3.627 \(\int \frac {1}{(d+e x)^2 (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\)

Optimal. Leaf size=348 \[ -\frac {3 b^2-10 a c}{2 a^2 e \left (b^2-4 a c\right ) (d+e x)}-\frac {\sqrt {c} \left (\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 e \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^2 e \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

[Out]

1/2*(10*a*c-3*b^2)/a^2/(-4*a*c+b^2)/e/(e*x+d)+1/2*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/(e*x+d)/(a+b*(e*x
+d)^2+c*(e*x+d)^4)-1/4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3-16*a*b*c+(-
10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)/e*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*arctan((e*
x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3-16*a*b*c-(-10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))
/a^2/(-4*a*c+b^2)^(3/2)/e*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]  time = 1.65, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1142, 1121, 1281, 1166, 205} \[ -\frac {3 b^2-10 a c}{2 a^2 e \left (b^2-4 a c\right ) (d+e x)}-\frac {\sqrt {c} \left (\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 e \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^2 e \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*b^2 - 10*a*c)/(2*a^2*(b^2 - 4*a*c)*e*(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*(d
+ e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (Sqrt[c]*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*c)*Sqrt[b^2 - 4*a*c])
*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b -
Sqrt[b^2 - 4*a*c]]*e) + (Sqrt[c]*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

Rule 205

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

Rule 1121

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[((d*x)^(m + 1)*(b^2 - 2*a
*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*d*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*
c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7)*
x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integer
Q[p] || IntegerQ[m])

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

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 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*(
f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 b^2+10 a c-3 b c x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{2 a \left (b^2-4 a c\right ) e}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {-b \left (3 b^2-13 a c\right )-c \left (3 b^2-10 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{2 a^2 \left (b^2-4 a c\right ) e}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (c \left (3 b^2-10 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {16 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^2 \left (b^2-4 a c\right ) e}-\frac {\left (c \left (3 b^2-10 a c-\frac {3 b^3}{\sqrt {b^2-4 a c}}+\frac {16 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{4 a^2 \left (b^2-4 a c\right ) e}\\ &=-\frac {3 b^2-10 a c}{2 a^2 \left (b^2-4 a c\right ) e (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\sqrt {c} \left (3 b^2-10 a c+\frac {3 b^3}{\sqrt {b^2-4 a c}}-\frac {16 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (3 b^2-10 a c-\frac {3 b^3}{\sqrt {b^2-4 a c}}+\frac {16 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

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Mathematica [A]  time = 1.63, size = 339, normalized size = 0.97 \[ \frac {\frac {2 (d+e x) \left (-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right )}{\left (4 a c-b^2\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}+16 a b c-3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {4}{d+e x}}{4 a^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4/(d + e*x) + (2*(d + e*x)*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b
*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b
^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sq
rt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*A
rcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c
]]))/(4*a^2*e)

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fricas [B]  time = 1.18, size = 4330, normalized size = 12.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*(3*b^2*c - 10*a*c^2)*e^4*x^4 + 8*(3*b^2*c - 10*a*c^2)*d*e^3*x^3 + 2*(3*b^2*c - 10*a*c^2)*d^4 + 2*(3*b^
3 - 11*a*b*c + 6*(3*b^2*c - 10*a*c^2)*d^2)*e^2*x^2 + 4*a*b^2 - 16*a^2*c + 2*(3*b^3 - 11*a*b*c)*d^2 + 4*(2*(3*b
^2*c - 10*a*c^2)*d^3 + (3*b^3 - 11*a*b*c)*d)*e*x + sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*e^6*x^5 + 5*(a^2*b^2*c -
4*a^3*c^2)*d*e^5*x^4 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^4*x^3 + (10*(a^2*b^2*c - 4*a^
3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*e^3*x^2 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*
b^3 - 4*a^3*b*c)*d^2)*e^2*x + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d
)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 -
64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 -
12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*
e^2))*log(-(189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*e*x - (189*b^6*c^3 - 1971*a*b^4*c^
4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*d + 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^
8*b^4*c^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*e^3*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2
*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)) - (27*b^11 - 486*a*b^9*c
+ 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)*e)*sqrt(-(9*b^7 - 105*a*b^5*c +
385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2*sqrt((81*b^8 - 91
8*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 -
64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))) - sqrt(1/2)*((a^2*b^2*c - 4
*a^3*c^2)*e^6*x^5 + 5*(a^2*b^2*c - 4*a^3*c^2)*d*e^5*x^4 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^
2)*e^4*x^3 + (10*(a^2*b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*e^3*x^2 + (a^3*b^2 - 4*a^4*c + 5*(a^
2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)*d^2)*e^2*x + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^
3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6
- 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^
2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4
*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))*log(-(189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*
e*x - (189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*d - 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6
*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*e^3*sqrt((81*b^8 - 918*a*b^6*c
+ 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c
^3)*e^4)) - (27*b^11 - 486*a*b^9*c + 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5
)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 + (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 -
64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 -
12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*
e^2))) - sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*e^6*x^5 + 5*(a^2*b^2*c - 4*a^3*c^2)*d*e^5*x^4 + (a^2*b^3 - 4*a^3*b
*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^4*x^3 + (10*(a^2*b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*e^
3*x^2 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)*d^2)*e^2*x + ((a^2*b^2*c
- 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*
b^3*c^2 - 420*a^3*b*c^3 - (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*
c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*
c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))*log(-(189*b^6*c^3 - 1971*a*b^4*c^4 +
5625*a^2*b^2*c^5 - 2500*a^3*c^6)*e*x - (189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*d + 1
/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c^3 + 2176*a^9*b^2*c^4 - 1280*a^10*c
^5)*e^3*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b
^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)) + (27*b^11 - 486*a*b^9*c + 3330*a^2*b^7*c^2 - 10549*a^3*b^5*c^3 +
14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 - (a^5*b^6
- 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^
2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4
*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))) + sqrt(1/2)*((a^2*b^2*c - 4*a^3*c^2)*e^6*x^5 + 5*(a^2*b^2*c - 4*a^3*c
^2)*d*e^5*x^4 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^4*x^3 + (10*(a^2*b^2*c - 4*a^3*c^2)*d
^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*e^3*x^2 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2)*d^4 + 3*(a^2*b^3 - 4*
a^3*b*c)*d^2)*e^2*x + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b^2 - 4*a^4*c)*d)*e)*sqr
t(-(9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3 - (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*
c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*
b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))*lo
g(-(189*b^6*c^3 - 1971*a*b^4*c^4 + 5625*a^2*b^2*c^5 - 2500*a^3*c^6)*e*x - (189*b^6*c^3 - 1971*a*b^4*c^4 + 5625
*a^2*b^2*c^5 - 2500*a^3*c^6)*d - 1/2*sqrt(1/2)*((3*a^5*b^10 - 55*a^6*b^8*c + 392*a^7*b^6*c^2 - 1344*a^8*b^4*c^
3 + 2176*a^9*b^2*c^4 - 1280*a^10*c^5)*e^3*sqrt((81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 6
25*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)*e^4)) + (27*b^11 - 486*a*b^9*c + 3330*
a^2*b^7*c^2 - 10549*a^3*b^5*c^3 + 14408*a^4*b^3*c^4 - 5200*a^5*b*c^5)*e)*sqrt(-(9*b^7 - 105*a*b^5*c + 385*a^2*
b^3*c^2 - 420*a^3*b*c^3 - (a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2*sqrt((81*b^8 - 918*a*b^6*
c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)/((a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*
c^3)*e^4)))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3)*e^2))))/((a^2*b^2*c - 4*a^3*c^2)*e^6*x^5 +
5*(a^2*b^2*c - 4*a^3*c^2)*d*e^5*x^4 + (a^2*b^3 - 4*a^3*b*c + 10*(a^2*b^2*c - 4*a^3*c^2)*d^2)*e^4*x^3 + (10*(a
^2*b^2*c - 4*a^3*c^2)*d^3 + 3*(a^2*b^3 - 4*a^3*b*c)*d)*e^3*x^2 + (a^3*b^2 - 4*a^4*c + 5*(a^2*b^2*c - 4*a^3*c^2
)*d^4 + 3*(a^2*b^3 - 4*a^3*b*c)*d^2)*e^2*x + ((a^2*b^2*c - 4*a^3*c^2)*d^5 + (a^2*b^3 - 4*a^3*b*c)*d^3 + (a^3*b
^2 - 4*a^4*c)*d)*e)

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giac [B]  time = 0.79, size = 847, normalized size = 2.43 \[ \frac {{\left (2 \, {\left (3 \, a^{3} b^{2} c - 10 \, a^{4} c^{2}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} {\left | a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2} \right |} e^{2} - {\left (a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2}\right )}^{2} {\left (3 \, b^{3} - 13 \, a b c\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} + {\left (3 \, a^{4} b^{7} - 31 \, a^{5} b^{5} c + 96 \, a^{6} b^{3} c^{2} - 80 \, a^{7} b c^{3}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} e^{4}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (x e + d\right )} \sqrt {\frac {a^{2} b^{3} e^{2} - 4 \, a^{3} b c e^{2} + \sqrt {{\left (a^{2} b^{3} e^{2} - 4 \, a^{3} b c e^{2}\right )}^{2} - 4 \, {\left (a^{3} b^{2} e^{4} - 4 \, a^{4} c e^{4}\right )} {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )}}}{a^{3} b^{2} e^{4} - 4 \, a^{4} c e^{4}}}}\right ) e^{\left (-3\right )}}{16 \, {\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} {\left | a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2} \right |} {\left | a \right |}} + \frac {{\left (2 \, {\left (3 \, a^{3} b^{2} c - 10 \, a^{4} c^{2}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} {\left | a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2} \right |} e^{2} + {\left (a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2}\right )}^{2} {\left (3 \, b^{3} - 13 \, a b c\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} - {\left (3 \, a^{4} b^{7} - 31 \, a^{5} b^{5} c + 96 \, a^{6} b^{3} c^{2} - 80 \, a^{7} b c^{3}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} e^{4}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (x e + d\right )} \sqrt {\frac {a^{2} b^{3} e^{2} - 4 \, a^{3} b c e^{2} - \sqrt {{\left (a^{2} b^{3} e^{2} - 4 \, a^{3} b c e^{2}\right )}^{2} - 4 \, {\left (a^{3} b^{2} e^{4} - 4 \, a^{4} c e^{4}\right )} {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )}}}{a^{3} b^{2} e^{4} - 4 \, a^{4} c e^{4}}}}\right ) e^{\left (-3\right )}}{16 \, {\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} {\left | a^{2} b^{2} e^{2} - 4 \, a^{3} c e^{2} \right |} {\left | a \right |}} - \frac {\frac {b^{2} c e^{\left (-1\right )}}{x e + d} - \frac {2 \, a c^{2} e^{\left (-1\right )}}{x e + d} + \frac {b^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a b c e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}}}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c + \frac {b}{{\left (x e + d\right )}^{2}} + \frac {a}{{\left (x e + d\right )}^{4}}\right )}} - \frac {e^{\left (-1\right )}}{{\left (x e + d\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*(3*a^3*b^2*c - 10*a^4*c^2)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*abs(a^2*b^2*e^2 - 4*a
^3*c*e^2)*e^2 - (a^2*b^2*e^2 - 4*a^3*c*e^2)^2*(3*b^3 - 13*a*b*c)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a) + (3*a^4*
b^7 - 31*a^5*b^5*c + 96*a^6*b^3*c^2 - 80*a^7*b*c^3)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*e^4)*arctan(2*sqrt(1/2
)*e^(-1)/((x*e + d)*sqrt((a^2*b^3*e^2 - 4*a^3*b*c*e^2 + sqrt((a^2*b^3*e^2 - 4*a^3*b*c*e^2)^2 - 4*(a^3*b^2*e^4
- 4*a^4*c*e^4)*(a^2*b^2*c - 4*a^3*c^2)))/(a^3*b^2*e^4 - 4*a^4*c*e^4))))*e^(-3)/((a^5*b^2*c - 4*a^6*c^2)*sqrt(b
^2 - 4*a*c)*abs(a^2*b^2*e^2 - 4*a^3*c*e^2)*abs(a)) + 1/16*(2*(3*a^3*b^2*c - 10*a^4*c^2)*sqrt(2*a*b - 2*sqrt(b^
2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*abs(a^2*b^2*e^2 - 4*a^3*c*e^2)*e^2 + (a^2*b^2*e^2 - 4*a^3*c*e^2)^2*(3*b^3 - 13
*a*b*c)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a) - (3*a^4*b^7 - 31*a^5*b^5*c + 96*a^6*b^3*c^2 - 80*a^7*b*c^3)*sqrt(
2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*e^4)*arctan(2*sqrt(1/2)*e^(-1)/((x*e + d)*sqrt((a^2*b^3*e^2 - 4*a^3*b*c*e^2 - s
qrt((a^2*b^3*e^2 - 4*a^3*b*c*e^2)^2 - 4*(a^3*b^2*e^4 - 4*a^4*c*e^4)*(a^2*b^2*c - 4*a^3*c^2)))/(a^3*b^2*e^4 - 4
*a^4*c*e^4))))*e^(-3)/((a^5*b^2*c - 4*a^6*c^2)*sqrt(b^2 - 4*a*c)*abs(a^2*b^2*e^2 - 4*a^3*c*e^2)*abs(a)) - 1/2*
(b^2*c*e^(-1)/(x*e + d) - 2*a*c^2*e^(-1)/(x*e + d) + b^3*e^(-1)/(x*e + d)^3 - 3*a*b*c*e^(-1)/(x*e + d)^3)/((a^
2*b^2 - 4*a^3*c)*(c + b/(x*e + d)^2 + a/(x*e + d)^4)) - e^(-1)/((x*e + d)*a^2)

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maple [C]  time = 0.03, size = 1304, normalized size = 3.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*c^2*e^2/(4*a*c-b^
2)*x^3+1/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*c*e^2/(
4*a*c-b^2)*x^3*b^2-3/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)
*d*c^2*e/(4*a*c-b^2)*x^2+3/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*
x+b*d^2+a)*d*c*e/(4*a*c-b^2)*x^2*b^2-3/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+
2*b*d*e*x+b*d^2+a)/(4*a*c-b^2)*x*c^2*d^2+3/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^
2+c*d^4+2*b*d*e*x+b*d^2+a)/(4*a*c-b^2)*x*b^2*c*d^2-3/2/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+
b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)/(4*a*c-b^2)*x*b*c+1/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*
e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)/(4*a*c-b^2)*x*b^3-1/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*
e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*d^3/e/(4*a*c-b^2)*c^2+1/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+
4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*d^3/e/(4*a*c-b^2)*b^2*c-3/2/a/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*
e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*d/e/(4*a*c-b^2)*b*c+1/2/a^2/(c*e^4*x^4+4*c*d*e^3*x^3+6*
c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)*d/e/(4*a*c-b^2)*b^3-1/4/a^2/(4*a*c-b^2)/e*sum((c*
e^2*(10*a*c-3*b^2)*_R^2+2*c*d*e*(10*a*c-3*b^2)*_R+10*a*c^2*d^2-3*b^2*c*d^2+13*a*b*c-3*b^3)/(2*_R^3*c*e^3+6*_R^
2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(-_R+x),_R=RootOf(_Z^4*c*e^4+4*_Z^3*c*d*e^3+c*d^4+b*d^2+(6*c*d^2*
e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+a))-1/a^2/e/(e*x+d)

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

[Out]

Timed out

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mupad [B]  time = 6.38, size = 10556, normalized size = 30.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 +
30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a
*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8
*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c
^3*e^12 - 3264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12
- 458752*a^11*b^2*c^8*e^12) + (-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2
- 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^1
1*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7
*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 - 9*b^4
*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800
*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(
a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b
^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b
^11*c^3*e^14 + 61440*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^
3*c^7*e^14) + 1048576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^
9*c^4*d*e^13 - 327680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) - 851968
*a^14*b*c^8*e^12 - 192*a^8*b^13*c^2*e^12 + 4672*a^9*b^11*c^3*e^12 - 47360*a^10*b^9*c^4*e^12 + 256000*a^11*b^7*
c^5*e^12 - 778240*a^12*b^5*c^6*e^12 + 1261568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d
*e^11 - 3264*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*
d*e^11 - 458752*a^11*b^2*c^8*d*e^11)*1i + (-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*
a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2)
- 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^
2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204
800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c
^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 458752*a^11*b^2*c^8*e^12) + (-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) +
26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-
(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6
*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c
^5*e^2)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b
^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2
*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 -
1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 25
6*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^14 + 61440*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a
^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14) + 1048576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^
11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^13 - 327680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 157
2864*a^15*b^3*c^7*d*e^13) + 851968*a^14*b*c^8*e^12 + 192*a^8*b^13*c^2*e^12 - 4672*a^9*b^11*c^3*e^12 + 47360*a^
10*b^9*c^4*e^12 - 256000*a^11*b^7*c^5*e^12 + 778240*a^12*b^5*c^6*e^12 - 1261568*a^13*b^3*c^7*e^12) + 204800*a^
12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 3264*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6
*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11 - 458752*a^11*b^2*c^8*d*e^11)*1i)/((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^
9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25
*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 40
96*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144
*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3264*a^7*b^10*c^4*e^12 + 30112*a
^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 458752*a^11*b^2*c^8*e^12) + (-(9*b^13 -
9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 -
44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))
/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840
*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^
6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^
9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*
b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2
)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^14 + 61440*a^12*b^9*c^4*e^14 - 3
27680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14) + 1048576*a^16*b*c^8*d*e^13 +
256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^13 - 327680*a^13*b^7*c^5*d*e^13
+ 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) + 851968*a^14*b*c^8*e^12 + 192*a^8*b^13*c^2*e^12 -
4672*a^9*b^11*c^3*e^12 + 47360*a^10*b^9*c^4*e^12 - 256000*a^11*b^7*c^5*e^12 + 778240*a^12*b^5*c^6*e^12 - 1261
568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 3264*a^7*b^10*c^4*d*e^11 + 30112*a
^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11 - 458752*a^11*b^2*c^8*d*e^11) - (-(
9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b
^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9
)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^
2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3
264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 458752*a
^11*b^2*c^8*e^12) + (-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^
3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*
b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^
2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 - 9*b^4*(-(4*a*c -
b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^
5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^
2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2
- 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^1
4 + 61440*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14)
+ 1048576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^1
3 - 327680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) - 851968*a^14*b*c^8
*e^12 - 192*a^8*b^13*c^2*e^12 + 4672*a^9*b^11*c^3*e^12 - 47360*a^10*b^9*c^4*e^12 + 256000*a^11*b^7*c^5*e^12 -
778240*a^12*b^5*c^6*e^12 + 1261568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 326
4*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11 - 45
8752*a^11*b^2*c^8*d*e^11) + 128000*a^10*c^9*e^10 + 504*a^6*b^8*c^5*e^10 - 8112*a^7*b^6*c^6*e^10 + 48704*a^8*b^
4*c^7*e^10 - 129280*a^9*b^2*c^8*e^10))*(-(9*b^13 - 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2
*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 - 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 2
13*a*b^11*c + 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 +
240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*2i - atan(
((-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a
^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^
2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^
3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12
- 3264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 4587
52*a^11*b^2*c^8*e^12) + (-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 1065
6*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 5
1*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^
2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 + 9*b^4*(-(4*a
*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^
3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^1
2*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*
e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3
*e^14 + 61440*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e
^14) + 1048576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d
*e^13 - 327680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) - 851968*a^14*b
*c^8*e^12 - 192*a^8*b^13*c^2*e^12 + 4672*a^9*b^11*c^3*e^12 - 47360*a^10*b^9*c^4*e^12 + 256000*a^11*b^7*c^5*e^1
2 - 778240*a^12*b^5*c^6*e^12 + 1261568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 -
3264*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11
- 458752*a^11*b^2*c^8*d*e^11)*1i + (-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9
*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a
*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240
*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^1
2*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12
+ 365568*a^10*b^4*c^7*e^12 - 458752*a^11*b^2*c^8*e^12) + (-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a
^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c
- b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 -
24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)
))^(1/2)*((-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3
+ 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4
*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a
^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*
b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^14 + 61440*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5
*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14) + 1048576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11
*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^13 - 327680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^
15*b^3*c^7*d*e^13) + 851968*a^14*b*c^8*e^12 + 192*a^8*b^13*c^2*e^12 - 4672*a^9*b^11*c^3*e^12 + 47360*a^10*b^9*
c^4*e^12 - 256000*a^11*b^7*c^5*e^12 + 778240*a^12*b^5*c^6*e^12 - 1261568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*
d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 3264*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*
e^11 + 365568*a^10*b^4*c^7*d*e^11 - 458752*a^11*b^2*c^8*d*e^11)*1i)/((-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2
) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^
2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11
*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b
^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3264*a^7*b^10*c^4*e^12 + 30112*a^8*b^8*
c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 458752*a^11*b^2*c^8*e^12) + (-(9*b^13 + 9*b^4*
(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*
a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a
^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^
4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 207
7*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2
) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*
e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(1
048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^14 + 61440*a^12*b^9*c^4*e^14 - 327680*a
^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14) + 1048576*a^16*b*c^8*d*e^13 + 256*a^1
0*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^13 - 327680*a^13*b^7*c^5*d*e^13 + 98304
0*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) + 851968*a^14*b*c^8*e^12 + 192*a^8*b^13*c^2*e^12 - 4672*a
^9*b^11*c^3*e^12 + 47360*a^10*b^9*c^4*e^12 - 256000*a^11*b^7*c^5*e^12 + 778240*a^12*b^5*c^6*e^12 - 1261568*a^1
3*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 3264*a^7*b^10*c^4*d*e^11 + 30112*a^8*b^8*
c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11 - 458752*a^11*b^2*c^8*d*e^11) - (-(9*b^13
+ 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4
- 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2)
)/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 384
0*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*(x*(204800*a^12*c^9*e^12 + 144*a^6*b^12*c^3*e^12 - 3264*a^7
*b^10*c^4*e^12 + 30112*a^8*b^8*c^5*e^12 - 143360*a^9*b^6*c^6*e^12 + 365568*a^10*b^4*c^7*e^12 - 458752*a^11*b^2
*c^8*e^12) + (-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c
^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(
-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 128
0*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*((-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9
)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*
a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 409
6*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*
a^10*b^2*c^5*e^2)))^(1/2)*(x*(1048576*a^16*b*c^8*e^14 + 256*a^10*b^13*c^2*e^14 - 6144*a^11*b^11*c^3*e^14 + 614
40*a^12*b^9*c^4*e^14 - 327680*a^13*b^7*c^5*e^14 + 983040*a^14*b^5*c^6*e^14 - 1572864*a^15*b^3*c^7*e^14) + 1048
576*a^16*b*c^8*d*e^13 + 256*a^10*b^13*c^2*d*e^13 - 6144*a^11*b^11*c^3*d*e^13 + 61440*a^12*b^9*c^4*d*e^13 - 327
680*a^13*b^7*c^5*d*e^13 + 983040*a^14*b^5*c^6*d*e^13 - 1572864*a^15*b^3*c^7*d*e^13) - 851968*a^14*b*c^8*e^12 -
192*a^8*b^13*c^2*e^12 + 4672*a^9*b^11*c^3*e^12 - 47360*a^10*b^9*c^4*e^12 + 256000*a^11*b^7*c^5*e^12 - 778240*
a^12*b^5*c^6*e^12 + 1261568*a^13*b^3*c^7*e^12) + 204800*a^12*c^9*d*e^11 + 144*a^6*b^12*c^3*d*e^11 - 3264*a^7*b
^10*c^4*d*e^11 + 30112*a^8*b^8*c^5*d*e^11 - 143360*a^9*b^6*c^6*d*e^11 + 365568*a^10*b^4*c^7*d*e^11 - 458752*a^
11*b^2*c^8*d*e^11) + 128000*a^10*c^9*e^10 + 504*a^6*b^8*c^5*e^10 - 8112*a^7*b^6*c^6*e^10 + 48704*a^8*b^4*c^7*e
^10 - 129280*a^9*b^2*c^8*e^10))*(-(9*b^13 + 9*b^4*(-(4*a*c - b^2)^9)^(1/2) + 26880*a^6*b*c^6 + 2077*a^2*b^9*c^
2 - 10656*a^3*b^7*c^3 + 30240*a^4*b^5*c^4 - 44800*a^5*b^3*c^5 + 25*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^
11*c - 51*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^5*b^12*e^2 + 4096*a^11*c^6*e^2 - 24*a^6*b^10*c*e^2 + 240*a^
7*b^8*c^2*e^2 - 1280*a^8*b^6*c^3*e^2 + 3840*a^9*b^4*c^4*e^2 - 6144*a^10*b^2*c^5*e^2)))^(1/2)*2i - ((x*(3*b^3*d
- 20*a*c^2*d^3 + 6*b^2*c*d^3 - 11*a*b*c*d))/(a*(a*b^2 - 4*a^2*c)) - (x^4*(10*a*c^2*e^3 - 3*b^2*c*e^3))/(2*a*(
a*b^2 - 4*a^2*c)) - (2*x^3*(10*a*c^2*d*e^2 - 3*b^2*c*d*e^2))/(a*(a*b^2 - 4*a^2*c)) + (2*a*b^2 - 8*a^2*c + 3*b^
3*d^2 - 10*a*c^2*d^4 + 3*b^2*c*d^4 - 11*a*b*c*d^2)/(2*a*e*(a*b^2 - 4*a^2*c)) + (x^2*(3*b^3*e - 60*a*c^2*d^2*e
+ 18*b^2*c*d^2*e - 11*a*b*c*e))/(2*a*(a*b^2 - 4*a^2*c)))/(a*d + x*(a*e + 3*b*d^2*e + 5*c*d^4*e) + x^3*(b*e^3 +
10*c*d^2*e^3) + b*d^3 + c*d^5 + x^2*(10*c*d^3*e^2 + 3*b*d*e^2) + c*e^5*x^5 + 5*c*d*e^4*x^4)

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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/(e*x+d)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

Timed out

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