3.3.58 \(\int \frac {1}{x^4 (d+e x^2) (a+c x^4)^2} \, dx\) [258]

Optimal. Leaf size=751 \[ -\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (3 \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )}+\frac {c^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (3 \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )}-\frac {c^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )}+\frac {c^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (3 \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )}-\frac {c^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{11/4} \left (c d^2+a e^2\right )^2} \]

[Out]

-1/3/a^2/d/x^3+e/a^2/d^2/x-1/4*c^2*x*(-e*x^2+d)/a^2/(a*e^2+c*d^2)/(c*x^4+a)+e^(11/2)*arctan(x*e^(1/2)/d^(1/2))
/d^(5/2)/(a*e^2+c*d^2)^2-1/4*c^(5/4)*(2*a*e^2+c*d^2)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+d*c^(1/2
))/a^(11/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/4*c^(5/4)*(2*a*e^2+c*d^2)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2
)+d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/8*c^(5/4)*(2*a*e^2+c*d^2)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2
)+x^2*c^(1/2))*(e*a^(1/2)+d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/8*c^(5/4)*(2*a*e^2+c*d^2)*ln(a^(1/4)*c
^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/16*c^(5/4)*arct
an(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+3*d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)*2^(1/2)-1/16*c^(5/4)*arctan(1
+c^(1/4)*x*2^(1/2)/a^(1/4))*(-e*a^(1/2)+3*d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)*2^(1/2)+1/32*c^(5/4)*ln(-a^(1/4)*c
^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+3*d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)*2^(1/2)-1/32*c^(5/4)*ln(a
^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(e*a^(1/2)+3*d*c^(1/2))/a^(11/4)/(a*e^2+c*d^2)*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.44, antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1350, 211, 1193, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {c^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 a e^2+c d^2\right )}{2 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac {c^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {c} d-\sqrt {a} e\right )}{8 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )}-\frac {c^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 a e^2+c d^2\right )}{2 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac {c^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {c} d-\sqrt {a} e\right )}{8 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )}+\frac {c^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac {c^{5/4} \left (\sqrt {a} e+3 \sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )}-\frac {c^{5/4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac {c^{5/4} \left (\sqrt {a} e+3 \sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{11/4} \left (a e^2+c d^2\right )}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {e}{a^2 d^2 x}-\frac {1}{3 a^2 d x^3}+\frac {e^{11/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/(a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (e^(11/2)*ArcTa
n[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c
^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d - Sqrt
[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d - S
qrt[a]*e)*(c*d^2 + 2*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) +
(c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4
)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*
x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*
e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^
2)^2)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

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

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

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

Rule 1193

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

Rule 1350

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q,
0] || IntegersQ[m, q])

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\int \left (\frac {1}{a^2 d x^4}-\frac {e}{a^2 d^2 x^2}+\frac {e^6}{d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {c^2 \left (d-e x^2\right )}{a \left (c d^2+a e^2\right ) \left (a+c x^4\right )^2}-\frac {c^2 \left (c d^2+2 a e^2\right ) \left (d-e x^2\right )}{a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}+\frac {e^6 \int \frac {1}{d+e x^2} \, dx}{d^2 \left (c d^2+a e^2\right )^2}-\frac {c^2 \int \frac {d-e x^2}{\left (a+c x^4\right )^2} \, dx}{a \left (c d^2+a e^2\right )}-\frac {\left (c^2 \left (c d^2+2 a e^2\right )\right ) \int \frac {d-e x^2}{a+c x^4} \, dx}{a^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^2 \int \frac {-3 d+e x^2}{a+c x^4} \, dx}{4 a^2 \left (c d^2+a e^2\right )}-\frac {\left (c \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 a^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 a^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^2 \left (c d^2+a e^2\right )}-\frac {\left (c \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^2 \left (c d^2+a e^2\right )}-\frac {\left (c \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 a^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 a^2 \left (c d^2+a e^2\right )^2}+\frac {\left (c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac {\left (c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^2 \left (c d^2+a e^2\right )}-\frac {\left (c \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^2 \left (c d^2+a e^2\right )}+\frac {\left (c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}+\frac {\left (c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}-\frac {\left (c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac {\left (c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}+\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}-\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}+\frac {\left (c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}\\ &=-\frac {1}{3 a^2 d x^3}+\frac {e}{a^2 d^2 x}-\frac {c^2 x \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}+\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}-\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}+\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}+\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}-\frac {c^{5/4} \left (\frac {3 \sqrt {c} d}{\sqrt {a}}+e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )}-\frac {c^{5/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (c d^2+2 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{9/4} \left (c d^2+a e^2\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 513, normalized size = 0.68 \begin {gather*} \frac {1}{96} \left (-\frac {32}{a^2 d x^3}+\frac {96 e}{a^2 d^2 x}-\frac {24 c^2 x \left (d-e x^2\right )}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {96 e^{11/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \left (c d^2+a e^2\right )^2}+\frac {6 \sqrt {2} c^{5/4} \left (7 c^{3/2} d^3-5 \sqrt {a} c d^2 e+11 a \sqrt {c} d e^2-9 a^{3/2} e^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4} \left (c d^2+a e^2\right )^2}+\frac {6 \sqrt {2} c^{5/4} \left (-7 c^{3/2} d^3+5 \sqrt {a} c d^2 e-11 a \sqrt {c} d e^2+9 a^{3/2} e^3\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4} \left (c d^2+a e^2\right )^2}+\frac {3 \sqrt {2} c^{5/4} \left (7 c^{3/2} d^3+5 \sqrt {a} c d^2 e+11 a \sqrt {c} d e^2+9 a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4} \left (c d^2+a e^2\right )^2}-\frac {3 \sqrt {2} c^{5/4} \left (7 c^{3/2} d^3+5 \sqrt {a} c d^2 e+11 a \sqrt {c} d e^2+9 a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{11/4} \left (c d^2+a e^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32/(a^2*d*x^3) + (96*e)/(a^2*d^2*x) - (24*c^2*x*(d - e*x^2))/(a^2*(c*d^2 + a*e^2)*(a + c*x^4)) + (96*e^(11/2
)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 - 5*Sqrt[a]*c*d
^2*e + 11*a*Sqrt[c]*d*e^2 - 9*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^
2) + (6*Sqrt[2]*c^(5/4)*(-7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*ArcTan[1 + (
Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2 + a*e^2)^2) + (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^
2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c
*d^2 + a*e^2)^2) - (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)
*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2))/96

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Maple [A]
time = 0.23, size = 355, normalized size = 0.47

method	result	size
default	\(\frac {e^{6} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {d e}}-\frac {c^{2} \left (\frac {\left (-\frac {1}{4} a \,e^{3}-\frac {1}{4} c \,d^{2} e \right ) x^{3}+\left (\frac {1}{4} d \,e^{2} a +\frac {1}{4} c \,d^{3}\right ) x}{c \,x^{4}+a}+\frac {\left (11 d \,e^{2} a +7 c \,d^{3}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}+\frac {\left (-9 a \,e^{3}-5 c \,d^{2} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} a^{2}}-\frac {1}{3 a^{2} d \,x^{3}}+\frac {e}{a^{2} d^{2} x}\)	\(355\)
risch	\(\text {Expression too large to display}\)	\(2000\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*e^6/(a*e^2+c*d^2)^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-c^2/(a*e^2+c*d^2)^2/a^2*(((-1/4*a*e^3-1/4*c*d^2*
e)*x^3+(1/4*d*e^2*a+1/4*c*d^3)*x)/(c*x^4+a)+1/32*(11*a*d*e^2+7*c*d^3)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c)^(1/
4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(
2^(1/2)/(a/c)^(1/4)*x-1))+1/32*(-9*a*e^3-5*c*d^2*e)/c/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)
^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4
)*x-1)))-1/3/a^2/d/x^3+e/a^2/d^2/x

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Maxima [A]
time = 0.54, size = 527, normalized size = 0.70 \begin {gather*} -\frac {c^{2} {\left (\frac {2 \, \sqrt {2} {\left (7 \, c^{\frac {3}{2}} d^{3} - 5 \, \sqrt {a} c d^{2} e + 11 \, a \sqrt {c} d e^{2} - 9 \, a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (7 \, c^{\frac {3}{2}} d^{3} - 5 \, \sqrt {a} c d^{2} e + 11 \, a \sqrt {c} d e^{2} - 9 \, a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (7 \, c^{\frac {3}{2}} d^{3} + 5 \, \sqrt {a} c d^{2} e + 11 \, a \sqrt {c} d e^{2} + 9 \, a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, c^{\frac {3}{2}} d^{3} + 5 \, \sqrt {a} c d^{2} e + 11 \, a \sqrt {c} d e^{2} + 9 \, a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )}} + \frac {\arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {11}{2}}}{{\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4}\right )} \sqrt {d}} + \frac {3 \, {\left (5 \, c^{2} d^{2} e + 4 \, a c e^{3}\right )} x^{6} - 4 \, a c d^{3} - {\left (7 \, c^{2} d^{3} + 4 \, a c d e^{2}\right )} x^{4} - 4 \, a^{2} d e^{2} + 12 \, {\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2}}{12 \, {\left ({\left (a^{2} c^{2} d^{4} + a^{3} c d^{2} e^{2}\right )} x^{7} + {\left (a^{3} c d^{4} + a^{4} d^{2} e^{2}\right )} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*c^2*(2*sqrt(2)*(7*c^(3/2)*d^3 - 5*sqrt(a)*c*d^2*e + 11*a*sqrt(c)*d*e^2 - 9*a^(3/2)*e^3)*arctan(1/2*sqrt(
2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*
sqrt(2)*(7*c^(3/2)*d^3 - 5*sqrt(a)*c*d^2*e + 11*a*sqrt(c)*d*e^2 - 9*a^(3/2)*e^3)*arctan(1/2*sqrt(2)*(2*sqrt(c)
*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(7*c^(3
/2)*d^3 + 5*sqrt(a)*c*d^2*e + 11*a*sqrt(c)*d*e^2 + 9*a^(3/2)*e^3)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x
+ sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(7*c^(3/2)*d^3 + 5*sqrt(a)*c*d^2*e + 11*a*sqrt(c)*d*e^2 + 9*a^(3/2)*e^3
)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a
^4*e^4) + arctan(x*e^(1/2)/sqrt(d))*e^(11/2)/((c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4)*sqrt(d)) + 1/12*(3*(5*c^
2*d^2*e + 4*a*c*e^3)*x^6 - 4*a*c*d^3 - (7*c^2*d^3 + 4*a*c*d*e^2)*x^4 - 4*a^2*d*e^2 + 12*(a*c*d^2*e + a^2*e^3)*
x^2)/((a^2*c^2*d^4 + a^3*c*d^2*e^2)*x^7 + (a^3*c*d^4 + a^4*d^2*e^2)*x^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4891 vs. \(2 (572) = 1144\).
time = 63.54, size = 9816, normalized size = 13.07 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(28*c^3*d^5*x^4 + 16*a*c^2*d^5 - 24*(a^2*c*x^7 + a^3*x^3)*sqrt(-e/d)*e^5*log((x^2*e + 2*d*x*sqrt(-e/d)
- d)/(x^2*e + d)) - 3*(a^2*c^3*d^6*x^7 + a^3*c^2*d^6*x^3 + (a^4*c*d^2*x^7 + a^5*d^2*x^3)*e^4 + 2*(a^3*c^2*d^4*
x^7 + a^4*c*d^4*x^3)*e^2)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^
3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 196
79*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(
a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a
^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 +
6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-2401*c^8*d^8*x - 10290*a*c^7*d^6*x*e^2 - 11968*a^2*c^6*d^
4*x*e^4 + 1458*a^3*c^5*d^2*x*e^6 + 6561*a^4*c^4*x*e^8 + (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5
*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 + (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6
*e^5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 +
19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12
)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 5
6*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d
^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)
*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^
4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4
+ 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^1
4 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) + 3*(a^2*
c^3*d^6*x^7 + a^3*c^2*d^6*x^3 + (a^4*c*d^2*x^7 + a^5*d^2*x^3)*e^4 + 2*(a^3*c^2*d^4*x^7 + a^4*c*d^4*x^3)*e^2)*s
qrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e
^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3
*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*
d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*
c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c
*d^2*e^6 + a^9*e^8))*log(-2401*c^8*d^8*x - 10290*a*c^7*d^6*x*e^2 - 11968*a^2*c^6*d^4*x*e^4 + 1458*a^3*c^5*d^2*
x*e^6 + 6561*a^4*c^4*x*e^8 - (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*
e^6 - 891*a^7*c^3*d*e^8 + (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7
+ 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*
a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c
^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^
17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5
+ (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12
642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^
10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 7
0*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d
^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) - 3*(a^2*c^3*d^6*x^7 + a^3*c^2*d^6*x
^3 + (a^4*c*d^2*x^7 + a^5*d^2*x^3)*e^4 + 2*(a^3*c^2*d^4*x^7 + a^4*c*d^4*x^3)*e^2)*sqrt((70*c^5*d^5*e + 236*a*c
^4*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*
e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^
7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12
*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2
*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-2
401*c^8*d^8*x - 10290*a*c^7*d^6*x*e^2 - 11968*a^2*c^6*d^4*x*e^4 + 1458*a^3*c^5*d^2*x*e^6 + 6561*a^4*c^4*x*e^8
+ (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 - (
5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^1...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 5.36, size = 628, normalized size = 0.84 \begin {gather*} -\frac {{\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 11 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 9 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{3} c^{3} d^{4} + 2 \, \sqrt {2} a^{4} c^{2} d^{2} e^{2} + \sqrt {2} a^{5} c e^{4}\right )}} - \frac {{\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 11 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 9 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{3} c^{3} d^{4} + 2 \, \sqrt {2} a^{4} c^{2} d^{2} e^{2} + \sqrt {2} a^{5} c e^{4}\right )}} - \frac {{\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 11 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 9 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} a^{3} c^{3} d^{4} + 2 \, \sqrt {2} a^{4} c^{2} d^{2} e^{2} + \sqrt {2} a^{5} c e^{4}\right )}} + \frac {{\left (7 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + 11 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 9 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} a^{3} c^{3} d^{4} + 2 \, \sqrt {2} a^{4} c^{2} d^{2} e^{2} + \sqrt {2} a^{5} c e^{4}\right )}} + \frac {\arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {11}{2}}}{{\left (c^{2} d^{6} + 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4}\right )} \sqrt {d}} + \frac {c^{2} x^{3} e - c^{2} d x}{4 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )} {\left (c x^{4} + a\right )}} + \frac {3 \, x^{2} e - d}{3 \, a^{2} d^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4)*c*d^2*e - 9*(a*c^3)^(3/4)*a*e^3
)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2
+ sqrt(2)*a^5*c*e^4) - 1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4)*c*d^2*e
- 9*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*
sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) - 1/16*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 +
5*(a*c^3)^(3/4)*c*d^2*e + 9*(a*c^3)^(3/4)*a*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^3*c^3
*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + 1/16*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2
*d*e^2 + 5*(a*c^3)^(3/4)*c*d^2*e + 9*(a*c^3)^(3/4)*a*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2
)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + arctan(x*e^(1/2)/sqrt(d))*e^(11/2)/((c^2*d^6
+ 2*a*c*d^4*e^2 + a^2*d^2*e^4)*sqrt(d)) + 1/4*(c^2*x^3*e - c^2*d*x)/((a^2*c*d^2 + a^3*e^2)*(c*x^4 + a)) + 1/3*
(3*x^2*e - d)/(a^2*d^2*x^3)

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Mupad [B]
time = 5.22, size = 2500, normalized size = 3.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((x*(4917248*a^10*c^18*d^36*e^5 + 50677760*a^11*c^17*d^34*e^7 + 230498304*a^12*c^16*d^32*e^9 + 607559680*
a^13*c^15*d^30*e^11 + 1026486272*a^14*c^14*d^28*e^13 + 1166602240*a^15*c^13*d^26*e^15 + 923508736*a^16*c^12*d^
24*e^17 + 539500544*a^17*c^11*d^22*e^19 + 259409920*a^18*c^10*d^20*e^21 + 109709312*a^19*c^9*d^18*e^23 + 34537
472*a^20*c^8*d^16*e^25 + 5308416*a^21*c^7*d^14*e^27) - ((81*a^3*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)
^(1/2) + 70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5 + 236*a^7*c^4*d^3*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31
*a^2*c*d^2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*e^8 + a^11*c^4*d^8 + 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^
13*c^2*d^4*e^4)))^(1/2)*((x*(1787297792*a^19*c^13*d^31*e^12 - 147587072*a^15*c^17*d^39*e^4 - 698089472*a^16*c^
16*d^37*e^6 - 1660157952*a^17*c^15*d^35*e^8 - 1588068352*a^18*c^14*d^33*e^10 - 12845056*a^14*c^18*d^41*e^2 + 7
839678464*a^20*c^12*d^29*e^14 + 11879841792*a^21*c^11*d^27*e^16 + 10631249920*a^22*c^10*d^25*e^18 + 6274940928
*a^23*c^9*d^23*e^20 + 2652110848*a^24*c^8*d^21*e^22 + 891027456*a^25*c^7*d^19*e^24 + 234881024*a^26*c^6*d^17*e
^26 + 33554432*a^27*c^5*d^15*e^28) + ((81*a^3*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) + 70*a^6*c^
5*d^5*e + 198*a^8*c^3*d*e^5 + 236*a^7*c^4*d^3*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^2*e^4*(-a
^11*c^5)^(1/2))/(256*(a^15*e^8 + a^11*c^4*d^8 + 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^13*c^2*d^4*e^4)))^
(1/2)*(x*((81*a^3*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) + 70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5
+ 236*a^7*c^4*d^3*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*e
^8 + a^11*c^4*d^8 + 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^13*c^2*d^4*e^4)))^(1/2)*(134217728*a^20*c^16*d
^42*e^3 + 1342177280*a^21*c^15*d^40*e^5 + 5905580032*a^22*c^14*d^38*e^7 + 14763950080*a^23*c^13*d^36*e^9 + 221
45925120*a^24*c^12*d^34*e^11 + 17716740096*a^25*c^11*d^32*e^13 - 17716740096*a^27*c^9*d^28*e^17 - 22145925120*
a^28*c^8*d^26*e^19 - 14763950080*a^29*c^7*d^24*e^21 - 5905580032*a^30*c^6*d^22*e^23 - 1342177280*a^31*c^5*d^20
*e^25 - 134217728*a^32*c^4*d^18*e^27) + 29360128*a^17*c^17*d^42*e^2 + 239075328*a^18*c^16*d^40*e^4 + 708837376
*a^19*c^15*d^38*e^6 + 465567744*a^20*c^14*d^36*e^8 - 2726297600*a^21*c^13*d^34*e^10 - 9084862464*a^22*c^12*d^3
2*e^12 - 13614710784*a^23*c^11*d^30*e^14 - 10745806848*a^24*c^10*d^28*e^16 - 2403336192*a^25*c^9*d^26*e^18 + 3
879731200*a^26*c^8*d^24*e^20 + 4517265408*a^27*c^7*d^22*e^22 + 2294284288*a^28*c^6*d^20*e^24 + 603979776*a^29*
c^5*d^18*e^26 + 67108864*a^30*c^4*d^16*e^28))*((81*a^3*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) +
70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5 + 236*a^7*c^4*d^3*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^
2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*e^8 + a^11*c^4*d^8 + 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^13*c^2*d^
4*e^4)))^(1/2) + 7225344*a^12*c^18*d^39*e^3 + 76972032*a^13*c^17*d^37*e^5 + 367607808*a^14*c^16*d^35*e^7 + 103
6910592*a^15*c^15*d^33*e^9 + 1876983808*a^16*c^14*d^31*e^11 + 2115436544*a^17*c^13*d^29*e^13 + 1052803072*a^18
*c^12*d^27*e^15 - 848429056*a^19*c^11*d^25*e^17 - 2105458688*a^20*c^10*d^23*e^19 - 1909030912*a^21*c^9*d^21*e^
21 - 959037440*a^22*c^8*d^19*e^23 - 262144000*a^23*c^7*d^17*e^25 - 30408704*a^24*c^6*d^15*e^27))*((81*a^3*e^6*
(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) + 70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5 + 236*a^7*c^4*d^3*e^3
- 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*e^8 + a^11*c^4*d^8 + 4*
a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^13*c^2*d^4*e^4)))^(1/2)*1i + (x*(4917248*a^10*c^18*d^36*e^5 + 506777
60*a^11*c^17*d^34*e^7 + 230498304*a^12*c^16*d^32*e^9 + 607559680*a^13*c^15*d^30*e^11 + 1026486272*a^14*c^14*d^
28*e^13 + 1166602240*a^15*c^13*d^26*e^15 + 923508736*a^16*c^12*d^24*e^17 + 539500544*a^17*c^11*d^22*e^19 + 259
409920*a^18*c^10*d^20*e^21 + 109709312*a^19*c^9*d^18*e^23 + 34537472*a^20*c^8*d^16*e^25 + 5308416*a^21*c^7*d^1
4*e^27) - ((81*a^3*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) + 70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5
+ 236*a^7*c^4*d^3*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*
e^8 + a^11*c^4*d^8 + 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^13*c^2*d^4*e^4)))^(1/2)*((x*(1787297792*a^19*
c^13*d^31*e^12 - 147587072*a^15*c^17*d^39*e^4 - 698089472*a^16*c^16*d^37*e^6 - 1660157952*a^17*c^15*d^35*e^8 -
1588068352*a^18*c^14*d^33*e^10 - 12845056*a^14*c^18*d^41*e^2 + 7839678464*a^20*c^12*d^29*e^14 + 11879841792*a
^21*c^11*d^27*e^16 + 10631249920*a^22*c^10*d^25*e^18 + 6274940928*a^23*c^9*d^23*e^20 + 2652110848*a^24*c^8*d^2
1*e^22 + 891027456*a^25*c^7*d^19*e^24 + 234881024*a^26*c^6*d^17*e^26 + 33554432*a^27*c^5*d^15*e^28) - ((81*a^3
*e^6*(-a^11*c^5)^(1/2) - 49*c^3*d^6*(-a^11*c^5)^(1/2) + 70*a^6*c^5*d^5*e + 198*a^8*c^3*d*e^5 + 236*a^7*c^4*d^3
*e^3 - 129*a*c^2*d^4*e^2*(-a^11*c^5)^(1/2) - 31*a^2*c*d^2*e^4*(-a^11*c^5)^(1/2))/(256*(a^15*e^8 + a^11*c^4*d^8
+ 4*a^14*c*d^2*e^6 + 4*a^12*c^3*d^6*e^2 + 6*a^...

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