3} x^{4}-105 a d \,e^{4} x^{6}-10 b \,d^{4} e \,x^{2}+50 b \,d^{3} e^{2} x^{4}+75 b \,d^{2} e^{3} x^{6}-30 c \,d^{4} e \,x^{4}-45 c \,d^{3} e^{2} x^{6}+15 d^{5} e \,x^{6}}{30 d^{5} e \,x^{5} \left (e \,x^{2}+d \right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^2,x)
 

Output:

( - 105*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e**3*x**5 - 105* 
sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e**4*x**7 + 75*sqrt(e)*sqr 
t(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*d**2*e**2*x**5 + 75*sqrt(e)*sqrt(d)*a 
tan((e*x)/(sqrt(e)*sqrt(d)))*b*d*e**3*x**7 - 45*sqrt(e)*sqrt(d)*atan((e*x) 
/(sqrt(e)*sqrt(d)))*c*d**3*e*x**5 - 45*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e) 
*sqrt(d)))*c*d**2*e**2*x**7 + 15*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt( 
d)))*d**5*x**5 + 15*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d**4*e*x 
**7 - 6*a*d**4*e + 14*a*d**3*e**2*x**2 - 70*a*d**2*e**3*x**4 - 105*a*d*e** 
4*x**6 - 10*b*d**4*e*x**2 + 50*b*d**3*e**2*x**4 + 75*b*d**2*e**3*x**6 - 30 
*c*d**4*e*x**4 - 45*c*d**3*e**2*x**6 + 15*d**5*e*x**6)/(30*d**5*e*x**5*(d 
+ e*x**2))