3.253 \(\int \frac {x^6}{(d+e x^2) (a+c x^4)^2} \, dx\)

Optimal. Leaf size=687 \[ -\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}+\frac {d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}-\frac {d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}-\frac {x \left (a e+c d x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}+\frac {d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}-\frac {d^{5/2} \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (a e^2+c d^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.60, antiderivative size = 687, 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 = {1314, 1276, 1168, 1162, 617, 204, 1165, 628, 1288, 205} \[ -\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} \sqrt [4]{a} c^{5/4} \left (a e^2+c d^2\right )}-\frac {x \left (a e+c d x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}-\frac {d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}-\frac {d^{5/2} \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (a e^2+c d^2\right )^2}-\frac {d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2}+\frac {d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^6/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

Rule 204

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

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 617

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 628

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 1162

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 1165

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 1168

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 1276

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

Rule 1288

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

Rule 1314

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

Rubi steps

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

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Mathematica [A]  time = 0.37, size = 428, normalized size = 0.62 \[ -\frac {\frac {\sqrt {2} \left (a^{3/2} e^3+5 \sqrt {a} c d^2 e+a \sqrt {c} d e^2-3 c^{3/2} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{\sqrt [4]{a} c^{5/4}}-\frac {\sqrt {2} \left (a^{3/2} e^3+5 \sqrt {a} c d^2 e+a \sqrt {c} d e^2-3 c^{3/2} d^3\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{\sqrt [4]{a} c^{5/4}}+\frac {2 \sqrt {2} \left (a^{3/2} e^3+5 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} c^{5/4}}-\frac {2 \sqrt {2} \left (a^{3/2} e^3+5 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 c^{3/2} d^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} c^{5/4}}+\frac {8 \left (a e^2+c d^2\right ) \left (a e x+c d x^3\right )}{c \left (a+c x^4\right )}+32 d^{5/2} \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

-1/32*((8*(c*d^2 + a*e^2)*(a*e*x + c*d*x^3))/(c*(a + c*x^4)) + 32*d^(5/2)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
+ (2*Sqrt[2]*(3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x
)/a^(1/4)])/(a^(1/4)*c^(5/4)) - (2*Sqrt[2]*(3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)
*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(1/4)*c^(5/4)) + (Sqrt[2]*(-3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + a
*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(a^(1/4)*c^(5/4)) - (Sqr
t[2]*(-3*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(a^(1/4)*c^(5/4)))/(c*d^2 + a*e^2)^2

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fricas [B]  time = 22.08, size = 9822, normalized size = 14.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

[-1/16*(4*(c^2*d^3 + a*c*d*e^2)*x^3 + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2
+ a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4
*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 54
0*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 2
8*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 +
8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*
c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*x + (45*a*c
^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 - (3*a*c^9*d^11 + 11*a^2*
c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 -
558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6
*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 +
56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c*
d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt
(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5
*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*
a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 +
4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) - (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^
3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (
c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^
5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(
a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*
c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c
^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18
*a^3*c*d^2*e^6 - a^4*e^8)*x - (45*a*c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7
- a^5*c*e^9 - (3*a*c^9*d^11 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 -
a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*
c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4
*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5
*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 +
4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d
^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*
d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d
^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)))
+ (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^
5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c
^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*
e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^
10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))
/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c
^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*x + (45*a*c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*
a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 + (3*a*c^9*d^11 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6
*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8
*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^1
4*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^
4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4
*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2
+ 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16
+ 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^1
0 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4
+ 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) - (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2
+ a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^
4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 5
40*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 +
28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 +
8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4
*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*x - (45*a*
c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 + (3*a*c^9*d^11 + 11*a^2
*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 -
558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^
6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8
+ 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c
*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqr
t(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^
5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70
*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 +
4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) - 8*(c^2*d^2*x^4 + a*c*d^2)*sqrt(-d*
e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) + 4*(a*c*d^2*e + a^2*e^3)*x)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 +
a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4), -1/16*(4*(c^2*d^3 + a*c*d*e^2)*x^3 + 16*(c^2*d^2*
x^4 + a*c*d^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c
^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2
+ 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d
^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d
^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*
d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2
*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*
x + (45*a*c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 - (3*a*c^9*d^1
1 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*
c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2
*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^
9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5
*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^
2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e
^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^1
0*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/
(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) - (a*c^3*d^4 + 2*a^2*c^2*d
^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^
2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12
- 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 +
a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^
8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^
2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d
^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*x - (45*a*c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c
^2*d^2*e^7 - a^5*c*e^9 - (3*a*c^9*d^11 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5
*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6
+ 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e
^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^1
4 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 + (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4
*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 54
0*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 2
8*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 +
8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*
c^2*e^8))) + (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(
-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*
e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a
^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*
a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*
c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^
8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)*x + (45*a*c^5*d^8*e - 146*a^2*c^4*d^
6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 + (3*a*c^9*d^11 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*
d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*
a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a
^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28
*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (
c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^
5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(
a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*
c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c
^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) - (a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*
c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-(30*c^2*d^5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2
+ 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*
d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*
d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7
*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^
2*e^6 + a^4*c^2*e^8))*log(-(81*c^4*d^8 - 270*a*c^3*d^6*e^2 - 112*a^2*c^2*d^4*e^4 - 18*a^3*c*d^2*e^6 - a^4*e^8)
*x - (45*a*c^5*d^8*e - 146*a^2*c^4*d^6*e^3 - 76*a^3*c^3*d^4*e^5 - 14*a^4*c^2*d^2*e^7 - a^5*c*e^9 + (3*a*c^9*d^
11 + 11*a^2*c^8*d^9*e^2 + 14*a^3*c^7*d^7*e^4 + 6*a^4*c^6*d^5*e^6 - a^5*c^5*d^3*e^8 - a^6*c^4*d*e^10)*sqrt(-(81
*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*e^8 + 18*a^5*c*d^
2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^10*e^6 + 70*a^5*c
^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))*sqrt(-(30*c^2*d^
5*e - 4*a*c*d^3*e^3 - 2*a^2*d*e^5 - (c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c
^2*e^8)*sqrt(-(81*c^6*d^12 - 558*a*c^5*d^10*e^2 + 799*a^2*c^4*d^8*e^4 + 540*a^3*c^3*d^6*e^6 + 143*a^4*c^2*d^4*
e^8 + 18*a^5*c*d^2*e^10 + a^6*e^12)/(a*c^13*d^16 + 8*a^2*c^12*d^14*e^2 + 28*a^3*c^11*d^12*e^4 + 56*a^4*c^10*d^
10*e^6 + 70*a^5*c^9*d^8*e^8 + 56*a^6*c^8*d^6*e^10 + 28*a^7*c^7*d^4*e^12 + 8*a^8*c^6*d^2*e^14 + a^9*c^5*e^16)))
/(c^6*d^8 + 4*a*c^5*d^6*e^2 + 6*a^2*c^4*d^4*e^4 + 4*a^3*c^3*d^2*e^6 + a^4*c^2*e^8))) + 4*(a*c*d^2*e + a^2*e^3)
*x)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)]

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giac [A]  time = 0.59, size = 595, normalized size = 0.87 \[ -\frac {d^{\frac {5}{2}} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} - \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 c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} + \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} - \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 c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} + \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} + \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 c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{3} + \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 c^{5} d^{4} + 2 \, \sqrt {2} a^{2} c^{4} d^{2} e^{2} + \sqrt {2} a^{3} c^{3} e^{4}\right )}} - \frac {c d x^{3} + a x e}{4 \, {\left (c x^{4} + a\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 852, normalized size = 1.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

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

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maxima [A]  time = 2.08, size = 476, normalized size = 0.69 \[ -\frac {d^{3} e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {d e}} - \frac {c d x^{3} + a e x}{4 \, {\left (a c^{2} d^{2} + a^{2} c e^{2} + {\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, \sqrt {a} c^{2} d^{3} + 5 \, a c^{\frac {3}{2}} d^{2} e - a^{\frac {3}{2}} c d e^{2} + a^{2} \sqrt {c} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, \sqrt {a} c^{2} d^{3} + 5 \, a c^{\frac {3}{2}} d^{2} e - a^{\frac {3}{2}} c d e^{2} + a^{2} \sqrt {c} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (3 \, \sqrt {a} c^{2} d^{3} - 5 \, a c^{\frac {3}{2}} d^{2} e - a^{\frac {3}{2}} c d e^{2} - a^{2} \sqrt {c} e^{3}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (3 \, \sqrt {a} c^{2} d^{3} - 5 \, a c^{\frac {3}{2}} d^{2} e - a^{\frac {3}{2}} c d e^{2} - a^{2} \sqrt {c} e^{3}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{32 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(e*x^2+d)/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.82, size = 17909, normalized size = 26.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((a + c*x^4)^2*(d + e*x^2)),x)

[Out]

atan(((((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*
c^3*d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^
2*d^2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*
e^7 + 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4
*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (x*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(
1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^
4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))
^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 3
27680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^
5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c
^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2)
+ 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a
^4*c^6*d^2*e^6)))^(1/2) + (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c
^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8
+ 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1
/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4
*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^
(1/2))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d
^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2
*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) - (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d
^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4
*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a
^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)
^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*1i
- (((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*
d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^
2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7
+ 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c
^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (x*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2)
- 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-
a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/
2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 32768
0*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^
8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d
^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9
*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c
^6*d^2*e^6)))^(1/2) - (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d
^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*
a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2)
- 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a
*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2
))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e
+ 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8
*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e
^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6
*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c
^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/
2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*1i)/(((
(432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*
e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^
6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 40
9600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d
^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (x*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2
*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^
5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(
65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^
5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 +
a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(
-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2
*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d
^2*e^6)))^(1/2) + (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e
^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^
4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*
a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5
)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*(
-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 3
1*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6
*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) - (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11
+ 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2
+ 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d
*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/
(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (((432*a*
c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*e^10 +
48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (
((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 409600*a^
5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2
+ 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (x*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^
3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2
))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(65536*a
^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d
^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 + a^4*c*e
^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)
^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*
e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)
))^(1/2) - (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 56
32*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e
^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3
*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2)
)/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*(-(a^3*e
^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2
*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 +
6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^
2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^
2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 -
4*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a
*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (a^4*d^3*e^9 + 1
08*a*c^3*d^9*e^3 + 18*a^3*c*d^5*e^7 + 93*a^2*c^2*d^7*e^5)/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*
c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6))))*(-(a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) - 2*a^3*c^3*d*e^5 - 4
*a^2*c^4*d^3*e^3 + 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*
c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*2i - ((d*x^3)/(4*(a
*e^2 + c*d^2)) + (a*e*x)/(4*c*(a*e^2 + c*d^2)))/(a + c*x^4) + atan(((((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10
*e^4 + 12000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^
8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*
a^8*c^4*d*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^
5*e^11 + 24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d
^2*e^6)) - (x*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a
*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 +
4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*
e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 +
589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d
^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^
4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8
+ a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (x*(1152*a*c^9*d^13*e^2
+ 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d
^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*
d^2*e^6)))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^
5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*
a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5
)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2
*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6
)))^(1/2) - (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 +
118*a^4*c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*
((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 3
1*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6
*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*1i - (((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12
000*a^3*c^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c
*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d
*e^15 + 221184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 +
24576*a^7*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6))
+ (x*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*
e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^
8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327
680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a
^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 +
4*a^3*c^2*d^2*e^6)))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3
- 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^
5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) - (x*(1152*a*c^9*d^13*e^2 + 1152*a^
7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 +
4864*a^6*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6))
)*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e +
31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d
^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) +
2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*
c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)
+ (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*
c^2*d^4*e^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e^6
*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d
^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*
a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*1i)/((((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c
^5*d^8*e^6 - 1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*
a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 2
21184*a^3*c^9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7
*c^5*d^3*e^13)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (x*((a^
3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*
c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2
+ 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c
^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^
4*e^13 + 327680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2
*d^2*e^6)))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c
^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4
*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e
^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6
*c^4*d^3*e^12))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e
^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2
*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 +
6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^
3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2
))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) - (x*(a^
6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e
^9))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e^6*(-a*c^5)
^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-
a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d
^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (((432*a*c^7*d^12*e^2 + 13040*a^2*c^6*d^10*e^4 + 12000*a^3*c^5*d^8*e^6 -
1056*a^4*c^4*d^6*e^8 - 400*a^5*c^3*d^4*e^10 + 48*a^6*c^2*d^2*e^12)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^
2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((45056*a^2*c^10*d^13*e^3 - 4096*a^8*c^4*d*e^15 + 221184*a^3*c^
9*d^11*e^5 + 430080*a^4*c^8*d^9*e^7 + 409600*a^5*c^7*d^7*e^9 + 184320*a^6*c^6*d^5*e^11 + 24576*a^7*c^5*d^3*e^1
3)/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (x*((a^3*e^6*(-a*c^
5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*
(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7
*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5
- 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327
680*a^8*c^6*d^2*e^15))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*
((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 3
1*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6
*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) - (x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*
a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^1
2))/(128*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e^6*(-a*c^5)^
(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a
*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^
4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*
a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) + 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c
^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4*c^6*d^2*e^6)))^(1/2) + (x*(a^6*e^13 - 288
*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(128*(c
^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c
^3*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2)
+ 9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a
^4*c^6*d^2*e^6)))^(1/2) + (a^4*d^3*e^9 + 108*a*c^3*d^9*e^3 + 18*a^3*c*d^5*e^7 + 93*a^2*c^2*d^7*e^5)/(128*(c^5*
d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6))))*((a^3*e^6*(-a*c^5)^(1/2) - 9*c^3
*d^6*(-a*c^5)^(1/2) + 2*a^3*c^3*d*e^5 + 4*a^2*c^4*d^3*e^3 - 30*a*c^5*d^5*e + 31*a*c^2*d^4*e^2*(-a*c^5)^(1/2) +
9*a^2*c*d^2*e^4*(-a*c^5)^(1/2))/(256*(a*c^9*d^8 + a^5*c^5*e^8 + 4*a^2*c^8*d^6*e^2 + 6*a^3*c^7*d^4*e^4 + 4*a^4
*c^6*d^2*e^6)))^(1/2)*2i - (atan(-(((-d^5*e)^(1/2)*(((-d^5*e)^(1/2)*(((27*a*c^7*d^12*e^2)/16 + (815*a^2*c^6*d^
10*e^4)/16 + (375*a^3*c^5*d^8*e^6)/8 - (33*a^4*c^4*d^6*e^8)/8 - (25*a^5*c^3*d^4*e^10)/16 + (3*a^6*c^2*d^2*e^12
)/16)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((x*(1152*a*c^9*d
^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a
^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*
a^3*c^2*d^2*e^6)) + (((176*a^2*c^10*d^13*e^3 - 16*a^8*c^4*d*e^15 + 864*a^3*c^9*d^11*e^5 + 1680*a^4*c^8*d^9*e^7
+ 1600*a^5*c^7*d^7*e^9 + 720*a^6*c^6*d^5*e^11 + 96*a^7*c^5*d^3*e^13)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^
2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (x*(-d^5*e)^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3
- 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589
824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(512*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)*(c^5*d^8 + a^4*c*e^8
+ 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^
2*e^2)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2))))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (
x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*
d^4*e^9))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*1i)/(a^2*e^4
+ c^2*d^4 + 2*a*c*d^2*e^2) - ((-d^5*e)^(1/2)*(((-d^5*e)^(1/2)*(((27*a*c^7*d^12*e^2)/16 + (815*a^2*c^6*d^10*e^4
)/16 + (375*a^3*c^5*d^8*e^6)/8 - (33*a^4*c^4*d^6*e^8)/8 - (25*a^5*c^3*d^4*e^10)/16 + (3*a^6*c^2*d^2*e^12)/16)/
(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (((x*(1152*a*c^9*d^13*e^
2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5
*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^
2*d^2*e^6)) - (((176*a^2*c^10*d^13*e^3 - 16*a^8*c^4*d*e^15 + 864*a^3*c^9*d^11*e^5 + 1680*a^4*c^8*d^9*e^7 + 160
0*a^5*c^7*d^7*e^9 + 720*a^6*c^6*d^5*e^11 + 96*a^7*c^5*d^3*e^13)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*
a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (x*(-d^5*e)^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 3276
80*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^
7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(512*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)*(c^5*d^8 + a^4*c*e^8 + 4*a
*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)
))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2))))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x*(a^6
*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^
9))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*1i)/(a^2*e^4 + c^2*
d^4 + 2*a*c*d^2*e^2))/(((a^4*d^3*e^9)/128 + (27*a*c^3*d^9*e^3)/32 + (9*a^3*c*d^5*e^7)/64 + (93*a^2*c^2*d^7*e^5
)/128)/(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6) + ((-d^5*e)^(1/2)*(((-d
^5*e)^(1/2)*(((27*a*c^7*d^12*e^2)/16 + (815*a^2*c^6*d^10*e^4)/16 + (375*a^3*c^5*d^8*e^6)/8 - (33*a^4*c^4*d^6*e
^8)/8 - (25*a^5*c^3*d^4*e^10)/16 + (3*a^6*c^2*d^2*e^12)/16)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*
c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 +
25472*a^3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(256*(c^5*d^8
+ a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (((176*a^2*c^10*d^13*e^3 - 16*a^8*c^
4*d*e^15 + 864*a^3*c^9*d^11*e^5 + 1680*a^4*c^8*d^9*e^7 + 1600*a^5*c^7*d^7*e^9 + 720*a^6*c^6*d^5*e^11 + 96*a^7*
c^5*d^3*e^13)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (x*(-d^5*e
)^(1/2)*(65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 -
327680*a^5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(512*(a
^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e
^6)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d
^2*e^2))))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 1
7*a^2*c^4*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 +
6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6))))/(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2) + ((-d^5*e)^(1/2)*(((-d^5*e)^(1/
2)*(((27*a*c^7*d^12*e^2)/16 + (815*a^2*c^6*d^10*e^4)/16 + (375*a^3*c^5*d^8*e^6)/8 - (33*a^4*c^4*d^6*e^8)/8 - (
25*a^5*c^3*d^4*e^10)/16 + (3*a^6*c^2*d^2*e^12)/16)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e
^4 + 4*a^3*c^2*d^2*e^6)) - (((x*(1152*a*c^9*d^13*e^2 + 1152*a^7*c^3*d*e^14 + 21248*a^2*c^8*d^11*e^4 + 25472*a^
3*c^7*d^9*e^6 - 5632*a^4*c^6*d^7*e^8 - 7296*a^5*c^5*d^5*e^10 + 4864*a^6*c^4*d^3*e^12))/(256*(c^5*d^8 + a^4*c*e
^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) - (((176*a^2*c^10*d^13*e^3 - 16*a^8*c^4*d*e^15
+ 864*a^3*c^9*d^11*e^5 + 1680*a^4*c^8*d^9*e^7 + 1600*a^5*c^7*d^7*e^9 + 720*a^6*c^6*d^5*e^11 + 96*a^7*c^5*d^3*e
^13)/(2*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)) + (x*(-d^5*e)^(1/2)*(
65536*a^9*c^5*e^17 - 65536*a^2*c^12*d^14*e^3 - 327680*a^3*c^11*d^12*e^5 - 589824*a^4*c^10*d^10*e^7 - 327680*a^
5*c^9*d^8*e^9 + 327680*a^6*c^8*d^6*e^11 + 589824*a^7*c^7*d^4*e^13 + 327680*a^8*c^6*d^2*e^15))/(512*(a^2*e^4 +
c^2*d^4 + 2*a*c*d^2*e^2)*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3*d^4*e^4 + 4*a^3*c^2*d^2*e^6)))*(-d
^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))*(-d^5*e)^(1/2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))
)/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x*(a^6*e^13 - 288*a*c^5*d^10*e^3 + 20*a^5*c*d^2*e^11 + 17*a^2*c^4
*d^8*e^5 + 148*a^3*c^3*d^6*e^7 + 118*a^4*c^2*d^4*e^9))/(256*(c^5*d^8 + a^4*c*e^8 + 4*a*c^4*d^6*e^2 + 6*a^2*c^3
*d^4*e^4 + 4*a^3*c^2*d^2*e^6))))/(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))*(-d^5*e)^(1/2)*1i)/(a^2*e^4 + c^2*d^4 +
2*a*c*d^2*e^2)

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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(x**6/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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