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

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*(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*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]])

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Rubi [A]  time = 1.43676, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(d+4 e x) \left (-256 a e^3+13 d^4-3 d^2 (d+4 e x)^2\right )}{2 \left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}-\frac{24 e \left (-d^2 \sqrt{d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}+\frac{24 e \left (d^2 \sqrt{d^4-64 a e^3}+128 a e^3+d^4\right ) \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}}\right )}{\left (d^4-64 a e^3\right )^{3/2} \left (256 a e^3+5 d^4\right ) \sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}} \]

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]

((d + 4*e*x)*(13*d^4 - 256*a*e^3 - 3*d^2*(d + 4*e*x)^2))/(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 + 1
28*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*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]])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [C]  time = 0.281656, 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*Roo
tSum[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)

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Maple [C]  time = 0.051, size = 288, normalized size = 0.8 \[{1 \left ( 12\,{\frac{{d}^{2}{e}^{3}{x}^{3}}{ \left ( 256\,{e}^{3}a+5\,{d}^{4} \right ) \left ( 64\,{e}^{3}a-{d}^{4} \right ) }}+9\,{\frac{{d}^{3}{e}^{2}{x}^{2}}{ \left ( 256\,{e}^{3}a+5\,{d}^{4} \right ) \left ( 64\,{e}^{3}a-{d}^{4} \right ) }}+{\frac{ex}{256\,{e}^{3}a+5\,{d}^{4}}}+{\frac{d \left ( 128\,{e}^{3}a-5\,{d}^{4} \right ) }{131072\,{a}^{2}{e}^{6}+512\,a{d}^{4}{e}^{3}-40\,{d}^{8}}} \right ) \left ({e}^{3}{x}^{4}+d{e}^{2}{x}^{3}-{\frac{{d}^{3}x}{8}}+a{e}^{2} \right ) ^{-1}}+48\,{e}^{2}\sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}{e}^{3}+8\,{{\it \_Z}}^{3}d{e}^{2}-{\it \_Z}\,{d}^{3}+8\,a{e}^{2} \right ) }{\frac{ \left ( 2\,{d}^{2}e{{\it \_R}}^{2}+{d}^{3}{\it \_R}+32\,a{e}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ( 256\,{e}^{3}a+5\,{d}^{4} \right ) \left ( 64\,{e}^{3}a-{d}^{4} \right ) \left ( 32\,{{\it \_R}}^{3}{e}^{3}+24\,{{\it \_R}}^{2}d{e}^{2}-{d}^{3} \right ) }} \]

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)+48*e^2*sum((2*_R^2*d^2*e+_
R*d^3+32*a*e^2)/(256*a*e^3+5*d^4)/(64*a*e^3-d^4)/(32*_R^3*e^3+24*_R^2*d*e^2-d^3)
*ln(x-_R),_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]  time = 0., size = 0, normalized size = 0. \[ -\frac{48 \, e^{2} \int \frac{2 \, d^{2} e x^{2} + d^{3} x + 32 \, a e^{2}}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}} - \frac{96 \, d^{2} e^{3} x^{3} + 72 \, d^{3} e^{2} x^{2} - 5 \, d^{5} + 128 \, a d e^{3} - 8 \,{\left (d^{4} e - 64 \, a e^{4}\right )} x}{40 \, a d^{8} e^{2} - 512 \, a^{2} d^{4} e^{5} - 131072 \, a^{3} e^{8} + 8 \,{\left (5 \, d^{8} e^{3} - 64 \, a d^{4} e^{6} - 16384 \, a^{2} e^{9}\right )} x^{4} + 8 \,{\left (5 \, d^{9} e^{2} - 64 \, a d^{5} e^{5} - 16384 \, a^{2} d e^{8}\right )} x^{3} -{\left (5 \, d^{11} - 64 \, a d^{7} e^{3} - 16384 \, a^{2} d^{3} e^{6}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-48*e^2*integrate((2*d^2*e*x^2 + d^3*x + 32*a*e^2)/(8*e^3*x^4 + 8*d*e^2*x^3 - d^
3*x + 8*a*e^2), x)/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6) - (96*d^2*e^3*x^3 + 72
*d^3*e^2*x^2 - 5*d^5 + 128*a*d*e^3 - 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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Fricas [A]  time = 0.41681, size = 5785, normalized size = 16.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)*sq
rt((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 - 70931694131085
312*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 - 115200
000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78082
505441280*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 - 5153
9607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 6
5536*a^2*e^10)/(15625*d^36 + 1800000*a*d^32*e^3 - 115200000*a^2*d^28*e^6 - 21135
360000*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 - 518814677073
0811392*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 - 5
1539607552*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 + 382
5205248*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 - 115
200000*a^2*d^28*e^6 - 21135360000*a^3*d^24*e^9 - 150994944000*a^4*d^20*e^12 + 78
082505441280*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))*lo
g(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 + 780825054
41280*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 - 1
167360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 51539607
552*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 - 211353600
00*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 - 5188146770730811
392*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 - 51539
607552*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 + 3825205
248*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 - 1152000
00*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 + 3825
205248*a^4*d^8*e^12 - 51539607552*a^5*d^4*e^15 - 4398046511104*a^6*e^18))*log(88
4736*a*d^5*e^6 + 226492416*a^2*d*e^9 + 3538944*(a*d^4*e^7 + 256*a^2*e^10)*x + 13
824*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^1
1 - 268435456*a^4*e^14 + (125*d^30 + 59200*a*d^26*e^3 - 3624960*a^2*d^22*e^6 - 5
66493184*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 + 5
12*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 + 7808250544128
0*a^5*d^16*e^15 + 2744381022928896*a^6*d^12*e^18 - 70931694131085312*a^7*d^8*e^2
1 - 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 - 11673
60*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 + 274438
1022928896*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 - 11
67360*a^2*d^16*e^6 + 31195136*a^3*d^12*e^9 + 3825205248*a^4*d^8*e^12 - 515396075
52*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 - 163
84*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^2
4 - 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 + 7808250544
1280*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 + 38252052
48*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 - 56649
3184*a^3*d^18*e^9 + 19797114880*a^4*d^14*e^12 + 1906965479424*a^5*d^10*e^15 - 30
786325577728*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 - 4398046511104*a^6*e^18)*sqrt((d^8*e^4 + 512*a*d^4*e^7 + 65536*a^2*e^1
0)/(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 + 2744381022
928896*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 - 116736
0*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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Sympy [A]  time = 44.8092, size = 580, normalized size = 1.7 \[ \frac{128 a d e^{3} - 5 d^{5} + 72 d^{3} e^{2} x^{2} + 96 d^{2} e^{3} x^{3} + x \left (512 a e^{4} - 8 d^{4} e\right )}{131072 a^{3} e^{8} + 512 a^{2} d^{4} e^{5} - 40 a d^{8} e^{2} + x^{4} \left (131072 a^{2} e^{9} + 512 a d^{4} e^{6} - 40 d^{8} e^{3}\right ) + x^{3} \left (131072 a^{2} d e^{8} + 512 a d^{5} e^{5} - 40 d^{9} e^{2}\right ) + x \left (- 16384 a^{2} d^{3} e^{6} - 64 a d^{7} e^{3} + 5 d^{11}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1152921504606846976 a^{9} e^{27} - 40532396646334464 a^{8} d^{4} e^{24} - 791648371998720 a^{7} d^{8} e^{21} + 44324062494720 a^{6} d^{12} e^{18} - 96636764160 a^{5} d^{16} e^{15} - 15250489344 a^{4} d^{20} e^{12} + 163577856 a^{3} d^{24} e^{9} + 1290240 a^{2} d^{28} e^{6} - 28800 a d^{32} e^{3} + 125 d^{36}\right ) + t^{2} \left (6184752906240 a^{5} d^{2} e^{17} - 265751101440 a^{4} d^{6} e^{14} + 3548381184 a^{3} d^{10} e^{11} - 12976128 a^{2} d^{14} e^{8} + 18432 a d^{18} e^{5} - 576 d^{22} e^{2}\right ) + 84934656 a^{2} e^{10}, \left ( t \mapsto t \log{\left (x + \frac{- 2251799813685248 t^{3} a^{7} d^{2} e^{21} - 30786325577728 t^{3} a^{6} d^{6} e^{18} + 1906965479424 t^{3} a^{5} d^{10} e^{15} + 19797114880 t^{3} a^{4} d^{14} e^{12} - 566493184 t^{3} a^{3} d^{18} e^{9} - 3624960 t^{3} a^{2} d^{22} e^{6} + 59200 t^{3} a d^{26} e^{3} + 125 t^{3} d^{30} + 77309411328 t a^{4} e^{14} - 8455716864 t a^{3} d^{4} e^{11} - 17694720 t a^{2} d^{8} e^{8} - 156672 t a d^{12} e^{5} - 576 t d^{16} e^{2} + 56623104 a^{2} d e^{9} + 221184 a d^{5} e^{6}}{226492416 a^{2} e^{10} + 884736 a d^{4} e^{7}} \right )} \right )\right )} \]

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]

(128*a*d*e**3 - 5*d**5 + 72*d**3*e**2*x**2 + 96*d**2*e**3*x**3 + x*(512*a*e**4 -
8*d**4*e))/(131072*a**3*e**8 + 512*a**2*d**4*e**5 - 40*a*d**8*e**2 + x**4*(1310
72*a**2*e**9 + 512*a*d**4*e**6 - 40*d**8*e**3) + x**3*(131072*a**2*d*e**8 + 512*
a*d**5*e**5 - 40*d**9*e**2) + x*(-16384*a**2*d**3*e**6 - 64*a*d**7*e**3 + 5*d**1
1)) + RootSum(_t**4*(1152921504606846976*a**9*e**27 - 40532396646334464*a**8*d**
4*e**24 - 791648371998720*a**7*d**8*e**21 + 44324062494720*a**6*d**12*e**18 - 96
636764160*a**5*d**16*e**15 - 15250489344*a**4*d**20*e**12 + 163577856*a**3*d**24
*e**9 + 1290240*a**2*d**28*e**6 - 28800*a*d**32*e**3 + 125*d**36) + _t**2*(61847
52906240*a**5*d**2*e**17 - 265751101440*a**4*d**6*e**14 + 3548381184*a**3*d**10*
e**11 - 12976128*a**2*d**14*e**8 + 18432*a*d**18*e**5 - 576*d**22*e**2) + 849346
56*a**2*e**10, Lambda(_t, _t*log(x + (-2251799813685248*_t**3*a**7*d**2*e**21 -
30786325577728*_t**3*a**6*d**6*e**18 + 1906965479424*_t**3*a**5*d**10*e**15 + 19
797114880*_t**3*a**4*d**14*e**12 - 566493184*_t**3*a**3*d**18*e**9 - 3624960*_t*
*3*a**2*d**22*e**6 + 59200*_t**3*a*d**26*e**3 + 125*_t**3*d**30 + 77309411328*_t
*a**4*e**14 - 8455716864*_t*a**3*d**4*e**11 - 17694720*_t*a**2*d**8*e**8 - 15667
2*_t*a*d**12*e**5 - 576*_t*d**16*e**2 + 56623104*a**2*d*e**9 + 221184*a*d**5*e**
6)/(226492416*a**2*e**10 + 884736*a*d**4*e**7))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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