3.44 \(\int \frac {1}{(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4)^2} \, dx\)

Optimal. Leaf size=342 \[ \frac {2 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac {24 e \left (-d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}+\frac {24 e \left (d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \]

[Out]

2*e*(1/4*d/e+x)*(13*d^4-256*a*e^3-48*d^2*e^2*(1/4*d/e+x)^2)/(-16384*a^2*e^6-64*a*d^4*e^3+5*d^8)/(8*e^3*x^4+8*d
*e^2*x^3-d^3*x+8*a*e^2)-24*e*arctanh((4*e*x+d)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2))^(1/2))*(d^4+128*a*e^3-d^2*(-64*
a*e^3+d^4)^(1/2))/(-64*a*e^3+d^4)^(3/2)/(256*a*e^3+5*d^4)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2))^(1/2)+24*e*arctanh((
4*e*x+d)/(3*d^2+2*(-64*a*e^3+d^4)^(1/2))^(1/2))*(d^4+128*a*e^3+d^2*(-64*a*e^3+d^4)^(1/2))/(-64*a*e^3+d^4)^(3/2
)/(256*a*e^3+5*d^4)/(3*d^2+2*(-64*a*e^3+d^4)^(1/2))^(1/2)

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Rubi [A]  time = 0.53, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1106, 1092, 1166, 208} \[ \frac {2 e \left (\frac {d}{4 e}+x\right ) \left (-256 a e^3-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2+13 d^4\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac {24 e \left (-d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}+\frac {24 e \left (d^2 \sqrt {d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \]

Antiderivative was successfully verified.

[In]

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-2),x]

[Out]

(2*e*(d/(4*e) + x)*(13*d^4 - 256*a*e^3 - 48*d^2*e^2*(d/(4*e) + x)^2))/((5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)*
(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) - (24*e*(d^4 + 128*a*e^3 - d^2*Sqrt[d^4 - 64*a*e^3])*ArcTanh[(d +
4*e*x)/Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]])/((d^4 - 64*a*e^3)^(3/2)*(5*d^4 + 256*a*e^3)*Sqrt[3*d^2 - 2*Sqrt
[d^4 - 64*a*e^3]]) + (24*e*(d^4 + 128*a*e^3 + d^2*Sqrt[d^4 - 64*a*e^3])*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 + 2*Sqr
t[d^4 - 64*a*e^3]]])/((d^4 - 64*a*e^3)^(3/2)*(5*d^4 + 256*a*e^3)*Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])

Rule 208

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

Rule 1092

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^
4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p + 1)
*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{\left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4\right )^2} \, dx,x,\frac {d}{4 e}+x\right )\\ &=\frac {2 e \left (\frac {d}{4 e}+x\right ) \left (13 d^4-256 a e^3-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (5 d^8-64 a d^4 e^3-16384 a^2 e^6\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac {4 \operatorname {Subst}\left (\int \frac {9 d^4 e^2-\frac {1}{2} e^3 \left (\frac {5 d^4}{e}+256 a e^2\right )-2 \left (9 d^4 e^2-e^3 \left (\frac {5 d^4}{e}+256 a e^2\right )\right )+24 d^2 e^4 x^2}{\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac {d}{4 e}+x\right )}{e \left (5 d^8-64 a d^4 e^3-16384 a^2 e^6\right )}\\ &=\frac {2 e \left (\frac {d}{4 e}+x\right ) \left (13 d^4-256 a e^3-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (5 d^8-64 a d^4 e^3-16384 a^2 e^6\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}+\frac {\left (48 e^3 \left (d^4+128 a e^3-d^2 \sqrt {d^4-64 a e^3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}+e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right )}-\frac {\left (48 e^3 \left (d^4+128 a e^3+d^2 \sqrt {d^4-64 a e^3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}-e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right )}\\ &=\frac {2 e \left (\frac {d}{4 e}+x\right ) \left (13 d^4-256 a e^3-48 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{\left (5 d^8-64 a d^4 e^3-16384 a^2 e^6\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac {24 e \left (d^4+128 a e^3-d^2 \sqrt {d^4-64 a e^3}\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right ) \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}+\frac {24 e \left (d^4+128 a e^3+d^2 \sqrt {d^4-64 a e^3}\right ) \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (5 d^4+256 a e^3\right ) \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\\ \end {align*}

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Mathematica [C]  time = 0.19, size = 234, normalized size = 0.68 \[ \frac {48 e^2 \text {RootSum}\left [8 \text {$\#$1}^4 e^3+8 \text {$\#$1}^3 d e^2-\text {$\#$1} d^3+8 a e^2\& ,\frac {2 \text {$\#$1}^2 d^2 e \log (x-\text {$\#$1})+32 a e^2 \log (x-\text {$\#$1})+\text {$\#$1} d^3 \log (x-\text {$\#$1})}{32 \text {$\#$1}^3 e^3+24 \text {$\#$1}^2 d e^2-d^3}\& \right ]}{16384 a^2 e^6+64 a d^4 e^3-5 d^8}+\frac {(d+4 e x) \left (-128 a e^3+5 d^4-12 d^3 e x-24 d^2 e^2 x^2\right )}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-2),x]

[Out]

((d + 4*e*x)*(5*d^4 - 128*a*e^3 - 12*d^3*e*x - 24*d^2*e^2*x^2))/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)*(8*a*e^2
- d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) + (48*e^2*RootSum[8*a*e^2 - d^3*#1 + 8*d*e^2*#1^3 + 8*e^3*#1^4 & , (32*a*
e^2*Log[x - #1] + d^3*Log[x - #1]*#1 + 2*d^2*e*Log[x - #1]*#1^2)/(-d^3 + 24*d*e^2*#1^2 + 32*e^3*#1^3) & ])/(-5
*d^8 + 64*a*d^4*e^3 + 16384*a^2*e^6)

________________________________________________________________________________________

fricas [B]  time = 0.73, size = 4285, normalized size = 12.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="fricas")

[Out]

-(96*d^2*e^3*x^3 + 72*d^3*e^2*x^2 - 5*d^5 + 128*a*d*e^3 + 12*sqrt(2)*(40*a*d^8*e^2 - 512*a^2*d^4*e^5 - 131072*
a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4*e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2 - 64*a*d^5*e^5 - 16384*a^2*d*e^8)*x^
3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 + (125*d
^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a
^5*d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*
d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5
*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^2
4 - 73786976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^
9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))*log(884736*a*d^5*e^6 + 22649
2416*a^2*d*e^9 + 3538944*(a*d^4*e^7 + 256*a^2*e^10)*x + 13824*sqrt(2)*(d^16*e^2 - 128*a*d^12*e^5 - 61440*a^2*d
^8*e^8 + 8388608*a^3*d^4*e^11 - 268435456*a^4*e^14 - (125*d^30 + 59200*a*d^26*e^3 - 3624960*a^2*d^22*e^6 - 566
493184*a^3*d^18*e^9 + 19797114880*a^4*d^14*e^12 + 1906965479424*a^5*d^10*e^15 - 30786325577728*a^6*d^6*e^18 -
2251799813685248*a^7*d^2*e^21)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d^32*e^
3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*d^16*e
^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24 - 737
86976294838206464*a^9*e^27)))*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 + (125*d^24 - 4800*a*d^20*e^3
- 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046
511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d^32*e^3 - 115200000*
a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*d^16*e^15 + 27443810
22928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24 - 73786976294838206
464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^
8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))) - 12*sqrt(2)*(40*a*d^8*e^2 - 512*a^2*d^4*e^5 - 1
31072*a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4*e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2 - 64*a*d^5*e^5 - 16384*a^2*d*e
^8)*x^3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 +
(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 5153960
7552*a^5*d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800
000*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 780825054412
80*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d
^4*e^24 - 73786976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d
^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))*log(884736*a*d^5*e^6 +
226492416*a^2*d*e^9 + 3538944*(a*d^4*e^7 + 256*a^2*e^10)*x - 13824*sqrt(2)*(d^16*e^2 - 128*a*d^12*e^5 - 61440
*a^2*d^8*e^8 + 8388608*a^3*d^4*e^11 - 268435456*a^4*e^14 - (125*d^30 + 59200*a*d^26*e^3 - 3624960*a^2*d^22*e^6
- 566493184*a^3*d^18*e^9 + 19797114880*a^4*d^14*e^12 + 1906965479424*a^5*d^10*e^15 - 30786325577728*a^6*d^6*e
^18 - 2251799813685248*a^7*d^2*e^21)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d
^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*
d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24
- 73786976294838206464*a^9*e^27)))*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 + (125*d^24 - 4800*a*d^
20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4
398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d^32*e^3 - 1152
00000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*d^16*e^15 + 27
44381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24 - 73786976294
838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*
a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))) + 12*sqrt(2)*(40*a*d^8*e^2 - 512*a^2*d^4*e
^5 - 131072*a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4*e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2 - 64*a*d^5*e^5 - 16384*a
^2*d*e^8)*x^3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*
e^8 - (125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 5
1539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36
+ 1800000*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 780825
05441280*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392
*a^8*d^4*e^24 - 73786976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136
*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))*log(884736*a*d^5
*e^6 + 226492416*a^2*d*e^9 + 3538944*(a*d^4*e^7 + 256*a^2*e^10)*x + 13824*sqrt(2)*(d^16*e^2 - 128*a*d^12*e^5 -
61440*a^2*d^8*e^8 + 8388608*a^3*d^4*e^11 - 268435456*a^4*e^14 + (125*d^30 + 59200*a*d^26*e^3 - 3624960*a^2*d^
22*e^6 - 566493184*a^3*d^18*e^9 + 19797114880*a^4*d^14*e^12 + 1906965479424*a^5*d^10*e^15 - 30786325577728*a^6
*d^6*e^18 - 2251799813685248*a^7*d^2*e^21)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 18000
00*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 7808250544128
0*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^
4*e^24 - 73786976294838206464*a^9*e^27)))*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 - (125*d^24 - 480
0*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^
15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d^32*e^3
- 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*d^16*e^1
5 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24 - 73786
976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 38252
05248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))) - 12*sqrt(2)*(40*a*d^8*e^2 - 512*a^2
*d^4*e^5 - 131072*a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4*e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2 - 64*a*d^5*e^5 - 1
6384*a^2*d*e^8)*x^3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^
2*d^2*e^8 - (125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^
12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625
*d^36 + 1800000*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 +
78082505441280*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730
811392*a^8*d^4*e^24 - 73786976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31
195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))*log(884736
*a*d^5*e^6 + 226492416*a^2*d*e^9 + 3538944*(a*d^4*e^7 + 256*a^2*e^10)*x - 13824*sqrt(2)*(d^16*e^2 - 128*a*d^12
*e^5 - 61440*a^2*d^8*e^8 + 8388608*a^3*d^4*e^11 - 268435456*a^4*e^14 + (125*d^30 + 59200*a*d^26*e^3 - 3624960*
a^2*d^22*e^6 - 566493184*a^3*d^18*e^9 + 19797114880*a^4*d^14*e^12 + 1906965479424*a^5*d^10*e^15 - 307863255777
28*a^6*d^6*e^18 - 2251799813685248*a^7*d^2*e^21)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 +
1800000*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 7808250
5441280*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*
a^8*d^4*e^24 - 73786976294838206464*a^9*e^27)))*sqrt((d^10*e^2 + 160*a*d^6*e^5 + 40960*a^2*d^2*e^8 - (125*d^24
- 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607552*a^5*
d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^10)/(15625*d^36 + 1800000*a*d^3
2*e^3 - 115200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082505441280*a^5*d^
16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^21 - 5188146770730811392*a^8*d^4*e^24 -
73786976294838206464*a^9*e^27)))/(125*d^24 - 4800*a*d^20*e^3 - 1167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 +
3825205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))) - 8*(d^4*e - 64*a*e^4)*x)/(40*
a*d^8*e^2 - 512*a^2*d^4*e^5 - 131072*a^3*e^8 + 8*(5*d^8*e^3 - 64*a*d^4*e^6 - 16384*a^2*e^9)*x^4 + 8*(5*d^9*e^2
- 64*a*d^5*e^5 - 16384*a^2*d*e^8)*x^3 - (5*d^11 - 64*a*d^7*e^3 - 16384*a^2*d^3*e^6)*x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.02, size = 288, normalized size = 0.84 \[ \frac {384 e^{2} \left (2 \RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )^{2} d^{2} e +\RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right ) d^{3}+32 a \,e^{2}\right ) \ln \left (-\RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )+x \right )}{\left (2048 a \,e^{3}+40 d^{4}\right ) \left (64 a \,e^{3}-d^{4}\right ) \left (32 \RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )^{3} e^{3}+24 \RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 e^{2} d \,\textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )^{2} d \,e^{2}-d^{3}\right )}+\frac {\frac {12 d^{2} e^{3} x^{3}}{\left (256 a \,e^{3}+5 d^{4}\right ) \left (64 a \,e^{3}-d^{4}\right )}+\frac {9 d^{3} e^{2} x^{2}}{\left (256 a \,e^{3}+5 d^{4}\right ) \left (64 a \,e^{3}-d^{4}\right )}+\frac {e x}{256 a \,e^{3}+5 d^{4}}+\frac {\left (128 a \,e^{3}-5 d^{4}\right ) d}{131072 a^{2} e^{6}+512 a \,d^{4} e^{3}-40 d^{8}}}{e^{3} x^{4}+d \,e^{2} x^{3}-\frac {1}{8} d^{3} x +a \,e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x)

[Out]

(12*d^2*e^3/(256*a*e^3+5*d^4)/(64*a*e^3-d^4)*x^3+9*d^3*e^2/(256*a*e^3+5*d^4)/(64*a*e^3-d^4)*x^2+e/(256*a*e^3+5
*d^4)*x+1/8*d*(128*a*e^3-5*d^4)/(16384*a^2*e^6+64*a*d^4*e^3-5*d^8))/(e^3*x^4+d*e^2*x^3-1/8*d^3*x+a*e^2)+384*e^
2/(2048*a*e^3+40*d^4)/(64*a*e^3-d^4)*sum((2*_R^2*d^2*e+_R*d^3+32*a*e^2)/(32*_R^3*e^3+24*_R^2*d*e^2-d^3)*ln(-_R
+x),_R=RootOf(8*_Z^4*e^3+8*_Z^3*d*e^2-_Z*d^3+8*a*e^2))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 7.11, size = 10351, normalized size = 30.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^2,x)

[Out]

((8*e*x)/(256*a*e^3 + 5*d^4) - (5*d^5 - 128*a*d*e^3)/((64*a*e^3 - d^4)*(256*a*e^3 + 5*d^4)) + (72*d^3*e^2*x^2)
/((64*a*e^3 - d^4)*(256*a*e^3 + 5*d^4)) + (96*d^2*e^3*x^3)/((64*a*e^3 - d^4)*(256*a*e^3 + 5*d^4)))/(8*a*e^2 -
d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3) + atan((((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5
+ 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*
(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e
^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*
e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*(((1536*(68719476736*a^5*e^24 + 2
0*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^
20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e
^12) - ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d
^11*e^20 - 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^
16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*
d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24)
)/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*((288*(d^22*e^2
+ d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440
*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 115292150460684
6976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 -
96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a
^8*d^4*e^24))^(1/2))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^
8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9
)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d
^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 7916483719987
20*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^
3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 123
20768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d
^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*1i + ((288*(d^22*e^2 +
d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a
^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 11529215046068469
76*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 9
6636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8
*d^4*e^24))^(1/2)*((1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^2
0 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^
12) - ((1536*(68719476736*a^5*e^24 + 20*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e
^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 1232
0768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19
922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20 - 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d
^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*
e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^
4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 +
2097152*a^3*d^4*e^9))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e
^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^
9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*
d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998
720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/
2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2
*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 +
1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 443240
62494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (6144*x*(7864
32*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*
e^6 + 2097152*a^3*d^4*e^9))*1i)/(((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528
*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*
e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163
577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 7
91648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*(((1536*(68719476736*a^5*e^24 + 20*d^20*e
^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 171
79869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - (
(1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20
- 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 -
118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^1
2 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d^
16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*((288*(d^22*e^2 + d^4*e^
2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6
*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9
*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 9663676
4160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e
^24))^(1/2))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160
384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2))
)/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9
- 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d
^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^3*d*e^19
+ 196608*a^2*d^5*e^16))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^
3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d^16 + 26
8435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9)) - ((288*(d^22*e^2 + d^4*e^2*(-(
64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14
- 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27
- 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*
a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))
^(1/2)*((1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 1717986
9184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - ((153
6*(68719476736*a^5*e^24 + 20*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e^18 - 21474
83648*a^4*d^4*e^21))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^
8*e^9 + 134217728*a^4*d^4*e^12) + ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*
d^15*e^17 - 654311424*a^4*d^11*e^20 - 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 17179
869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (61
44*x*(25*d^22*e^9 - 2240*a*d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21
- 17179869184*a^5*d^2*e^24))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3
*d^4*e^9))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 616038
4*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/
(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 -
15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8
*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - 32*a*d
^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^17 + 256
*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2
*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^
6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (6144*x*(786432*a^2*e^17
+ 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 20971
52*a^3*d^4*e^9)) + (113246208*a*d^2*e^14)/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*
e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12)))*((288*(d^22*e^2 + d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) -
32*a*d^18*e^5 + 22528*a^2*d^14*e^8 - 6160384*a^3*d^10*e^11 + 461373440*a^4*d^6*e^14 - 10737418240*a^5*d^2*e^1
7 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290
240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 4432406249
4720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*2i + atan(((-(288*(
d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 -
461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 115292
1504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^
20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 4053239664
6334464*a^8*d^4*e^24))^(1/2)*(((1536*(68719476736*a^5*e^24 + 20*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*
e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 11
8784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 +
24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20 - 25769803776*a^5*d^7*e^23 + 1099511627
776*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*
e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*
d^10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3
- 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*(-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^
18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*
a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*
d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6
*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2))*(-(288*(d^4*e^2*(-(64*a*e^
3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*
e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*
e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764
160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^
24))^(1/2) + (1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 17
179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) +
(6144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 15
9744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*1i + (-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^
18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*
a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*
d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6
*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*((1536*(96*d^13*e^10 + 3072
*a*d^9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 -
118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - ((1536*(68719476736*a^5*e^24 + 20*d^20*
e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 17
179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) +
((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^2
0 - 25769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3
- 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^
12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d
^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*(-(288*(d^4*e^2*(-(64*a
*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d
^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a
^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636
764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4
*e^24))^(1/2))*(-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6
160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/
2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e
^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^
7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (6144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14
))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*1i)/((-(288*(d^
4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 46
1373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 11529215
04606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20
*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 405323966463
34464*a^8*d^4*e^24))^(1/2)*(((1536*(68719476736*a^5*e^24 + 20*d^20*e^9 - 7936*a*d^16*e^12 + 770048*a^2*d^12*e^
15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 1187
84*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - ((1536*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24
576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20 - 25769803776*a^5*d^7*e^23 + 109951162777
6*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^
9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^12 - 118784*a^2*d^14*e^15 + 12320768*a^3*d^
10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 -
159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*(-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18
*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*
e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^
28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d
^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2))*(-(288*(d^4*e^2*(-(64*a*e^3
- d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^
14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^
27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 9663676416
0*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24
))^(1/2) + (1536*(96*d^13*e^10 + 3072*a*d^9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 1717
9869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6
144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(25*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 1597
44*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9)) - (-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^
5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5
*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*
e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12
*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*((1536*(96*d^13*e^10 + 3072*a*d^
9*e^13 - 50331648*a^3*d*e^19 + 196608*a^2*d^5*e^16))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 11878
4*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) - ((1536*(68719476736*a^5*e^24 + 20*d^20*e^9 -
7936*a*d^16*e^12 + 770048*a^2*d^12*e^15 - 5242880*a^3*d^8*e^18 - 2147483648*a^4*d^4*e^21))/(25*d^20 - 1717986
9184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + ((153
6*(25*d^27*e^8 - 3840*a*d^23*e^11 + 24576*a^2*d^19*e^14 + 19922944*a^3*d^15*e^17 - 654311424*a^4*d^11*e^20 - 2
5769803776*a^5*d^7*e^23 + 1099511627776*a^6*d^3*e^26))/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118
784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 134217728*a^4*d^4*e^12) + (6144*x*(25*d^22*e^9 - 2240*a*d^18*e^12 -
118784*a^2*d^14*e^15 + 12320768*a^3*d^10*e^18 + 134217728*a^4*d^6*e^21 - 17179869184*a^5*d^2*e^24))/(25*d^16 +
268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9))*(-(288*(d^4*e^2*(-(64*a*e^3
- d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^
14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^
27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 - 15250489344*a^4*d^20*e^12 - 9663676416
0*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24
))^(1/2))*(-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8 + 616038
4*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^(1/2)))/
(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^24*e^9 -
15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720*a^7*d^8
*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2) + (6144*x*(786432*a^2*e^17 + 96*d^8*e^11 + 9216*a*d^4*e^14))/(2
5*d^16 + 268435456*a^4*e^12 - 640*a*d^12*e^3 - 159744*a^2*d^8*e^6 + 2097152*a^3*d^4*e^9)) + (113246208*a*d^2*e
^14)/(25*d^20 - 17179869184*a^5*e^15 - 2240*a*d^16*e^3 - 118784*a^2*d^12*e^6 + 12320768*a^3*d^8*e^9 + 13421772
8*a^4*d^4*e^12)))*(-(288*(d^4*e^2*(-(64*a*e^3 - d^4)^9)^(1/2) - d^22*e^2 + 32*a*d^18*e^5 - 22528*a^2*d^14*e^8
+ 6160384*a^3*d^10*e^11 - 461373440*a^4*d^6*e^14 + 10737418240*a^5*d^2*e^17 + 256*a*e^5*(-(64*a*e^3 - d^4)^9)^
(1/2)))/(125*d^36 + 1152921504606846976*a^9*e^27 - 28800*a*d^32*e^3 + 1290240*a^2*d^28*e^6 + 163577856*a^3*d^2
4*e^9 - 15250489344*a^4*d^20*e^12 - 96636764160*a^5*d^16*e^15 + 44324062494720*a^6*d^12*e^18 - 791648371998720
*a^7*d^8*e^21 - 40532396646334464*a^8*d^4*e^24))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**2,x)

[Out]

Timed out

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