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

[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]])

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Rubi [A]  time = 0.532286, 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, 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 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 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 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]

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]

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*}

Mathematica [C]  time = 0.17972, size = 234, normalized size = 0.68 \[ \frac{48 e^2 \text{RootSum}\left [8 \text{$\#$1}^3 d e^2+8 \text{$\#$1}^4 e^3-\text{$\#$1} d^3+8 a e^2\& ,\frac{2 \text{$\#$1}^2 d^2 e \log (x-\text{$\#$1})+32 a e^2 \log (x-\text{$\#$1})+\text{$\#$1} d^3 \log (x-\text{$\#$1})}{24 \text{$\#$1}^2 d e^2+32 \text{$\#$1}^3 e^3-d^3}\& \right ]}{16384 a^2 e^6+64 a d^4 e^3-5 d^8}+\frac{(d+4 e x) \left (-128 a e^3-24 d^2 e^2 x^2-12 d^3 e x+5 d^4\right )}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right ) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )} \]

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)

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Maple [C]  time = 0.015, size = 288, normalized size = 0.8 \begin{align*}{ \left ( 12\,{\frac{{d}^{2}{e}^{3}{x}^{3}}{ \left ( 256\,a{e}^{3}+5\,{d}^{4} \right ) \left ( 64\,a{e}^{3}-{d}^{4} \right ) }}+9\,{\frac{{d}^{3}{e}^{2}{x}^{2}}{ \left ( 256\,a{e}^{3}+5\,{d}^{4} \right ) \left ( 64\,a{e}^{3}-{d}^{4} \right ) }}+{\frac{ex}{256\,a{e}^{3}+5\,{d}^{4}}}+{\frac{d \left ( 128\,a{e}^{3}-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}}+384\,{\frac{{e}^{2}}{ \left ( 2048\,a{e}^{3}+40\,{d}^{4} \right ) \left ( 64\,a{e}^{3}-{d}^{4} \right ) }\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\,{{\it \_R}}^{2}{d}^{2}e+{\it \_R}\,{d}^{3}+32\,a{e}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}{e}^{3}+24\,{{\it \_R}}^{2}d{e}^{2}-{d}^{3}}}} \end{align*}

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(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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

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]

Exception raised: AttributeError

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Fricas [B]  time = 2.36008, size = 12891, normalized size = 37.69 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 14.0486, size = 580, normalized size = 1.7 \begin{align*} \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 )} \end{align*}

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*(131072*a**2*e**9 + 512*a*d**4*e**6 - 40*d**8*e**3) + x**3*(13107
2*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**11)) + Root
Sum(_t**4*(1152921504606846976*a**9*e**27 - 40532396646334464*a**8*d**4*e**24 - 791648371998720*a**7*d**8*e**2
1 + 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) + _t**2*(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) + 84934656*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 + 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))))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError