\(\int \frac {1}{(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4)^2} \, dx\) [38]

Optimal result

Integrand size = 29, antiderivative size = 746 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} c+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}+\sqrt {2} d x}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}-\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}-\sqrt {2} \sqrt [4]{c} d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}+\sqrt {2} \sqrt [4]{c} d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}} \]

[Out]

-1/16*(c/d+x)*(c^3-4*a*d^2-c*d^2*(c/d+x)^2)/a/c/(4*a*d^2+c^3)/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)-1/64*d*arcta
nh((c*2^(1/2)+d*x*2^(1/2)+c^(1/4)*(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2))/c^(1/4)/(c^(3/2)-(4*a*d^2+c^3)^(1/2))^(
1/2))*(c^3+12*a*d^2+c^(3/2)*(4*a*d^2+c^3)^(1/2))/a/c^(7/4)/(4*a*d^2+c^3)^(3/2)*2^(1/2)/(c^(3/2)-(4*a*d^2+c^3)^
(1/2))^(1/2)+1/64*d*arctanh((-(d*x+c)*2^(1/2)+c^(1/4)*(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2))/c^(1/4)/(c^(3/2)-(4
*a*d^2+c^3)^(1/2))^(1/2))*(c^3+12*a*d^2+c^(3/2)*(4*a*d^2+c^3)^(1/2))/a/c^(7/4)/(4*a*d^2+c^3)^(3/2)*2^(1/2)/(c^
(3/2)-(4*a*d^2+c^3)^(1/2))^(1/2)-1/128*d*ln(d^2*(c/d+x)^2+c^(1/2)*(4*a*d^2+c^3)^(1/2)-c^(1/4)*d*(c/d+x)*2^(1/2
)*(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2))*(c^3+12*a*d^2-c^(3/2)*(4*a*d^2+c^3)^(1/2))/a/c^(7/4)/(4*a*d^2+c^3)^(3/2
)*2^(1/2)/(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)+1/128*d*ln(d^2*(c/d+x)^2+c^(1/2)*(4*a*d^2+c^3)^(1/2)+c^(1/4)*d*(
c/d+x)*2^(1/2)*(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2))*(c^3+12*a*d^2-c^(3/2)*(4*a*d^2+c^3)^(1/2))/a/c^(7/4)/(4*a*
d^2+c^3)^(3/2)*2^(1/2)/(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1120, 1106, 1183, 648, 632, 212, 642} \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=-\frac {d \left (c^{3/2} \sqrt {4 a d^2+c^3}+12 a d^2+c^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}+\sqrt {2} c+\sqrt {2} d x}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{32 \sqrt {2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}+\frac {d \left (c^{3/2} \sqrt {4 a d^2+c^3}+12 a d^2+c^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}-\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{32 \sqrt {2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}-\frac {d \left (-c^{3/2} \sqrt {4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt {c} \sqrt {4 a d^2+c^3}-\sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )}{64 \sqrt {2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {d \left (-c^{3/2} \sqrt {4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt {c} \sqrt {4 a d^2+c^3}+\sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )}{64 \sqrt {2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )} \]

[In]

Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]

[Out]

-1/16*((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(a*c*(c^3 + 4*a*d^2)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^
2*x^4)) - (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*ArcTanh[(Sqrt[2]*c + c^(1/4)*Sqrt[c^(3/2) + Sqrt[c
^3 + 4*a*d^2]] + Sqrt[2]*d*x)/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])])/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a
*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*ArcTanh[(
c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] - Sqrt[2]*(c + d*x))/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])
])/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*(c^3 + 12*a*d^2 - c^(
3/2)*Sqrt[c^3 + 4*a*d^2])*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] - Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^
2]]*(c/d + x) + d^2*(c/d + x)^2])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^
2]]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] + Sqrt[2]*c^(1/4)*d*S
qrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*S
qrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]])

Rule 212

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1106

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

Rule 1120

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - b*(d/(8*e)) + (c - 3*(d^2/(8*e
)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^2} \, dx,x,\frac {c}{d}+x\right ) \\ & = -\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}+\frac {\text {Subst}\left (\int \frac {4 c^4-2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac {c^3}{d^2}\right ) d^2\right )+2 c^2 d^2 x^2}{c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4} \, dx,x,\frac {c}{d}+x\right )}{32 a c^2 \left (c^3+4 a d^2\right )} \\ & = -\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}+\frac {d \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (4 c^4-2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac {c^3}{d^2}\right ) d^2\right )\right )}{d}-\left (4 c^4-2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 c^{5/2} \sqrt {c^3+4 a d^2}-2 \left (4 c^4-4 c \left (4 a+\frac {c^3}{d^2}\right ) d^2\right )\right ) x}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 \sqrt {2} a c^{11/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (4 c^4-2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac {c^3}{d^2}\right ) d^2\right )\right )}{d}+\left (4 c^4-2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 c^{5/2} \sqrt {c^3+4 a d^2}-2 \left (4 c^4-4 c \left (4 a+\frac {c^3}{d^2}\right ) d^2\right )\right ) x}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 \sqrt {2} a c^{11/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}} \\ & = -\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac {\left (d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}+2 x}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\left (d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}+2 x}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}+\frac {\left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac {c}{d}+x\right )}{64 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}} \\ & = -\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}-\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}+\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}-\frac {\left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {2 \sqrt {c} \left (c^{3/2}-\sqrt {c^3+4 a d^2}\right )}{d^2}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}+2 \left (\frac {c}{d}+x\right )\right )}{32 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}-\frac {\left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {2 \sqrt {c} \left (c^{3/2}-\sqrt {c^3+4 a d^2}\right )}{d^2}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}+2 \left (\frac {c}{d}+x\right )\right )}{32 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}} \\ & = -\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\sqrt {2} c^{3/4}-\sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}\right )+\sqrt {2} d x}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}-\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\sqrt {2} c^{3/4}+\sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}\right )+\sqrt {2} d x}{\sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}-\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}-\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \log \left (\sqrt {c} \sqrt {c^3+4 a d^2}+\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{64 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.07 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\frac {\frac {4 (c+d x) (4 a d+c x (2 c+d x))}{4 a c+x^2 (2 c+d x)^2}+\text {RootSum}\left [4 a c+4 c^2 \text {$\#$1}^2+4 c d \text {$\#$1}^3+d^2 \text {$\#$1}^4\&,\frac {2 c^3 \log (x-\text {$\#$1})+12 a d^2 \log (x-\text {$\#$1})+2 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}+c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{2 c^2 \text {$\#$1}+3 c d \text {$\#$1}^2+d^2 \text {$\#$1}^3}\&\right ]}{64 a c \left (c^3+4 a d^2\right )} \]

[In]

Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]

[Out]

((4*(c + d*x)*(4*a*d + c*x*(2*c + d*x)))/(4*a*c + x^2*(2*c + d*x)^2) + RootSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3
+ d^2*#1^4 & , (2*c^3*Log[x - #1] + 12*a*d^2*Log[x - #1] + 2*c^2*d*Log[x - #1]*#1 + c*d^2*Log[x - #1]*#1^2)/(
2*c^2*#1 + 3*c*d*#1^2 + d^2*#1^3) & ])/(64*a*c*(c^3 + 4*a*d^2))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.31

method	result	size
default	\(\frac {\frac {d^{2} x^{3}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {3 d c \,x^{2}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {\left (2 a \,d^{2}+c^{3}\right ) x}{32 c \left (4 a \,d^{2}+c^{3}\right ) a}+\frac {d}{64 a \,d^{2}+16 c^{3}}}{\frac {1}{4} d^{2} x^{4}+c d \,x^{3}+c^{2} x^{2}+a c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\left (\textit {\_R}^{2} c \,d^{2}+2 \textit {\_R} \,c^{2} d +12 a \,d^{2}+2 c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}}{64 a c \left (4 a \,d^{2}+c^{3}\right )}\)	\(230\)
risch	\(\frac {\frac {d^{2} x^{3}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {3 d c \,x^{2}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {\left (2 a \,d^{2}+c^{3}\right ) x}{32 c \left (4 a \,d^{2}+c^{3}\right ) a}+\frac {d}{64 a \,d^{2}+16 c^{3}}}{\frac {1}{4} d^{2} x^{4}+c d \,x^{3}+c^{2} x^{2}+a c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\left (\frac {d^{2} \textit {\_R}^{2}}{4 a \,d^{2}+c^{3}}+\frac {2 c d \textit {\_R}}{4 a \,d^{2}+c^{3}}+\frac {12 a \,d^{2}+2 c^{3}}{c \left (4 a \,d^{2}+c^{3}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}}{64 a}\)	\(252\)

[In]

int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x,method=_RETURNVERBOSE)

[Out]

(1/64*d^2/a/(4*a*d^2+c^3)*x^3+3/64*d/a*c/(4*a*d^2+c^3)*x^2+1/32/c*(2*a*d^2+c^3)/(4*a*d^2+c^3)/a*x+1/16*d/(4*a*
d^2+c^3))/(1/4*d^2*x^4+c*d*x^3+c^2*x^2+a*c)+1/64/a/c/(4*a*d^2+c^3)*sum((_R^2*c*d^2+2*_R*c^2*d+12*a*d^2+2*c^3)/
(_R^3*d^2+3*_R^2*c*d+2*_R*c^2)*ln(x-_R),_R=RootOf(_Z^4*d^2+4*_Z^3*c*d+4*_Z^2*c^2+4*a*c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3222 vs. \(2 (608) = 1216\).

Time = 0.37 (sec) , antiderivative size = 3222, normalized size of antiderivative = 4.32 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="fricas")

[Out]

1/64*(4*c*d^2*x^3 + 12*c^2*d*x^2 + 16*a*c*d + (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*
(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3
*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(
a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4
096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^
5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2
*d^8 + (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6
+ 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7
*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 1
2*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25
+ 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c
^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*
c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*
a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 3
60*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^
13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^
6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x - (5*a^2*c^8*d^4
+ 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 + (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*
a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4
+ 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2
+ 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*
d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 +
6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + (4
*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2
*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6
*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 +
1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 +
48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 +
324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^1
3*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*
a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^
12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)
*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*
c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c
^5*d^4 + 64*a^6*c^2*d^6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*
c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^
8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4
*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)
))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7
+ (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x - (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19
+ 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 +
1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*
a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 4
8*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2
+ 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c
^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + 8*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a^3*c^2*d^2 +
(a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\int { \frac {1}{{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{2}} \,d x } \]

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="maxima")

[Out]

1/16*(c*d^2*x^3 + 3*c^2*d*x^2 + 4*a*c*d + 2*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^
2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2) + 1/16*integrate((c*d^2*x^2 +
2*c^2*d*x + 2*c^3 + 12*a*d^2)/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)/(a*c^4 + 4*a^2*c*d^2)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=-\frac {\frac {{\left (c d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} - 2 \, c^{2} d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )} + 2 \, c^{3} + 12 \, a d^{2}\right )} \log \left (x + \sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{3} - 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}} - \frac {{\left (c d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )} + 2 \, c^{3} + 12 \, a d^{2}\right )} \log \left (x - \sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{3} + 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}} + \frac {{\left (c d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} - 2 \, c^{2} d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )} + 2 \, c^{3} + 12 \, a d^{2}\right )} \log \left (x + \sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{3} - 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}} - \frac {{\left (c d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )} + 2 \, c^{3} + 12 \, a d^{2}\right )} \log \left (x - \sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{3} + 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}}}{64 \, {\left (a c^{4} + 4 \, a^{2} c d^{2}\right )}} + \frac {c d^{2} x^{3} + 3 \, c^{2} d x^{2} + 2 \, c^{3} x + 4 \, a d^{2} x + 4 \, a c d}{16 \, {\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )} {\left (a c^{4} + 4 \, a^{2} c d^{2}\right )}} \]

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="giac")

[Out]

-1/64*((c*d^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)^2 - 2*c^2*d*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^
4) + c/d) + 2*c^3 + 12*a*d^2)*log(x + sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)/(d^2*(sqrt((c^2*d^2 + 2*sq
rt(-a*c)*d^3)/d^4) + c/d)^3 - 3*c*d*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)^2 + 2*c^2*(sqrt((c^2*d^2 +
2*sqrt(-a*c)*d^3)/d^4) + c/d)) - (c*d^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*d*(sqrt((c^2*
d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d) + 2*c^3 + 12*a*d^2)*log(x - sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)/
(d^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)^3 + 3*c*d*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)^2
+ 2*c^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)) + (c*d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/
d)^2 - 2*c^2*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d) + 2*c^3 + 12*a*d^2)*log(x + sqrt((c^2*d^2 - 2*sq
rt(-a*c)*d^3)/d^4) + c/d)/(d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)^3 - 3*c*d*(sqrt((c^2*d^2 - 2*sqr
t(-a*c)*d^3)/d^4) + c/d)^2 + 2*c^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)) - (c*d^2*(sqrt((c^2*d^2 - 2
*sqrt(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d) + 2*c^3 + 12*a*d^2)*lo
g(x - sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)/(d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d)^3 + 3*
c*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d)))/
(a*c^4 + 4*a^2*c*d^2) + 1/16*(c*d^2*x^3 + 3*c^2*d*x^2 + 2*c^3*x + 4*a*d^2*x + 4*a*c*d)/((d^2*x^4 + 4*c*d*x^3 +
4*c^2*x^2 + 4*a*c)*(a*c^4 + 4*a^2*c*d^2))

Mupad [B] (verification not implemented)

Time = 12.09 (sec) , antiderivative size = 5844, normalized size of antiderivative = 7.83 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/(4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^2,x)

[Out]

(d/(4*(4*a*d^2 + c^3)) + (d^2*x^3)/(16*a*(4*a*d^2 + c^3)) + (x*(2*a*d^2 + c^3))/(8*a*c*(4*a*d^2 + c^3)) + (3*c
*d*x^2)/(16*a*(4*a*d^2 + c^3)))/(4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3) - atan(((-(a^3*c^11 + 10*c^3*d^3*(-a
^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2
+ 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d
^9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*
c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4
*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 6
4*a^9*c^7*d^6)))^(1/2) - (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c
^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*
d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^
7*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10
+ c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*1i + (-(a^3*c^11 + 10*c^3*d^3*(-a^
9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2
+ 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^
9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c
^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*
c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64
*a^9*c^7*d^6)))^(1/2) + (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^
5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d
^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7
*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 +
c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*1i)/((9*a*d^8 + c^3*d^6)/(512*(a^3*
c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) - (-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c
^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1
/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a
^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 +
16*a^4*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^
9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) - (4096*a^3*c^8*d^
6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 +
10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 1
2*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)
/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 +
8*a^3*c^5*d^2 + 16*a^4*c^2*d^4))) + (-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d
^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*
((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c
^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*
a^4*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^
7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (4096*a^3*c^8*d^6 +
65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 + 10*c
^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^
7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(10
24*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a
^3*c^5*d^2 + 16*a^4*c^2*d^4)))))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 +
72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*2i -
atan(((-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2)
)/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 2097
152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d
^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 - 1
0*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12
*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) - (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5
*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a
^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 +
64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 +
16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))
*1i + (-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))
/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 20971
52*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^
6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 - 10
*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*
a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*
c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^
4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 +
64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 1
6*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*
1i)/((9*a*d^8 + c^3*d^6)/(512*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) - (-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7
)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*
a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1
024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^
10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d
^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*
c^7*d^6)))^(1/2) - (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2
+ 16*a^5*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-
a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5
+ 2304*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*
d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4))) + (-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1
/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*
c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/(1024*
(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))
/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 +
60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*
d^6)))^(1/2) + (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 1
6*a^5*c^2*d^4)))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*
c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 23
04*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6
+ 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))))*(-(a^3*c^11 - 10*c^3*d^3*(-a^9*c^7)^(1/2)
+ 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 - 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10
*d^4 + 64*a^9*c^7*d^6)))^(1/2)*2i