3.3.55 \(\int \frac {x^2}{(d+e x^2) (a+c x^4)^2} \, dx\) [255]

Optimal. Leaf size=685 \[ \frac {x \left (a e+c d x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\sqrt {d} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{c} d e \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} d+3 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{c} d e \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} d+3 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{c} d e \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} d-3 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{c} d e \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} d-3 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (c d^2+a e^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 685, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {1330, 1193, 1182, 1176, 631, 210, 1179, 642, 1185, 211} \begin {gather*} \frac {\sqrt [4]{c} d e \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{c} d e \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {a} e+\sqrt {c} d\right )}{8 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (a e^2+c d^2\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {a} e+\sqrt {c} d\right )}{8 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (a e^2+c d^2\right )}+\frac {\sqrt [4]{c} d e \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{c} d e \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^2+c d^2\right )^2}+\frac {\left (\sqrt {c} d-3 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac {\left (\sqrt {c} d-3 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \sqrt [4]{c} \left (a e^2+c d^2\right )}-\frac {\sqrt {d} e^{5/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (a e^2+c d^2\right )^2}+\frac {x \left (a e+c d x^2\right )}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*e + c*d*x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) - (Sqrt[d]*e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 +
a*e^2)^2 + (c^(1/4)*d*e*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(
c*d^2 + a*e^2)^2) - ((Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(5/4)*c^(
1/4)*(c*d^2 + a*e^2)) - (c^(1/4)*d*e*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[
2]*a^(3/4)*(c*d^2 + a*e^2)^2) + ((Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]
*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)) + (c^(1/4)*d*e*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) + ((Sqrt[c]*d - 3*Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(5/4)*c^(1/4)*(c*d^2 + a*e^2)) - (c^(1/4)*d*e*(Sqrt[c]*d + Sqrt[
a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) - ((Sqrt[c
]*d - 3*Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(5/4)*c^(1/4)*(c*d^2
+ a*e^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 1185

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

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 1330

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

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 428, normalized size = 0.62 \begin {gather*} \frac {\frac {8 \left (c d^2+a e^2\right ) \left (a e x+c d x^3\right )}{a \left (a+c x^4\right )}-32 \sqrt {d} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {2 \sqrt {2} \left (c^{3/2} d^3-\sqrt {a} c d^2 e+5 a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/4} \sqrt [4]{c}}+\frac {2 \sqrt {2} \left (c^{3/2} d^3-\sqrt {a} c d^2 e+5 a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/4} \sqrt [4]{c}}+\frac {\sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+5 a \sqrt {c} d e^2-3 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^{5/4} \sqrt [4]{c}}-\frac {\sqrt {2} \left (c^{3/2} d^3+\sqrt {a} c d^2 e+5 a \sqrt {c} d e^2-3 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^{5/4} \sqrt [4]{c}}}{32 \left (c d^2+a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*(c*d^2 + a*e^2)*(a*e*x + c*d*x^3))/(a*(a + c*x^4)) - 32*Sqrt[d]*e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (2*S
qrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/(a^(5/4)*c^(1/4)) + (2*Sqrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*ArcTa
n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(5/4)*c^(1/4)) + (Sqrt[2]*(c^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*
d*e^2 - 3*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(5/4)*c^(1/4)) - (Sqrt[2]*(c
^(3/2)*d^3 + Sqrt[a]*c*d^2*e + 5*a*Sqrt[c]*d*e^2 - 3*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(a^(5/4)*c^(1/4)))/(32*(c*d^2 + a*e^2)^2)

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Maple [A]
time = 0.23, size = 339, normalized size = 0.49

method	result	size
default	\(-\frac {d \,e^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {d e}}+\frac {\frac {\frac {c d \left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{4 a}+\left (\frac {1}{4} a \,e^{3}+\frac {1}{4} c \,d^{2} e \right ) x}{c \,x^{4}+a}+\frac {\frac {\left (3 a^{2} e^{3}-a \,d^{2} e c \right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (5 a c d \,e^{2}+c^{2} d^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a}}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(339\)
risch	\(\frac {\frac {c d \,x^{3}}{4 a \left (a \,e^{2}+c \,d^{2}\right )}+\frac {e x}{4 a \,e^{2}+4 c \,d^{2}}}{c \,x^{4}+a}+\frac {\sqrt {-d e}\, e^{2} \ln \left (\left (-4096 \left (-d e \right )^{\frac {5}{2}} a^{5} c \,e^{8}+4096 \left (-d e \right )^{\frac {5}{2}} a^{4} c^{2} d^{2} e^{6}-3552 \left (-d e \right )^{\frac {3}{2}} a^{5} c d \,e^{9}+5248 \left (-d e \right )^{\frac {3}{2}} a^{4} c^{2} d^{3} e^{7}+704 \left (-d e \right )^{\frac {3}{2}} a^{3} c^{3} d^{5} e^{5}+128 \left (-d e \right )^{\frac {3}{2}} a^{2} c^{4} d^{7} e^{3}+32 \left (-d e \right )^{\frac {3}{2}} a \,c^{5} d^{9} e -81 \sqrt {-d e}\, a^{6} e^{12}-54 \sqrt {-d e}\, a^{5} c \,d^{2} e^{10}+81 \sqrt {-d e}\, a^{4} c^{2} d^{4} e^{8}+12 \sqrt {-d e}\, a^{3} c^{3} d^{6} e^{6}-31 \sqrt {-d e}\, a^{2} c^{4} d^{8} e^{4}+10 \sqrt {-d e}\, a \,c^{5} d^{10} e^{2}-\sqrt {-d e}\, c^{6} d^{12}\right ) x -81 a^{6} d \,e^{12}-598 a^{5} c \,d^{3} e^{10}-1071 a^{4} c^{2} d^{5} e^{8}-692 a^{3} c^{3} d^{7} e^{6}-159 a^{2} c^{4} d^{9} e^{4}-22 a \,c^{5} d^{11} e^{2}-c^{6} d^{13}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {\sqrt {-d e}\, e^{2} \ln \left (\left (4096 \left (-d e \right )^{\frac {5}{2}} a^{5} c \,e^{8}-4096 \left (-d e \right )^{\frac {5}{2}} a^{4} c^{2} d^{2} e^{6}+3552 \left (-d e \right )^{\frac {3}{2}} a^{5} c d \,e^{9}-5248 \left (-d e \right )^{\frac {3}{2}} a^{4} c^{2} d^{3} e^{7}-704 \left (-d e \right )^{\frac {3}{2}} a^{3} c^{3} d^{5} e^{5}-128 \left (-d e \right )^{\frac {3}{2}} a^{2} c^{4} d^{7} e^{3}-32 \left (-d e \right )^{\frac {3}{2}} a \,c^{5} d^{9} e +81 \sqrt {-d e}\, a^{6} e^{12}+54 \sqrt {-d e}\, a^{5} c \,d^{2} e^{10}-81 \sqrt {-d e}\, a^{4} c^{2} d^{4} e^{8}-12 \sqrt {-d e}\, a^{3} c^{3} d^{6} e^{6}+31 \sqrt {-d e}\, a^{2} c^{4} d^{8} e^{4}-10 \sqrt {-d e}\, a \,c^{5} d^{10} e^{2}+\sqrt {-d e}\, c^{6} d^{12}\right ) x -81 a^{6} d \,e^{12}-598 a^{5} c \,d^{3} e^{10}-1071 a^{4} c^{2} d^{5} e^{8}-692 a^{3} c^{3} d^{7} e^{6}-159 a^{2} c^{4} d^{9} e^{4}-22 a \,c^{5} d^{11} e^{2}-c^{6} d^{13}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{9} c \,e^{8}+4 a^{8} c^{2} d^{2} e^{6}+6 a^{7} c^{3} d^{4} e^{4}+4 a^{6} c^{4} d^{6} e^{2}+a^{5} c^{5} d^{8}\right ) \textit {\_Z}^{4}+\left (60 a^{5} c d \,e^{5}-8 a^{4} c^{2} d^{3} e^{3}-4 a^{3} c^{3} d^{5} e \right ) \textit {\_Z}^{2}+81 a^{2} e^{4}+18 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{11} c \,e^{15}-10 a^{10} c^{2} d^{2} e^{13}-18 a^{9} c^{3} d^{4} e^{11}-10 a^{8} c^{4} d^{6} e^{9}+10 a^{7} c^{5} d^{8} e^{7}+18 a^{6} c^{6} d^{10} e^{5}+10 a^{5} c^{7} d^{12} e^{3}+2 a^{4} c^{8} d^{14} e \right ) \textit {\_R}^{5}+\left (-111 a^{7} c d \,e^{12}-58 a^{6} c^{2} d^{3} e^{10}+239 a^{5} c^{3} d^{5} e^{8}+212 a^{4} c^{4} d^{7} e^{6}+31 a^{3} c^{5} d^{9} e^{4}+6 a^{2} c^{6} d^{11} e^{2}+a \,c^{7} d^{13}\right ) \textit {\_R}^{3}+\left (-162 a^{4} e^{11}+216 a^{3} c \,d^{2} e^{9}-108 a^{2} c^{2} d^{4} e^{7}+24 a \,c^{3} d^{6} e^{5}-2 c^{4} d^{8} e^{3}\right ) \textit {\_R} \right ) x +\left (-13 a^{9} c d \,e^{13}-66 a^{8} c^{2} d^{3} e^{11}-135 a^{7} c^{3} d^{5} e^{9}-140 a^{6} c^{4} d^{7} e^{7}-75 a^{5} c^{5} d^{9} e^{5}-18 a^{4} c^{6} d^{11} e^{3}-a^{3} c^{7} d^{13} e \right ) \textit {\_R}^{4}+\left (-777 a^{5} c \,d^{2} e^{10}-805 a^{4} c^{2} d^{4} e^{8}-58 a^{3} c^{3} d^{6} e^{6}-42 a^{2} c^{4} d^{8} e^{4}-13 a \,c^{5} d^{10} e^{2}-c^{6} d^{12}\right ) \textit {\_R}^{2}-1296 a^{2} d \,e^{9}-864 a c \,d^{3} e^{7}-80 c^{2} d^{5} e^{5}\right )\right )}{16}\)	\(1392\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*e^3/(a*e^2+c*d^2)^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/(a*e^2+c*d^2)^2*((1/4*c*d*(a*e^2+c*d^2)/a*x^3+(1/
4*a*e^3+1/4*c*d^2*e)*x)/(c*x^4+a)+1/4/a*(1/8*(3*a^2*e^3-a*c*d^2*e)*(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))+1/8*(5*a*c*d*e^2+c^2*d^3)/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))))

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Maxima [A]
time = 0.53, size = 458, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {5}{2}}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {c d x^{3} + a x e}{4 \, {\left (a^{2} c d^{2} + {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{4} + a^{3} e^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (\sqrt {a} c^{2} d^{3} - a c^{\frac {3}{2}} d^{2} e + 5 \, a^{\frac {3}{2}} c d e^{2} + 3 \, a^{2} \sqrt {c} 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 (\sqrt {a} c^{2} d^{3} - a c^{\frac {3}{2}} d^{2} e + 5 \, a^{\frac {3}{2}} c d e^{2} + 3 \, a^{2} \sqrt {c} 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 (\sqrt {a} c^{2} d^{3} + a c^{\frac {3}{2}} d^{2} e + 5 \, a^{\frac {3}{2}} c d e^{2} - 3 \, a^{2} \sqrt {c} 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 (\sqrt {a} c^{2} d^{3} + a c^{\frac {3}{2}} d^{2} e + 5 \, a^{\frac {3}{2}} c d e^{2} - 3 \, a^{2} \sqrt {c} 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}}}}{32 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(d)*arctan(x*e^(1/2)/sqrt(d))*e^(5/2)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/4*(c*d*x^3 + a*x*e)/(a^2*c*
d^2 + (a*c^2*d^2 + a^2*c*e^2)*x^4 + a^3*e^2) + 1/32*(2*sqrt(2)*(sqrt(a)*c^2*d^3 - a*c^(3/2)*d^2*e + 5*a^(3/2)*
c*d*e^2 + 3*a^2*sqrt(c)*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)*(sqrt(a)*c^2*d^3 - a*c^(3/2)*d^2*e + 5*a^(3/2)*c*d*e^2 +
3*a^2*sqrt(c)*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)*(sqrt(a)*c^2*d^3 + a*c^(3/2)*d^2*e + 5*a^(3/2)*c*d*e^2 - 3*a^2*sqrt(c
)*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)*(sqrt(a)*c^2*d^3 + a
*c^(3/2)*d^2*e + 5*a^(3/2)*c*d*e^2 - 3*a^2*sqrt(c)*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*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4578 vs. \(2 (514) = 1028\).
time = 6.79, size = 9185, normalized size = 13.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*c^2*d^3*x^3 + 4*a*c*d*x^3*e^2 + 4*a*c*d^2*x*e + 4*a^2*x*e^3 + 8*(a*c*x^4 + a^2)*sqrt(-d*e)*e^2*log((x
^2*e - 2*sqrt(-d*e)*x - d)/(x^2*e + d)) + (a*c^3*d^4*x^4 + a^2*c^2*d^4 + (a^3*c*x^4 + a^4)*e^4 + 2*(a^2*c^2*d^
2*x^4 + a^3*c*d^2)*e^2)*sqrt((2*c^2*d^5*e + 4*a*c*d^3*e^3 - 30*a^2*d*e^5 + (a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 +
6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540
*a^3*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2
+ 28*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12
+ 8*a^12*c^2*d^2*e^14 + a^13*c*e^16)))/(a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6
+ a^6*e^8))*log(-c^4*d^8*x - 18*a*c^3*d^6*x*e^2 - 112*a^2*c^2*d^4*x*e^4 - 270*a^3*c*d^2*x*e^6 + 81*a^4*x*e^8
+ (a^2*c^4*d^8*e + 6*a^3*c^3*d^6*e^3 + 4*a^4*c^2*d^4*e^5 - 102*a^5*c*d^2*e^7 + 27*a^6*e^9 - (a^4*c^6*d^11 + 9*
a^5*c^5*d^9*e^2 + 26*a^6*c^4*d^7*e^4 + 34*a^7*c^3*d^5*e^6 + 21*a^8*c^2*d^3*e^8 + 5*a^9*c*d*e^10)*sqrt(-(c^6*d^
12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10
+ 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8
*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^14 + a^13*c*e^16)))*sqrt((2*c^2*d^5*e +
4*a*c*d^3*e^3 - 30*a^2*d*e^5 + (a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^
8)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558
*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6
+ 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^14 + a^13*c*e^16)))/(a^2
*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8))) - (a*c^3*d^4*x^4 + a^2*c^2*d^4
+ (a^3*c*x^4 + a^4)*e^4 + 2*(a^2*c^2*d^2*x^4 + a^3*c*d^2)*e^2)*sqrt((2*c^2*d^5*e + 4*a*c*d^3*e^3 - 30*a^2*d*e
^5 + (a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(-(c^6*d^12 + 18*a*
c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e
^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*
a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^14 + a^13*c*e^16)))/(a^2*c^4*d^8 + 4*a^3*c^3*d^6*e
^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8))*log(-c^4*d^8*x - 18*a*c^3*d^6*x*e^2 - 112*a^2*c^2*d^4*x*e
^4 - 270*a^3*c*d^2*x*e^6 + 81*a^4*x*e^8 - (a^2*c^4*d^8*e + 6*a^3*c^3*d^6*e^3 + 4*a^4*c^2*d^4*e^5 - 102*a^5*c*d
^2*e^7 + 27*a^6*e^9 - (a^4*c^6*d^11 + 9*a^5*c^5*d^9*e^2 + 26*a^6*c^4*d^7*e^4 + 34*a^7*c^3*d^5*e^6 + 21*a^8*c^2
*d^3*e^8 + 5*a^9*c*d*e^10)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 7
99*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^
4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^
14 + a^13*c*e^16)))*sqrt((2*c^2*d^5*e + 4*a*c*d^3*e^3 - 30*a^2*d*e^5 + (a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^
4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3
*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28
*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8
*a^12*c^2*d^2*e^14 + a^13*c*e^16)))/(a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a
^6*e^8))) + (a*c^3*d^4*x^4 + a^2*c^2*d^4 + (a^3*c*x^4 + a^4)*e^4 + 2*(a^2*c^2*d^2*x^4 + a^3*c*d^2)*e^2)*sqrt((
2*c^2*d^5*e + 4*a*c*d^3*e^3 - 30*a^2*d*e^5 - (a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^
2*e^6 + a^6*e^8)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 799*a^4*c^2
*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 + 8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^4 + 56*a^8
*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 + 28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^14 + a^13*
c*e^16)))/(a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8))*log(-c^4*d^8*x -
18*a*c^3*d^6*x*e^2 - 112*a^2*c^2*d^4*x*e^4 - 270*a^3*c*d^2*x*e^6 + 81*a^4*x*e^8 + (a^2*c^4*d^8*e + 6*a^3*c^3*d
^6*e^3 + 4*a^4*c^2*d^4*e^5 - 102*a^5*c*d^2*e^7 + 27*a^6*e^9 + (a^4*c^6*d^11 + 9*a^5*c^5*d^9*e^2 + 26*a^6*c^4*d
^7*e^4 + 34*a^7*c^3*d^5*e^6 + 21*a^8*c^2*d^3*e^8 + 5*a^9*c*d*e^10)*sqrt(-(c^6*d^12 + 18*a*c^5*d^10*e^2 + 143*a
^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 799*a^4*c^2*d^4*e^8 - 558*a^5*c*d^2*e^10 + 81*a^6*e^12)/(a^5*c^9*d^16 +
8*a^6*c^8*d^14*e^2 + 28*a^7*c^7*d^12*e^4 + 56*a^8*c^6*d^10*e^6 + 70*a^9*c^5*d^8*e^8 + 56*a^10*c^4*d^6*e^10 +
28*a^11*c^3*d^4*e^12 + 8*a^12*c^2*d^2*e^14 + a^...

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

[Out]

Timed out

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Giac [A]
time = 3.89, size = 603, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {5}{2}}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{2}\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^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e - \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{2}\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^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{2}\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^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e + \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} - 3 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{2}\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^{2} c^{4} d^{4} + 2 \, \sqrt {2} a^{3} c^{3} d^{2} e^{2} + \sqrt {2} a^{4} c^{2} e^{4}\right )}} + \frac {c d x^{3} + a x e}{4 \, {\left (c x^{4} + a\right )} {\left (a c d^{2} + a^{2} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.87, size = 2500, normalized size = 3.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((e*x)/(4*(a*e^2 + c*d^2)) + (c*d*x^3)/(4*a*(a*e^2 + c*d^2)))/(a + c*x^4) + atan(((((((53248*a^9*c^4*d*e^15 +
4096*a^3*c^10*d^13*e^3 + 73728*a^4*c^9*d^11*e^5 + 307200*a^5*c^8*d^9*e^7 + 573440*a^6*c^7*d^7*e^9 + 552960*a^7
*c^6*d^5*e^11 + 270336*a^8*c^5*d^3*e^13)/(256*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6
*a^4*c^2*d^4*e^4)) - (x*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3
*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^
5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2)*(65536*a^11*c^4*e^17 - 65536*a^
4*c^11*d^14*e^3 - 327680*a^5*c^10*d^12*e^5 - 589824*a^6*c^9*d^10*e^7 - 327680*a^7*c^8*d^8*e^9 + 327680*a^8*c^7
*d^6*e^11 + 589824*a^9*c^6*d^4*e^13 + 327680*a^10*c^5*d^2*e^15))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6
+ 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d
^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/
(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2) + (x*(128*a
*c^10*d^13*e^2 - 14208*a^7*c^4*d*e^14 + 768*a^2*c^9*d^11*e^4 + 3968*a^3*c^8*d^9*e^6 + 27136*a^4*c^7*d^7*e^8 +
30592*a^5*c^6*d^5*e^10 - 7424*a^6*c^5*d^3*e^12))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6
*e^2 + 6*a^4*c^2*d^4*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2
*d^3*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8
+ a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2) + (16*c^9*d^12*e^2 + 208*a*
c^8*d^10*e^4 + 672*a^2*c^7*d^8*e^6 + 928*a^3*c^6*d^6*e^8 + 12880*a^4*c^5*d^4*e^10 + 12432*a^5*c^4*d^2*e^12)/(2
56*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/
2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c
)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e
^4 + 4*a^8*c^2*d^2*e^6)))^(1/2) - (x*(81*a^4*c^3*e^13 + c^7*d^8*e^5 - 12*a*c^6*d^6*e^7 + 54*a^2*c^5*d^4*e^9 -
108*a^3*c^4*d^2*e^11))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4))
)*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*d*e^5
+ 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^6*c^4*
d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2)*1i - (((((53248*a^9*c^4*d*e^15 + 4096*a^3*c^10*d^13*e
^3 + 73728*a^4*c^9*d^11*e^5 + 307200*a^5*c^8*d^9*e^7 + 573440*a^6*c^7*d^7*e^9 + 552960*a^7*c^6*d^5*e^11 + 2703
36*a^8*c^5*d^3*e^13)/(256*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4)) +
(x*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*d*e^
5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^6*c^
4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2)*(65536*a^11*c^4*e^17 - 65536*a^4*c^11*d^14*e^3 - 32
7680*a^5*c^10*d^12*e^5 - 589824*a^6*c^9*d^10*e^7 - 327680*a^7*c^8*d^8*e^9 + 327680*a^8*c^7*d^6*e^11 + 589824*a
^9*c^6*d^4*e^13 + 327680*a^10*c^5*d^2*e^15))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2
+ 6*a^4*c^2*d^4*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3
*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^
5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2) - (x*(128*a*c^10*d^13*e^2 - 142
08*a^7*c^4*d*e^14 + 768*a^2*c^9*d^11*e^4 + 3968*a^3*c^8*d^9*e^6 + 27136*a^4*c^7*d^7*e^8 + 30592*a^5*c^6*d^5*e^
10 - 7424*a^6*c^5*d^3*e^12))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4
*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*
d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^
6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e^6)))^(1/2) + (16*c^9*d^12*e^2 + 208*a*c^8*d^10*e^4 + 672*a
^2*c^7*d^8*e^6 + 928*a^3*c^6*d^6*e^8 + 12880*a^4*c^5*d^4*e^10 + 12432*a^5*c^4*d^2*e^12)/(256*(a^6*e^8 + a^2*c^
4*d^8 + 4*a^5*c*d^2*e^6 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4)))*(-(c^3*d^6*(-a^5*c)^(1/2) - 9*a^3*e^6*(-a^5
*c)^(1/2) - 2*a^3*c^3*d^5*e - 4*a^4*c^2*d^3*e^3 + 30*a^5*c*d*e^5 + 9*a*c^2*d^4*e^2*(-a^5*c)^(1/2) + 31*a^2*c*d
^2*e^4*(-a^5*c)^(1/2))/(256*(a^9*c*e^8 + a^5*c^5*d^8 + 4*a^6*c^4*d^6*e^2 + 6*a^7*c^3*d^4*e^4 + 4*a^8*c^2*d^2*e
^6)))^(1/2) + (x*(81*a^4*c^3*e^13 + c^7*d^8*e^5 - 12*a*c^6*d^6*e^7 + 54*a^2*c^5*d^4*e^9 - 108*a^3*c^4*d^2*e^11
))/(128*(a^6*e^8 + a^2*c^4*d^8 + 4*a^5*c*d^2*e^...

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