\(\int \frac {1}{(d+e x^2)^2 (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\) [219]

Optimal result

Integrand size = 39, antiderivative size = 187 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3} \]

[Out]

-1/4*x/d/(-b*e+2*c*d)/(e*x^2+d)^2-1/8*(-3*b*e+10*c*d)*x/d^2/(-b*e+2*c*d)^2/(e*x^2+d)-1/8*(3*b^2*e^2-16*b*c*d*e
+28*c^2*d^2)*arctan(x*e^(1/2)/d^(1/2))/d^(5/2)/(-b*e+2*c*d)^3/e^(1/2)-c^(5/2)*arctanh(x*c^(1/2)*e^(1/2)/(-b*e+
c*d)^(1/2))/(-b*e+2*c*d)^3/e^(1/2)/(-b*e+c*d)^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1163, 425, 541, 536, 211, 214} \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3}-\frac {x (10 c d-3 b e)}{8 d^2 \left (d+e x^2\right ) (2 c d-b e)^2}-\frac {x}{4 d \left (d+e x^2\right )^2 (2 c d-b e)} \]

[In]

Int[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-1/4*x/(d*(2*c*d - b*e)*(d + e*x^2)^2) - ((10*c*d - 3*b*e)*x)/(8*d^2*(2*c*d - b*e)^2*(d + e*x^2)) - ((28*c^2*d
^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3) - (c^(5/2)*ArcTa
nh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^3)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1163

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right )^3 \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx \\ & = -\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}+\frac {\int \frac {e (7 c d-3 b e)-3 c e^2 x^2}{\left (d+e x^2\right )^2 \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{4 d e (2 c d-b e)} \\ & = -\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac {\int \frac {e^2 \left (18 c^2 d^2-13 b c d e+3 b^2 e^2\right )-c e^3 (10 c d-3 b e) x^2}{\left (d+e x^2\right ) \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{8 d^2 e^2 (2 c d-b e)^2} \\ & = -\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac {c^3 \int \frac {1}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx}{(2 c d-b e)^3}-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 (2 c d-b e)^3} \\ & = -\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\frac {1}{8} \left (-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {\frac {(-2 c d+b e) x \left (-b e \left (5 d+3 e x^2\right )+2 c d \left (7 d+5 e x^2\right )\right )}{d^2 \left (d+e x^2\right )^2}+\frac {8 c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {-c d+b e}}\right )}{\sqrt {e} \sqrt {-c d+b e}}}{(-2 c d+b e)^3}\right ) \]

[In]

Integrate[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

(-(((28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3)) - ((
(-2*c*d + b*e)*x*(-(b*e*(5*d + 3*e*x^2)) + 2*c*d*(7*d + 5*e*x^2)))/(d^2*(d + e*x^2)^2) + (8*c^(5/2)*ArcTan[(Sq
rt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*Sqrt[-(c*d) + b*e]))/(-2*c*d + b*e)^3)/8

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {c^{3} \arctan \left (\frac {x c e}{\sqrt {\left (b e -c d \right ) e c}}\right )}{\left (b e -2 c d \right )^{3} \sqrt {\left (b e -c d \right ) e c}}+\frac {\frac {\frac {e \left (3 b^{2} e^{2}-16 b c d e +20 c^{2} d^{2}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 b^{2} e^{2}-24 b c d e +28 c^{2} d^{2}\right ) x}{8 d}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (3 b^{2} e^{2}-16 b c d e +28 c^{2} d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{8 d^{2} \sqrt {e d}}}{\left (b e -2 c d \right )^{3}}\)	\(174\)
risch	\(\text {Expression too large to display}\)	\(1012\)

[In]

int(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x,method=_RETURNVERBOSE)

[Out]

-c^3/(b*e-2*c*d)^3/((b*e-c*d)*e*c)^(1/2)*arctan(x*c*e/((b*e-c*d)*e*c)^(1/2))+1/(b*e-2*c*d)^3*((1/8*e*(3*b^2*e^
2-16*b*c*d*e+20*c^2*d^2)/d^2*x^3+1/8*(5*b^2*e^2-24*b*c*d*e+28*c^2*d^2)/d*x)/(e*x^2+d)^2+1/8*(3*b^2*e^2-16*b*c*
d*e+28*c^2*d^2)/d^2/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (161) = 322\).

Time = 0.85 (sec) , antiderivative size = 1765, normalized size of antiderivative = 9.44 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[-1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 + 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d
^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2
- c*d + b*e)) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4
+ 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)
) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b
^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*
c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 +
(28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4 + 2*(28*c^2*d^3*
e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + 4*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^
2 + c^2*d^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)
/(c*e*x^2 - c*d + b*e)) + (28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 +
6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8
*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e
^3 + 3*b^2*d*e^4)*x^3 - 16*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e
*x*sqrt(-c/(c*d*e - b*e^2))) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3
*b^2*e^4)*x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x -
d)/(e*x^2 + d)) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2
*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d
^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^
2*d*e^4)*x^3 - 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c
/(c*d*e - b*e^2))) + (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*
x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (28*c^2*d^4*e - 2
4*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6
*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d
^5*e^4 - b^3*d^4*e^5)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x**2+d)**2/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\frac {c^{3} \arctan \left (\frac {c e x}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {-c^{2} d e + b c e^{2}}} - \frac {{\left (28 \, c^{2} d^{2} - 16 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, {\left (8 \, c^{3} d^{5} - 12 \, b c^{2} d^{4} e + 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \sqrt {d e}} - \frac {10 \, c d e x^{3} - 3 \, b e^{2} x^{3} + 14 \, c d^{2} x - 5 \, b d e x}{8 \, {\left (4 \, c^{2} d^{4} - 4 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (e x^{2} + d\right )}^{2}} \]

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

c^3*arctan(c*e*x/sqrt(-c^2*d*e + b*c*e^2))/((8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(-c^2*d
*e + b*c*e^2)) - 1/8*(28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*arctan(e*x/sqrt(d*e))/((8*c^3*d^5 - 12*b*c^2*d^4*e
+ 6*b^2*c*d^3*e^2 - b^3*d^2*e^3)*sqrt(d*e)) - 1/8*(10*c*d*e*x^3 - 3*b*e^2*x^3 + 14*c*d^2*x - 5*b*d*e*x)/((4*c^
2*d^4 - 4*b*c*d^3*e + b^2*d^2*e^2)*(e*x^2 + d)^2)

Mupad [B] (verification not implemented)

Time = 9.17 (sec) , antiderivative size = 6267, normalized size of antiderivative = 33.51 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\text {Too large to display} \]

[In]

int(1/((d + e*x^2)^2*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

((x*(5*b*e - 14*c*d))/(8*d*(b^2*e^2 + 4*c^2*d^2 - 4*b*c*d*e)) + (e*x^3*(3*b*e - 10*c*d))/(8*d^2*(b^2*e^2 + 4*c
^2*d^2 - 4*b*c*d*e)))/(d^2 + e^2*x^4 + 2*d*e*x^2) - (atan(((((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*
d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^
2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7
*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*
b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*
e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) - (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2
*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^1
3 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^
7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1
/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))
^(1/2)*1i)/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) + (((x*(9*b^4*c^3*e
^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4
*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3
504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^
12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*
b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) + (x*(-c^5*e*(b*e - c*d))^(1/2)
*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c
^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3
+ 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*
c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b
^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2)*1i)/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 -
7*b^3*c*d*e^4))/(((9*b^3*c^5*e^8)/32 - (35*c^8*d^3*e^5)/4 + (61*b*c^7*d^2*e^6)/8 - (39*b^2*c^6*d*e^7)/16)/(64*
c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 1
92*b*c^5*d^9*e) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*
d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^1
0*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^
5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^
6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*
e)) - (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 4
0960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d
^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^
3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3
*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3
*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) - (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 -
96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2
- 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 +
1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^
2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b
^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) + (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e
^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^
7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*
e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b
^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(b
^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2)*1i)
/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) - (atan(((((x*(9*b^4*c^3*e^10
+ 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^
4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504
*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12
- (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^
4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) - (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^
2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7
*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^
4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*
b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2
*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(16*(8*c^3*d^8*e - b^3*
d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 -
96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2
- 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 +
1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^
2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*
c^2*d^6*e^4 - 192*b*c^5*d^9*e) + (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8
- 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6
*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(
16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2
+ 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1
/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*
e^3)))/(((9*b^3*c^5*e^8)/32 - (35*c^8*d^3*e^5)/4 + (61*b*c^7*d^2*e^6)/8 - (39*b^2*c^6*d*e^7)/16)/(64*c^6*d^10
+ b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*
d^9*e) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))
/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^
6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5
*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 -
12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) - (x*(-d^
5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6
*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(
512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e
^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d
^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e
))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)) - (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*
e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5
*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^
8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d
^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160
*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) + (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*
d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b
^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^
7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*
(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^
2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2
*d^7*e^2 + 6*b^2*c*d^6*e^3))))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(8*(8*c^3*d^8*e - b^3*
d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3))