3.1.7 \(\int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx\)
Optimal. Leaf size=349 \[ -\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}} \]
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Rubi [A] time = 0.42, antiderivative size = 349, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used =
{1419, 1093, 207, 203} \begin {gather*} -\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]
[Out]
-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[
Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d
*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x
)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])
- (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt
[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]])
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 1093
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]
Rule 1419
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[(2*d)/e -
b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x^(n/2
) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2,
0] && IGtQ[n/2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !LtQ[(2*d)/e - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Rubi steps
\begin {align*} \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx &=\frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {b+2 d e} x^2}{e}+x^4} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {b+2 d e} x^2}{e}+x^4} \, dx}{2 e}\\ &=\frac {\int \frac {1}{-\frac {\sqrt {b-2 d e}}{2 e}-\frac {\sqrt {b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt {b-2 d e}}-\frac {\int \frac {1}{\frac {\sqrt {b-2 d e}}{2 e}-\frac {\sqrt {b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt {b-2 d e}}+\frac {\int \frac {1}{-\frac {\sqrt {b-2 d e}}{2 e}+\frac {\sqrt {b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt {b-2 d e}}-\frac {\int \frac {1}{\frac {\sqrt {b-2 d e}}{2 e}+\frac {\sqrt {b+2 d e}}{2 e}+x^2} \, dx}{2 \sqrt {b-2 d e}}\\ &=-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}-\sqrt {b+2 d e}}}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\right )}{\sqrt {2} \sqrt {b-2 d e} \sqrt {\sqrt {b-2 d e}+\sqrt {b+2 d e}}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 69, normalized size = 0.20 \begin {gather*} \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8 e^2-\text {$\#$1}^4 b+d^2\&,\frac {\text {$\#$1}^4 e \log (x-\text {$\#$1})+d \log (x-\text {$\#$1})}{2 \text {$\#$1}^7 e^2-\text {$\#$1}^3 b}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]
[Out]
RootSum[d^2 - b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(-(b*#1^3) + 2*e^2*#1^7) & ]/4
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^4}{d^2-b x^4+e^2 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]
[Out]
IntegrateAlgebraic[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8), x]
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fricas [B] time = 1.71, size = 3048, normalized size = 8.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^4+d)/(e^2*x^8-b*x^4+d^2),x, algorithm="fricas")
[Out]
-sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d
^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*arctan(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 - 12*b*d^4*e^2
+ 6*b^2*d^3*e - b^3*d^2)*x*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - (4*d^2*e^2
- 4*b*d*e + b^2)*x)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b
^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)) + (4*d^2*e^2 - 4*b*d*e + b^2 - (8*d^5*e^3 - 12*b*
d^4*e^2 + 6*b^2*d^3*e - b^3*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)))*sqrt(-
((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/
(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2))*sqrt((2*e^2*x^2 - sqrt(1/2)*(2*b*d*e - b^2 + (8*d^5*e^3 - 12*b*d^4*e^2 + 6*
b^2*d^3*e - b^3*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)))*sqrt(-((4*d^4*e^2
- 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 -
4*b*d^3*e + b^2*d^2)))/e^2))*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7
*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))/e) + sqrt(sqrt(1/2)*sqr
t(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b
)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*arctan(-1/4*(2*sqrt(1/2)*((8*d^5*e^3 - 12*b*d^4*e^2 + 6*b^2*d^3*e - b^3*
d^2)*x*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + (4*d^2*e^2 - 4*b*d*e + b^2)*x)*
sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5
*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e +
b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)) - (4*d^2*e^2 -
4*b*d*e + b^2 + (8*d^5*e^3 - 12*b*d^4*e^2 + 6*b^2*d^3*e - b^3*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^
2 + 6*b^2*d^5*e - b^3*d^4)))*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e
^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*sqrt(((4*d^4*e^2 - 4*b*d^
3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3
*e + b^2*d^2))*sqrt((2*e^2*x^2 - sqrt(1/2)*(2*b*d*e - b^2 - (8*d^5*e^3 - 12*b*d^4*e^2 + 6*b^2*d^3*e - b^3*d^2)
*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)))*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2
)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)
))/e^2))/e) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d
^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 -
4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)
*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4))
+ b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-
(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*log(e
*x - 1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e
- b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^
6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) + 1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e
^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^
2 - 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e
^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(
-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) - 1
/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2
*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e - (4*d^4*e^2 - 4*b*d^3*e + b^
2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(-((4*d^4
*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*
e^2 - 4*b*d^3*e + b^2*d^2))))
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^4+d)/(e^2*x^8-b*x^4+d^2),x, algorithm="giac")
[Out]
Timed out
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maple [C] time = 0.03, size = 55, normalized size = 0.16 \begin {gather*} \frac {\left (\RootOf \left (e^{2} \textit {\_Z}^{8}-b \,\textit {\_Z}^{4}+d^{2}\right )^{4} e +d \right ) \ln \left (-\RootOf \left (e^{2} \textit {\_Z}^{8}-b \,\textit {\_Z}^{4}+d^{2}\right )+x \right )}{8 \RootOf \left (e^{2} \textit {\_Z}^{8}-b \,\textit {\_Z}^{4}+d^{2}\right )^{7} e^{2}-4 \RootOf \left (e^{2} \textit {\_Z}^{8}-b \,\textit {\_Z}^{4}+d^{2}\right )^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^4+d)/(e^2*x^8-b*x^4+d^2),x)
[Out]
1/4*sum((_R^4*e+d)/(2*_R^7*e^2-_R^3*b)*ln(-_R+x),_R=RootOf(_Z^8*e^2-_Z^4*b+d^2))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e x^{4} + d}{e^{2} x^{8} - b x^{4} + d^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^4+d)/(e^2*x^8-b*x^4+d^2),x, algorithm="maxima")
[Out]
integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2), x)
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mupad [B] time = 4.03, size = 10337, normalized size = 29.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x)
[Out]
2*atan(((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 + ((b - 2*d*e)^5*(b +
2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*
e^2)))^(1/4)*((x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10
+ 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) +
4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(2
62144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6
*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2
- 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^7*e
^14 - 256*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*
d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (x*(3
2*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2)
+ 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*
((x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*
d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2
- 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^
15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 19660
8*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)
/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 + 256*b*d
^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b
^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4))/((x*(32*b*d^5*e^13
+ 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^
2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^
9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32
768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/
(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b
*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^1
2 - 196608*b^2*d^8*e^13)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^
2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*
b^4*d^3*e^10 + 64*b^3*d^4*e^11)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*
(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i - (x*(32*b*d^5*e^13 + 4*b^4*d^
2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*
d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 + 3
2768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^
6*e^12 - 65536*b^2*d^7*e^13) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*
d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14
- 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196608
*b^2*d^8*e^13)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6
*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^
10 - 64*b^3*d^4*e^11)*1i)*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 +
16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1
/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/
4) - atan(((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) + ((b^3 + ((b - 2*d*e)^5*(b
+ 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d
^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e
^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2
) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)
*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*
d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2
- 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14
- 256*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^
2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i + (x*(32*b
*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) +
4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x
*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5
*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4
*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15
+ 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b
^3*d^7*e^12 - 196608*b^2*d^8*e^13))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*
(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) + 256*d^7*e^14 + 256*b*d^6*e^13 -
16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512
*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i)/((x*(32*b*d^5*e^13 + 4*b^4*d
^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2
*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 +
32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d
^6*e^12 - 65536*b^2*d^7*e^13) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4
*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14
- 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 19660
8*b^2*d^8*e^13))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e
^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*b^4*d^3*e^10 +
64*b^3*d^4*e^11))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*
e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) - (x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e
^11 + 48*b^2*d^4*e^12) + ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 +
16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 102
4*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^
7*e^13) - ((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*
b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 -
4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13))*((b^3
+ ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*
b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11))*((b^
3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32
*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)))*((b^3 + ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(5
12*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*2i - atan(((x*(32*b*d^5*e^13 +
4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) + ((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2
- 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b
- 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e
^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 491
52*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) + x*(65536*d^9*e^15 + 32768*
b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^1
2 - 65536*b^2*d^7*e^13))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 +
16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*b^4*d^3
*e^10 + 64*b^3*d^4*e^11))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 +
16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i + (x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 2
4*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*
(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d
*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2
)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 4
9152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) - x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b
^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e
^13))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*
d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*
e^11))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3
*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*1i)/((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48
*b^2*d^4*e^12) + ((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e
^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^
2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d
^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 -
196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13) + x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b
^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*((b^3 - ((b -
2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^
3 + 24*b^2*d^4*e^2)))^(3/4) - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11))*((b^3 - ((b
- 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e
^3 + 24*b^2*d^4*e^2)))^(1/4) - (x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) - ((b^3
- ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*
b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(51
2*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^
9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 -
196608*b^2*d^8*e^13) - x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5
*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))
^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^
(3/4) - 256*d^7*e^14 - 256*b*d^6*e^13 + 16*b^4*d^3*e^10 + 64*b^3*d^4*e^11))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e)
)^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))
^(1/4)))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b
^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*2i - 2*atan(((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3
*e^11 + 48*b^2*d^4*e^12) - ((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2
+ 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2
) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)
*(262144*d^10*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*
d^6*e^11 - 196608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i + x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2
*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*
((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e
- 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^1
1)*1i)*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3
*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4) + (x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^
2*d^4*e^12) + ((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4
- 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e
^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10
*e^15 + 262144*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 19
6608*b^3*d^7*e^12 - 196608*b^2*d^8*e^13)*1i - x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b
^6*d^3*e^9 + 10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*((b^3 - ((b -
2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^
3 + 24*b^2*d^4*e^2)))^(3/4)*1i + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*((b^3
- ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b
*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4))/((x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) -
((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e
- 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*
e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 26214
4*b*d^9*e^14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*
e^12 - 196608*b^2*d^8*e^13)*1i + x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 +
10240*b^5*d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*((b^3 - ((b - 2*d*e)^5*(b
+ 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^
4*e^2)))^(3/4)*1i + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*((b^3 - ((b - 2*d*e
)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24
*b^2*d^4*e^2)))^(1/4)*1i - (x*(32*b*d^5*e^13 + 4*b^4*d^2*e^10 + 24*b^3*d^3*e^11 + 48*b^2*d^4*e^12) + ((b^3 - (
(b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^
5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*((((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b
^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)*(262144*d^10*e^15 + 262144*b*d^9*e^
14 - 4096*b^7*d^3*e^8 - 4096*b^6*d^4*e^9 + 49152*b^5*d^5*e^10 + 49152*b^4*d^6*e^11 - 196608*b^3*d^7*e^12 - 196
608*b^2*d^8*e^13)*1i - x*(65536*d^9*e^15 + 32768*b*d^8*e^14 - 1024*b^7*d^2*e^8 - 2048*b^6*d^3*e^9 + 10240*b^5*
d^4*e^10 + 20480*b^4*d^5*e^11 - 32768*b^3*d^6*e^12 - 65536*b^2*d^7*e^13))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^
(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(
3/4)*1i + 256*d^7*e^14 + 256*b*d^6*e^13 - 16*b^4*d^3*e^10 - 64*b^3*d^4*e^11)*1i)*((b^3 - ((b - 2*d*e)^5*(b + 2
*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e
^2)))^(1/4)*1i))*((b^3 - ((b - 2*d*e)^5*(b + 2*d*e))^(1/2) + 4*b*d^2*e^2 - 4*b^2*d*e)/(512*(b^4*d^2 + 16*d^6*e
^4 - 8*b^3*d^3*e - 32*b*d^5*e^3 + 24*b^2*d^4*e^2)))^(1/4)
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sympy [A] time = 8.25, size = 136, normalized size = 0.39 \begin {gather*} \operatorname {RootSum} {\left (t^{8} \left (65536 b^{4} d^{2} - 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} - 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \left (- 256 b^{3} + 1024 b^{2} d e - 1024 b d^{2} e^{2}\right ) + e^{2}, \left (t \mapsto t \log {\left (x + \frac {1024 t^{5} b^{2} d^{2} - 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} - 4 t b + 4 t d e}{e} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**4+d)/(e**2*x**8-b*x**4+d**2),x)
[Out]
RootSum(_t**8*(65536*b**4*d**2 - 524288*b**3*d**3*e + 1572864*b**2*d**4*e**2 - 2097152*b*d**5*e**3 + 1048576*d
**6*e**4) + _t**4*(-256*b**3 + 1024*b**2*d*e - 1024*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**
2*d**2 - 4096*_t**5*b*d**3*e + 4096*_t**5*d**4*e**2 - 4*_t*b + 4*_t*d*e)/e)))
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