3.1.35 \(\int \frac {A+B x+C x^2}{x^2 (a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=514 \[ -\frac {-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac {\sqrt {c} \left (A \left (3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (-\frac {A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {B \log (x)}{a^2}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

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Rubi [A]  time = 1.49, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1662, 1277, 1281, 1166, 205, 12, 1114, 740, 800, 634, 618, 206, 628} \begin {gather*} -\frac {-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac {\sqrt {c} \left (A \left (3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (-\frac {A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {B \log (x)}{a^2}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

-(3*A*b^2 - 10*a*A*c - a*b*C)/(2*a^2*(b^2 - 4*a*c)*x) + (B*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*
x^2 + c*x^4)) + (A*(b^2 - 2*a*c) - a*b*C + c*(A*b - 2*a*C)*x^2)/(2*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)) - (S
qrt[c]*(A*(3*b^3 - 16*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*a*c*Sqrt[b^2 - 4*a*c]) - a*(b^2 - 12*a*c + b*Sqrt[b
^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sq
rt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(3*A*b^2 - 10*a*A*c - a*b*C - (A*(3*b^3 - 16*a*b*c) - a*(b^2 - 12*a*c)*C
)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^2*(b^2 - 4*a*c)*Sqr
t[b + Sqrt[b^2 - 4*a*c]]) + (b*B*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^
(3/2)) + (B*Log[x])/a^2 - (B*Log[a + b*x^2 + c*x^4])/(4*a^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 206

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

Rule 618

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 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 634

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 740

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1114

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

Rule 1166

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

Rule 1277

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

Rule 1281

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

Rule 1662

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x],
k}, Int[(d*x)^m*Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(a + b*x^2 + c*x^4)^p, x] + Dist[1/d, Int[(d*
x)^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2 + 1}]*(a + b*x^2 + c*x^4)^p, x], x]] /; FreeQ[{
a, b, c, d, m, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac {B}{x \left (a+b x^2+c x^4\right )^2} \, dx+\int \frac {A+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}+B \int \frac {1}{x \left (a+b x^2+c x^4\right )^2} \, dx-\frac {\int \frac {-3 A b^2+10 a A c+a b C-3 c (A b-2 a C) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )+\frac {\int \frac {-A \left (3 b^3-13 a b c\right )+a \left (b^2-6 a c\right ) C-c \left (3 A b^2-10 a A c-a b C\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {B \operatorname {Subst}\left (\int \frac {-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}-\frac {\left (c \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (c \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {B \operatorname {Subst}\left (\int \left (\frac {-b^2+4 a c}{a x}+\frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {B \log (x)}{a^2}-\frac {B \operatorname {Subst}\left (\int \frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {B \log (x)}{a^2}-\frac {B \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}-\frac {\left (b B \left (b^2-6 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {\left (b B \left (b^2-6 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac {\sqrt {c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (3 A b^2-10 a A c-a b C-\frac {A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {B \log (x)}{a^2}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end {align*}

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Mathematica [A]  time = 2.03, size = 559, normalized size = 1.09 \begin {gather*} \frac {\frac {-4 a^2 c (B+C x)+2 a \left (b c x (3 A+x (B+C x))+2 A c^2 x^3+b^2 (B+C x)\right )-2 A b^2 x \left (b+c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}+16 a b c-3 b^3\right )+a C \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )+a C \left (b \sqrt {b^2-4 a c}+12 a c-b^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {B \left (b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt {b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {B \left (b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}+6 a b c-b^3\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {4 A}{x}+4 B \log (x)}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

((-4*A)/x + (-4*a^2*c*(B + C*x) - 2*A*b^2*x*(b + c*x^2) + 2*a*(2*A*c^2*x^3 + b^2*(B + C*x) + b*c*x*(3*A + x*(B
+ C*x))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(A*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a
*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b
- Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(A*(3*b^3 - 16*a*b
*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]) + a*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*C)*ArcTan[(
Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*B*Log[x
] - (B*(b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2]
)/(b^2 - 4*a*c)^(3/2) - (B*(-b^3 + 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2
- 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]

[Out]

IntegrateAlgebraic[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2), x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^2/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 11.55, size = 9015, normalized size = 17.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^2/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/4*B*log(abs(c*x^4 + b*x^2 + a))/a^2 + B*log(abs(x))/a^2 + 1/2*(C*a*b*c*x^4 - 3*A*b^2*c*x^4 + 10*A*a*c^2*x^4
+ B*a*b*c*x^3 + C*a*b^2*x^2 - 3*A*b^3*x^2 - 2*C*a^2*c*x^2 + 11*A*a*b*c*x^2 + B*a*b^2*x - 2*B*a^2*c*x - 2*A*a*
b^2 + 8*A*a^2*c)/((c*x^5 + b*x^3 + a*x)*(a^2*b^2 - 4*a^3*c)) - 1/16*((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^
2*(6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 2
2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 - 20*sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 6*(b^2 - 4*a*c)*b^2*c
^2 + 20*(b^2 - 4*a*c)*a*c^3)*A - (a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^2*(2*a*b^3*c^2 - 8*a^2*b*c^3 - sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*C + 2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^4*b^9*c - 49*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 - 6*sqrt(2)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^4*b^8*c^2 - 6*a^4*b^9*c^2 + 300*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 + 74*sqrt(
2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 + 98*a^
5*b^7*c^3 - 816*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 - 304*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^6*b^4*c^4 - 37*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 600*a^6*b^5*c^4 + 832*sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b*c^5 + 416*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 + 152*sqrt(2)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 + 1632*a^7*b^3*c^5 - 208*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
7*b*c^6 - 1664*a^8*b*c^6 + 6*(b^2 - 4*a*c)*a^4*b^7*c^2 - 74*(b^2 - 4*a*c)*a^5*b^5*c^3 + 304*(b^2 - 4*a*c)*a^6*
b^3*c^4 - 416*(b^2 - 4*a*c)*a^7*b*c^5)*A*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) - 2*(sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^5*b^8*c - 18*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^6*c^2 - 2*sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 - 2*a^5*b^8*c^2 + 120*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^4*c^3 + 2
8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 +
36*a^6*b^6*c^3 - 352*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^2*c^4 - 128*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^7*b^3*c^4 - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 240*a^7*b^4*c^4 + 384*sqrt(2)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*c^5 + 192*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b*c^5 + 64*sqrt(2)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 + 704*a^8*b^2*c^5 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*
c^6 - 768*a^9*c^6 + 2*(b^2 - 4*a*c)*a^5*b^6*c^2 - 28*(b^2 - 4*a*c)*a^6*b^4*c^3 + 128*(b^2 - 4*a*c)*a^7*b^2*c^4
- 192*(b^2 - 4*a*c)*a^8*c^5)*C*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + (6*a^8*b^12*c^4 - 128*a^9*b^10*c
^5 + 1088*a^10*b^8*c^6 - 4608*a^11*b^6*c^7 + 9728*a^12*b^4*c^8 - 8192*a^13*b^2*c^9 - 3*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^12*c^2 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a^9*b^10*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^11*c^3 - 544*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^8*c^4 - 104*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^9*b^9*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c^4 + 2304*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^6*c^5 + 672*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a^10*b^7*c^5 + 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^8*c^5 - 4
864*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^4*c^6 - 1920*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^6 - 336*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^1
0*b^6*c^6 + 4096*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^13*b^2*c^7 + 2048*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^7 + 960*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^11*b^4*c^7 - 1024*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^8 - 6*(b^2
- 4*a*c)*a^8*b^10*c^4 + 104*(b^2 - 4*a*c)*a^9*b^8*c^5 - 672*(b^2 - 4*a*c)*a^10*b^6*c^6 + 1920*(b^2 - 4*a*c)*a^
11*b^4*c^7 - 2048*(b^2 - 4*a*c)*a^12*b^2*c^8)*A - (2*a^9*b^11*c^4 - 56*a^10*b^9*c^5 + 576*a^11*b^7*c^6 - 2816*
a^12*b^5*c^7 + 6656*a^13*b^3*c^8 - 6144*a^14*b*c^9 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a^9*b^11*c^2 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^9*c^3 + 2*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^10*c^3 - 288*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^11*b^7*c^4 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^8*c^4 - sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^4 + 1408*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b
^2 - 4*a*c)*c)*a^12*b^5*c^5 + 384*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^6*c^5 + 24*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^5 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^13*b^3*c^6 - 1280*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b
^4*c^6 - 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^6 + 3072*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^14*b*c^7 + 1536*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^13*b^2*c^7 + 640*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^7 - 768*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^13*b*c^8 - 2*(b^2 - 4*a*c)*a^9*b^9*c^4 + 48*(b^2 - 4*a*c)*a
^10*b^7*c^5 - 384*(b^2 - 4*a*c)*a^11*b^5*c^6 + 1280*(b^2 - 4*a*c)*a^12*b^3*c^7 - 1536*(b^2 - 4*a*c)*a^13*b*c^8
)*C)*arctan(2*sqrt(1/2)*x/sqrt((a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3 + sqrt((a^4*b^5*c - 8*a^5*b^3*c^2 + 1
6*a^6*b*c^3)^2 - 4*(a^5*b^4*c - 8*a^6*b^2*c^2 + 16*a^7*c^3)*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)))/(a^4*
b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)))/((a^7*b^8*c - 16*a^8*b^6*c^2 - 2*a^7*b^7*c^2 + 96*a^9*b^4*c^3 + 24*a^8
*b^5*c^3 + a^7*b^6*c^3 - 256*a^10*b^2*c^4 - 96*a^9*b^3*c^4 - 12*a^8*b^4*c^4 + 256*a^11*c^5 + 128*a^10*b*c^5 +
48*a^9*b^2*c^5 - 64*a^10*c^6)*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*abs(c)) + 1/16*((a^4*b^4*c - 8*a^5*b
^2*c^2 + 16*a^6*c^3)^2*(6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*
a^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 -
6*(b^2 - 4*a*c)*b^2*c^2 + 20*(b^2 - 4*a*c)*a*c^3)*A - (a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^2*(2*a*b^3*c^2
- 8*a^2*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c
- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*C - 2*(3*sqrt(2
)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c - 49*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 - 6*sqrt(
2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 6*a^4*b^9*c^2 + 300*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^6*b^5*c^3 + 74*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 + 3*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c
)*a^4*b^7*c^3 - 98*a^5*b^7*c^3 - 816*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 - 304*sqrt(2)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 37*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 + 600*a^6*b^5*c^
4 + 832*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b*c^5 + 416*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^
2*c^5 + 152*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 - 1632*a^7*b^3*c^5 - 208*sqrt(2)*sqrt(b*c - sq
rt(b^2 - 4*a*c)*c)*a^7*b*c^6 + 1664*a^8*b*c^6 - 6*(b^2 - 4*a*c)*a^4*b^7*c^2 + 74*(b^2 - 4*a*c)*a^5*b^5*c^3 - 3
04*(b^2 - 4*a*c)*a^6*b^3*c^4 + 416*(b^2 - 4*a*c)*a^7*b*c^5)*A*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + 2*
(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^8*c - 18*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^6*c^2 -
2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + 2*a^5*b^8*c^2 + 120*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a^7*b^4*c^3 + 28*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^5*b^6*c^3 - 36*a^6*b^6*c^3 - 352*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^2*c^4 - 128*sqrt(2)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 - 14*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 240*a^7*b
^4*c^4 + 384*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*c^5 + 192*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8
*b*c^5 + 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 - 704*a^8*b^2*c^5 - 96*sqrt(2)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^8*c^6 + 768*a^9*c^6 - 2*(b^2 - 4*a*c)*a^5*b^6*c^2 + 28*(b^2 - 4*a*c)*a^6*b^4*c^3 - 128*(b^2
- 4*a*c)*a^7*b^2*c^4 + 192*(b^2 - 4*a*c)*a^8*c^5)*C*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + (6*a^8*b^12
*c^4 - 128*a^9*b^10*c^5 + 1088*a^10*b^8*c^6 - 4608*a^11*b^6*c^7 + 9728*a^12*b^4*c^8 - 8192*a^13*b^2*c^9 - 3*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^12*c^2 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^9*b^10*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^11*c^3
- 544*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^8*c^4 - 104*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b
^10*c^4 + 2304*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^6*c^5 + 672*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^5 + 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a^9*b^8*c^5 - 4864*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^4*c^6 - 1920*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^6 - 336*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^10*b^6*c^6 + 4096*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^13*b^2*c^7 +
2048*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^7 + 960*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^7 - 1024*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^12*b^2*c^8 - 6*(b^2 - 4*a*c)*a^8*b^10*c^4 + 104*(b^2 - 4*a*c)*a^9*b^8*c^5 - 672*(b^2 - 4*a*c)*a^10*b^6*c^6 +
1920*(b^2 - 4*a*c)*a^11*b^4*c^7 - 2048*(b^2 - 4*a*c)*a^12*b^2*c^8)*A - (2*a^9*b^11*c^4 - 56*a^10*b^9*c^5 + 576
*a^11*b^7*c^6 - 2816*a^12*b^5*c^7 + 6656*a^13*b^3*c^8 - 6144*a^14*b*c^9 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^9*b^11*c^2 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^9*c^3
+ 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^10*c^3 - 288*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^7*c^4 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10
*b^8*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^4 + 1408*sqrt(2)*sqrt(b^2 - 4*a
*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^5*c^5 + 384*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*a^11*b^6*c^5 + 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^5 - 3328*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^13*b^3*c^6 - 1280*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*a^12*b^4*c^6 - 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^6 + 307
2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^14*b*c^7 + 1536*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c - sqrt(b^2 - 4*a*c)*c)*a^13*b^2*c^7 + 640*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^
3*c^7 - 768*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^13*b*c^8 - 2*(b^2 - 4*a*c)*a^9*b^9*c^4
+ 48*(b^2 - 4*a*c)*a^10*b^7*c^5 - 384*(b^2 - 4*a*c)*a^11*b^5*c^6 + 1280*(b^2 - 4*a*c)*a^12*b^3*c^7 - 1536*(b^
2 - 4*a*c)*a^13*b*c^8)*C)*arctan(2*sqrt(1/2)*x/sqrt((a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3 - sqrt((a^4*b^5*
c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)^2 - 4*(a^5*b^4*c - 8*a^6*b^2*c^2 + 16*a^7*c^3)*(a^4*b^4*c^2 - 8*a^5*b^2*c^3
+ 16*a^6*c^4)))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)))/((a^7*b^8*c - 16*a^8*b^6*c^2 - 2*a^7*b^7*c^2 + 96
*a^9*b^4*c^3 + 24*a^8*b^5*c^3 + a^7*b^6*c^3 - 256*a^10*b^2*c^4 - 96*a^9*b^3*c^4 - 12*a^8*b^4*c^4 + 256*a^11*c^
5 + 128*a^10*b*c^5 + 48*a^9*b^2*c^5 - 64*a^10*c^6)*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*abs(c)) - 1/16*
((b^6*c - 10*a*b^4*c^2 - 2*b^5*c^2 + 24*a^2*b^2*c^3 + 12*a*b^3*c^3 + b^4*c^3 - 6*a*b^2*c^4 + (b^5*c - 10*a*b^3
*c^2 - 2*b^4*c^2 + 24*a^2*b*c^3 + 12*a*b^2*c^3 + b^3*c^3 - 6*a*b*c^4)*sqrt(b^2 - 4*a*c))*B*abs(a^4*b^4*c - 8*a
^5*b^2*c^2 + 16*a^6*c^3) - (a^4*b^10*c^2 - 18*a^5*b^8*c^3 - 2*a^4*b^9*c^3 + 120*a^6*b^6*c^4 + 28*a^5*b^7*c^4 +
a^4*b^8*c^4 - 352*a^7*b^4*c^5 - 128*a^6*b^5*c^5 - 14*a^5*b^6*c^5 + 384*a^8*b^2*c^6 + 192*a^7*b^3*c^6 + 64*a^6
*b^4*c^6 - 96*a^7*b^2*c^7 + (a^4*b^9*c^2 - 14*a^5*b^7*c^3 - 2*a^4*b^8*c^3 + 64*a^6*b^5*c^4 + 20*a^5*b^6*c^4 +
a^4*b^7*c^4 - 96*a^7*b^3*c^5 - 48*a^6*b^4*c^5 - 10*a^5*b^5*c^5 + 24*a^6*b^3*c^6)*sqrt(b^2 - 4*a*c))*B)*log(x^2
+ 1/2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3 + sqrt((a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)^2 - 4*(a^5*
b^4*c - 8*a^6*b^2*c^2 + 16*a^7*c^3)*(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3
+ 16*a^6*c^4))/((a^3*b^4 - 8*a^4*b^2*c - 2*a^3*b^3*c + 16*a^5*c^2 + 8*a^4*b*c^2 + a^3*b^2*c^2 - 4*a^4*c^3)*c^2
*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)) - 1/16*((b^6*c - 10*a*b^4*c^2 - 2*b^5*c^2 + 24*a^2*b^2*c^3 + 12*
a*b^3*c^3 + b^4*c^3 - 6*a*b^2*c^4 - (b^5*c - 10*a*b^3*c^2 - 2*b^4*c^2 + 24*a^2*b*c^3 + 12*a*b^2*c^3 + b^3*c^3
- 6*a*b*c^4)*sqrt(b^2 - 4*a*c))*B*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) - (a^4*b^10*c^2 - 18*a^5*b^8*c^3
- 2*a^4*b^9*c^3 + 120*a^6*b^6*c^4 + 28*a^5*b^7*c^4 + a^4*b^8*c^4 - 352*a^7*b^4*c^5 - 128*a^6*b^5*c^5 - 14*a^5
*b^6*c^5 + 384*a^8*b^2*c^6 + 192*a^7*b^3*c^6 + 64*a^6*b^4*c^6 - 96*a^7*b^2*c^7 - (a^4*b^9*c^2 - 14*a^5*b^7*c^3
- 2*a^4*b^8*c^3 + 64*a^6*b^5*c^4 + 20*a^5*b^6*c^4 + a^4*b^7*c^4 - 96*a^7*b^3*c^5 - 48*a^6*b^4*c^5 - 10*a^5*b^
5*c^5 + 24*a^6*b^3*c^6)*sqrt(b^2 - 4*a*c))*B)*log(x^2 + 1/2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3 - sqrt((
a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)^2 - 4*(a^5*b^4*c - 8*a^6*b^2*c^2 + 16*a^7*c^3)*(a^4*b^4*c^2 - 8*a^5*
b^2*c^3 + 16*a^6*c^4)))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4))/((a^3*b^4 - 8*a^4*b^2*c - 2*a^3*b^3*c + 16
*a^5*c^2 + 8*a^4*b*c^2 + a^3*b^2*c^2 - 4*a^4*c^3)*c^2*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3))

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maple [B]  time = 0.08, size = 2398, normalized size = 4.67

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/x^2/(c*x^4+b*x^2+a)^2,x)

[Out]

-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B/a*b^2-1/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^
(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*(-4*a*c+b^2)^(1/2)*b^2-1/a*c/(4*a*c-b^2)/(16*a*
c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*(-4*a
*c+b^2)^(1/2)*b^2-16/a*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2
)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*(-4*a*c+b^2)^(1/2)*b+3/a^2*c/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*(-4*a*c+b^2)^(1/2)*b^3+3/
a^2*c/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(
1/2))*c)^(1/2)*c*x)*A*(-4*a*c+b^2)^(1/2)*b^3-16/a*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*(-4*a*c+b^2)^(1/2)*b+B/a^2*ln(x)-1/(c*x^4+b
*x^2+a)/(4*a*c-b^2)*A/a*c^2*x^3+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A/a^2*b^3*x+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A/
a^2*b^2*c*x^3-3/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A/a*b*c*x+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B*c-1/2/(c*x^4+b*x^2+a)/
(4*a*c-b^2)*B/a*b*c*x^2-1/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(
1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*b^3+22/a*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)
^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*b^2-3/a^2*c/(4*a*c-b^2)/(16*a*c-4*b^2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*b^4+3/a^2*c/(
4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*c*x)*A*b^4-22/a*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A*b^2+1/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*b^3+6/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(-2*
c*x^2-b+(-4*a*c+b^2)^(1/2))*B*(-4*a*c+b^2)^(1/2)*b-6/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(2*c*x^2+b+(-4*a*c+b^2)^
(1/2))*B*(-4*a*c+b^2)^(1/2)*b+12*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
anh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*(-4*a*c+b^2)^(1/2)+12*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1
/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*(-4*a*c+b^2)^(1/2)
-4*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*c*x)*C*b+4*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*C*b+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*c*C-1/2/a/(c*x^4+b*x^2+a)*c/(4*a
*c-b^2)*x^3*C*b-1/a^2/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(-2*c*x^2-b+(-4*a*c+b^2)^(1/2))*B*(-4*a*c+b^2)^(1/2)*b^3+1/
a^2/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(2*c*x^2+b+(-4*a*c+b^2)^(1/2))*B*(-4*a*c+b^2)^(1/2)*b^3+8/a*c/(4*a*c-b^2)/(16
*a*c-4*b^2)*ln(2*c*x^2+b+(-4*a*c+b^2)^(1/2))*b^2*B+8/a*c/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(-2*c*x^2-b+(-4*a*c+b^2)
^(1/2))*b^2*B+40*c^3/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A-40*c^3/(4*a*c-b^2)/(16*a*c-4*b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*A-16*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(2*c*x^2+b+(-4*a*
c+b^2)^(1/2))*B-16*c^2/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(-2*c*x^2-b+(-4*a*c+b^2)^(1/2))*B-1/2/a/(c*x^4+b*x^2+a)/(4
*a*c-b^2)*x*C*b^2-1/a^2/(4*a*c-b^2)/(16*a*c-4*b^2)*ln(2*c*x^2+b+(-4*a*c+b^2)^(1/2))*b^4*B-1/a^2/(4*a*c-b^2)/(1
6*a*c-4*b^2)*ln(-2*c*x^2-b+(-4*a*c+b^2)^(1/2))*b^4*B-A/a^2/x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^2/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*(B*a*b*c*x^3 + (10*A*a*c^2 + (C*a*b - 3*A*b^2)*c)*x^4 - 2*A*a*b^2 + 8*A*a^2*c + (C*a*b^2 - 3*A*b^3 - (2*C*
a^2 - 11*A*a*b)*c)*x^2 + (B*a*b^2 - 2*B*a^2*c)*x)/((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (
a^3*b^2 - 4*a^4*c)*x) + 1/2*integrate((C*a*b^2 - 3*A*b^3 - 2*(B*b^2*c - 4*B*a*c^2)*x^3 + (10*A*a*c^2 + (C*a*b
- 3*A*b^2)*c)*x^2 - (6*C*a^2 - 13*A*a*b)*c - 2*(B*b^3 - 5*B*a*b*c)*x)/(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3
*c) + B*log(x)/a^2

________________________________________________________________________________________

mupad [B]  time = 2.47, size = 8684, normalized size = 16.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x)

[Out]

symsum(log(root(1572864*a^10*b^2*c^5*z^4 - 983040*a^9*b^4*c^4*z^4 + 327680*a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2
*z^4 + 6144*a^6*b^10*c*z^4 - 1048576*a^11*c^6*z^4 - 256*a^5*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^
7*b^4*c^4*z^3 + 327680*B*a^6*b^6*c^3*z^3 - 61440*B*a^5*b^8*c^2*z^3 + 6144*B*a^4*b^10*c*z^3 - 1048576*B*a^9*c^6
*z^3 - 256*B*a^3*b^12*z^3 - 2432*A*C*a^2*b^10*c*z^2 - 491520*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2
- 129024*A*C*a^4*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c^2*z^2 + 96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C
^2*a^3*b^9*c*z^2 + 1536*B^2*a^2*b^10*c*z^2 - 430080*A^2*a^6*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7
*c^6*z^2 - 61440*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a^5*b^5*c^3*z^2 - 4608*C^2*a^4*b^7*c^2*z^2 + 516096*B^2*a^6*b
^2*c^5*z^2 - 288768*B^2*a^5*b^4*c^4*z^2 + 88576*B^2*a^4*b^6*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a
^5*b^3*c^5*z^2 - 483840*A^2*a^4*b^5*c^4*z^2 + 170496*A^2*a^3*b^7*c^3*z^2 - 33232*A^2*a^2*b^9*c^2*z^2 - 64*B^2*
a*b^12*z^2 - 393216*B^2*a^7*c^6*z^2 - 16*C^2*a^2*b^11*z^2 - 144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36
864*A*B*C*a^3*b^4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z + 288*A*B*C*a*b^8*c^2*z + 3072*B*C^2*a^5*b*c^5*z - 138240*A
^2*B*a^4*b*c^6*z + 7344*A^2*B*a*b^7*c^3*z + 122880*A*B*C*a^5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*
b^5*c^3*z - 48*B*C^2*a^2*b^7*c^2*z + 131328*A^2*B*a^3*b^3*c^5*z - 46656*A^2*B*a^2*b^5*c^4*z + 61440*B^3*a^4*b^
2*c^5*z - 21504*B^3*a^3*b^4*c^4*z + 3328*B^3*a^2*b^6*c^3*z - 192*B^3*a*b^8*c^2*z - 432*A^2*B*b^9*c^2*z - 65536
*B^3*a^5*c^6*z - 5568*A*B^2*C*a^2*b^2*c^5 + 496*A*B^2*C*a*b^4*c^4 + 1104*B^2*C^2*a^2*b^3*c^4 - 3264*A^2*C^2*a^
2*b^2*c^5 - 3072*B^2*C^2*a^3*b*c^5 - 100*B^2*C^2*a*b^5*c^3 + 2070*A^2*C^2*a*b^4*c^4 - 1840*A*C^3*a^2*b^3*c^4 -
7680*A^2*B^2*a^2*b*c^6 + 3152*A^2*B^2*a*b^3*c^5 + 15200*A^3*C*a^2*b*c^6 - 6192*A^3*C*a*b^3*c^5 + 5472*A*C^3*a
^3*b*c^5 + 150*A*C^3*a*b^5*c^3 + 15360*A*B^2*C*a^3*c^6 - 144*B^4*a*b^4*c^4 + 4200*A^4*a*b^2*c^6 + 630*A^3*C*b^
5*c^4 + 360*C^4*a^3*b^2*c^4 - 25*C^4*a^2*b^4*c^3 + 1536*B^4*a^2*b^2*c^5 - 225*A^2*C^2*b^6*c^3 - 7200*A^2*C^2*a
^3*c^6 - 324*A^2*B^2*b^5*c^4 - 1296*C^4*a^4*c^5 - 4096*B^4*a^3*c^6 - 441*A^4*b^4*c^5 - 10000*A^4*a^2*c^7, z, k
)*(root(1572864*a^10*b^2*c^5*z^4 - 983040*a^9*b^4*c^4*z^4 + 327680*a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2*z^4 + 6
144*a^6*b^10*c*z^4 - 1048576*a^11*c^6*z^4 - 256*a^5*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^7*b^4*c^
4*z^3 + 327680*B*a^6*b^6*c^3*z^3 - 61440*B*a^5*b^8*c^2*z^3 + 6144*B*a^4*b^10*c*z^3 - 1048576*B*a^9*c^6*z^3 - 2
56*B*a^3*b^12*z^3 - 2432*A*C*a^2*b^10*c*z^2 - 491520*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2 - 129024
*A*C*a^4*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c^2*z^2 + 96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C^2*a^3*b
^9*c*z^2 + 1536*B^2*a^2*b^10*c*z^2 - 430080*A^2*a^6*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7*c^6*z^2
- 61440*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a^5*b^5*c^3*z^2 - 4608*C^2*a^4*b^7*c^2*z^2 + 516096*B^2*a^6*b^2*c^5*z
^2 - 288768*B^2*a^5*b^4*c^4*z^2 + 88576*B^2*a^4*b^6*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a^5*b^3*c
^5*z^2 - 483840*A^2*a^4*b^5*c^4*z^2 + 170496*A^2*a^3*b^7*c^3*z^2 - 33232*A^2*a^2*b^9*c^2*z^2 - 64*B^2*a*b^12*z
^2 - 393216*B^2*a^7*c^6*z^2 - 16*C^2*a^2*b^11*z^2 - 144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36864*A*B*
C*a^3*b^4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z + 288*A*B*C*a*b^8*c^2*z + 3072*B*C^2*a^5*b*c^5*z - 138240*A^2*B*a^4
*b*c^6*z + 7344*A^2*B*a*b^7*c^3*z + 122880*A*B*C*a^5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*b^5*c^3*
z - 48*B*C^2*a^2*b^7*c^2*z + 131328*A^2*B*a^3*b^3*c^5*z - 46656*A^2*B*a^2*b^5*c^4*z + 61440*B^3*a^4*b^2*c^5*z
- 21504*B^3*a^3*b^4*c^4*z + 3328*B^3*a^2*b^6*c^3*z - 192*B^3*a*b^8*c^2*z - 432*A^2*B*b^9*c^2*z - 65536*B^3*a^5
*c^6*z - 5568*A*B^2*C*a^2*b^2*c^5 + 496*A*B^2*C*a*b^4*c^4 + 1104*B^2*C^2*a^2*b^3*c^4 - 3264*A^2*C^2*a^2*b^2*c^
5 - 3072*B^2*C^2*a^3*b*c^5 - 100*B^2*C^2*a*b^5*c^3 + 2070*A^2*C^2*a*b^4*c^4 - 1840*A*C^3*a^2*b^3*c^4 - 7680*A^
2*B^2*a^2*b*c^6 + 3152*A^2*B^2*a*b^3*c^5 + 15200*A^3*C*a^2*b*c^6 - 6192*A^3*C*a*b^3*c^5 + 5472*A*C^3*a^3*b*c^5
+ 150*A*C^3*a*b^5*c^3 + 15360*A*B^2*C*a^3*c^6 - 144*B^4*a*b^4*c^4 + 4200*A^4*a*b^2*c^6 + 630*A^3*C*b^5*c^4 +
360*C^4*a^3*b^2*c^4 - 25*C^4*a^2*b^4*c^3 + 1536*B^4*a^2*b^2*c^5 - 225*A^2*C^2*b^6*c^3 - 7200*A^2*C^2*a^3*c^6 -
324*A^2*B^2*b^5*c^4 - 1296*C^4*a^4*c^5 - 4096*B^4*a^3*c^6 - 441*A^4*b^4*c^5 - 10000*A^4*a^2*c^7, z, k)*(root(
1572864*a^10*b^2*c^5*z^4 - 983040*a^9*b^4*c^4*z^4 + 327680*a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2*z^4 + 6144*a^6*
b^10*c*z^4 - 1048576*a^11*c^6*z^4 - 256*a^5*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^7*b^4*c^4*z^3 +
327680*B*a^6*b^6*c^3*z^3 - 61440*B*a^5*b^8*c^2*z^3 + 6144*B*a^4*b^10*c*z^3 - 1048576*B*a^9*c^6*z^3 - 256*B*a^3
*b^12*z^3 - 2432*A*C*a^2*b^10*c*z^2 - 491520*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2 - 129024*A*C*a^4
*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c^2*z^2 + 96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C^2*a^3*b^9*c*z^2
+ 1536*B^2*a^2*b^10*c*z^2 - 430080*A^2*a^6*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7*c^6*z^2 - 61440
*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a^5*b^5*c^3*z^2 - 4608*C^2*a^4*b^7*c^2*z^2 + 516096*B^2*a^6*b^2*c^5*z^2 - 288
768*B^2*a^5*b^4*c^4*z^2 + 88576*B^2*a^4*b^6*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a^5*b^3*c^5*z^2 -
483840*A^2*a^4*b^5*c^4*z^2 + 170496*A^2*a^3*b^7*c^3*z^2 - 33232*A^2*a^2*b^9*c^2*z^2 - 64*B^2*a*b^12*z^2 - 393
216*B^2*a^7*c^6*z^2 - 16*C^2*a^2*b^11*z^2 - 144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36864*A*B*C*a^3*b^
4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z + 288*A*B*C*a*b^8*c^2*z + 3072*B*C^2*a^5*b*c^5*z - 138240*A^2*B*a^4*b*c^6*z
+ 7344*A^2*B*a*b^7*c^3*z + 122880*A*B*C*a^5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*b^5*c^3*z - 48*B
*C^2*a^2*b^7*c^2*z + 131328*A^2*B*a^3*b^3*c^5*z - 46656*A^2*B*a^2*b^5*c^4*z + 61440*B^3*a^4*b^2*c^5*z - 21504*
B^3*a^3*b^4*c^4*z + 3328*B^3*a^2*b^6*c^3*z - 192*B^3*a*b^8*c^2*z - 432*A^2*B*b^9*c^2*z - 65536*B^3*a^5*c^6*z -
5568*A*B^2*C*a^2*b^2*c^5 + 496*A*B^2*C*a*b^4*c^4 + 1104*B^2*C^2*a^2*b^3*c^4 - 3264*A^2*C^2*a^2*b^2*c^5 - 3072
*B^2*C^2*a^3*b*c^5 - 100*B^2*C^2*a*b^5*c^3 + 2070*A^2*C^2*a*b^4*c^4 - 1840*A*C^3*a^2*b^3*c^4 - 7680*A^2*B^2*a^
2*b*c^6 + 3152*A^2*B^2*a*b^3*c^5 + 15200*A^3*C*a^2*b*c^6 - 6192*A^3*C*a*b^3*c^5 + 5472*A*C^3*a^3*b*c^5 + 150*A
*C^3*a*b^5*c^3 + 15360*A*B^2*C*a^3*c^6 - 144*B^4*a*b^4*c^4 + 4200*A^4*a*b^2*c^6 + 630*A^3*C*b^5*c^4 + 360*C^4*
a^3*b^2*c^4 - 25*C^4*a^2*b^4*c^3 + 1536*B^4*a^2*b^2*c^5 - 225*A^2*C^2*b^6*c^3 - 7200*A^2*C^2*a^3*c^6 - 324*A^2
*B^2*b^5*c^4 - 1296*C^4*a^4*c^5 - 4096*B^4*a^3*c^6 - 441*A^4*b^4*c^5 - 10000*A^4*a^2*c^7, z, k)*((x*(983040*B*
a^9*c^8 + 192*B*a^3*b^12*c^2 - 4736*B*a^4*b^10*c^3 + 48896*B*a^5*b^8*c^4 - 270336*B*a^6*b^6*c^5 + 843776*B*a^7
*b^4*c^6 - 1409024*B*a^8*b^2*c^7))/(16*(a^4*b^8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^
3)) - (10240*A*a^8*c^7 + 7168*C*a^8*b*c^6 - 48*A*a^3*b^10*c^2 + 832*A*a^4*b^8*c^3 - 5536*A*a^5*b^6*c^4 + 17280
*A*a^6*b^4*c^5 - 24064*A*a^7*b^2*c^6 + 16*C*a^4*b^9*c^2 - 304*C*a^5*b^7*c^3 + 2112*C*a^6*b^5*c^4 - 6400*C*a^7*
b^3*c^5)/(8*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)) + (root(1572864*a^10*b^2*c^5*z^4 - 983040*
a^9*b^4*c^4*z^4 + 327680*a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2*z^4 + 6144*a^6*b^10*c*z^4 - 1048576*a^11*c^6*z^4
- 256*a^5*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^7*b^4*c^4*z^3 + 327680*B*a^6*b^6*c^3*z^3 - 61440*B
*a^5*b^8*c^2*z^3 + 6144*B*a^4*b^10*c*z^3 - 1048576*B*a^9*c^6*z^3 - 256*B*a^3*b^12*z^3 - 2432*A*C*a^2*b^10*c*z^
2 - 491520*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2 - 129024*A*C*a^4*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c
^2*z^2 + 96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C^2*a^3*b^9*c*z^2 + 1536*B^2*a^2*b^10*c*z^2 - 43008
0*A^2*a^6*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7*c^6*z^2 - 61440*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a
^5*b^5*c^3*z^2 - 4608*C^2*a^4*b^7*c^2*z^2 + 516096*B^2*a^6*b^2*c^5*z^2 - 288768*B^2*a^5*b^4*c^4*z^2 + 88576*B^
2*a^4*b^6*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a^5*b^3*c^5*z^2 - 483840*A^2*a^4*b^5*c^4*z^2 + 1704
96*A^2*a^3*b^7*c^3*z^2 - 33232*A^2*a^2*b^9*c^2*z^2 - 64*B^2*a*b^12*z^2 - 393216*B^2*a^7*c^6*z^2 - 16*C^2*a^2*b
^11*z^2 - 144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36864*A*B*C*a^3*b^4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z
+ 288*A*B*C*a*b^8*c^2*z + 3072*B*C^2*a^5*b*c^5*z - 138240*A^2*B*a^4*b*c^6*z + 7344*A^2*B*a*b^7*c^3*z + 122880
*A*B*C*a^5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*b^5*c^3*z - 48*B*C^2*a^2*b^7*c^2*z + 131328*A^2*B*
a^3*b^3*c^5*z - 46656*A^2*B*a^2*b^5*c^4*z + 61440*B^3*a^4*b^2*c^5*z - 21504*B^3*a^3*b^4*c^4*z + 3328*B^3*a^2*b
^6*c^3*z - 192*B^3*a*b^8*c^2*z - 432*A^2*B*b^9*c^2*z - 65536*B^3*a^5*c^6*z - 5568*A*B^2*C*a^2*b^2*c^5 + 496*A*
B^2*C*a*b^4*c^4 + 1104*B^2*C^2*a^2*b^3*c^4 - 3264*A^2*C^2*a^2*b^2*c^5 - 3072*B^2*C^2*a^3*b*c^5 - 100*B^2*C^2*a
*b^5*c^3 + 2070*A^2*C^2*a*b^4*c^4 - 1840*A*C^3*a^2*b^3*c^4 - 7680*A^2*B^2*a^2*b*c^6 + 3152*A^2*B^2*a*b^3*c^5 +
15200*A^3*C*a^2*b*c^6 - 6192*A^3*C*a*b^3*c^5 + 5472*A*C^3*a^3*b*c^5 + 150*A*C^3*a*b^5*c^3 + 15360*A*B^2*C*a^3
*c^6 - 144*B^4*a*b^4*c^4 + 4200*A^4*a*b^2*c^6 + 630*A^3*C*b^5*c^4 + 360*C^4*a^3*b^2*c^4 - 25*C^4*a^2*b^4*c^3 +
1536*B^4*a^2*b^2*c^5 - 225*A^2*C^2*b^6*c^3 - 7200*A^2*C^2*a^3*c^6 - 324*A^2*B^2*b^5*c^4 - 1296*C^4*a^4*c^5 -
4096*B^4*a^3*c^6 - 441*A^4*b^4*c^5 - 10000*A^4*a^2*c^7, z, k)*x*(1310720*a^11*c^8 + 384*a^5*b^12*c^2 - 8960*a^
6*b^10*c^3 + 87040*a^7*b^8*c^4 - 450560*a^8*b^6*c^5 + 1310720*a^9*b^4*c^6 - 2031616*a^10*b^2*c^7))/(16*(a^4*b^
8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3))) + (5120*A*B*a^6*c^7 + 832*A*B*a^2*b^8*c^3
- 5392*A*B*a^3*b^6*c^4 + 15744*A*B*a^4*b^4*c^5 - 18944*A*B*a^5*b^2*c^6 + 16*B*C*a^2*b^9*c^2 - 304*B*C*a^3*b^7
*c^3 + 2064*B*C*a^4*b^5*c^4 - 5888*B*C*a^5*b^3*c^5 - 48*A*B*a*b^10*c^2 + 5888*B*C*a^6*b*c^6)/(8*(a^4*b^6 - 64*
a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)) + (x*(144*A^2*b^13*c^2 + 245760*B^2*a^7*c^8 + 33304*A^2*a^2*b^9*c^4
- 171768*A^2*a^3*b^7*c^5 + 492320*A^2*a^4*b^5*c^6 - 742016*A^2*a^5*b^3*c^7 - 128*B^2*a^2*b^10*c^3 + 2912*B^2*a
^3*b^8*c^4 - 26560*B^2*a^4*b^6*c^5 + 120832*B^2*a^5*b^4*c^6 - 273408*B^2*a^6*b^2*c^7 + 16*C^2*a^2*b^11*c^2 - 4
32*C^2*a^3*b^9*c^3 + 4616*C^2*a^4*b^7*c^4 - 24032*C^2*a^5*b^5*c^5 + 60800*C^2*a^6*b^3*c^6 - 276480*A*C*a^7*c^8
- 3408*A^2*a*b^11*c^3 + 458240*A^2*a^6*b*c^8 - 59904*C^2*a^7*b*c^7 + 2432*A*C*a^2*b^10*c^3 - 24816*A*C*a^3*b^
8*c^4 + 129952*A*C*a^4*b^6*c^5 - 365440*A*C*a^5*b^4*c^6 + 515584*A*C*a^6*b^2*c^7 - 96*A*C*a*b^12*c^2))/(16*(a^
4*b^8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3))) + (216*C^3*a^5*c^6 + 63*A^3*a^2*b^3*c
^6 - 30*C^3*a^4*b^2*c^5 + 4480*A*B^2*a^4*c^7 + 600*A^2*C*a^4*c^7 - 300*A^3*a^3*b*c^7 - 144*A*B^2*a*b^6*c^4 - 5
64*A*C^2*a^4*b*c^6 + 1408*B^2*C*a^4*b*c^6 + 1536*A*B^2*a^2*b^4*c^5 - 4984*A*B^2*a^3*b^2*c^6 + 105*A*C^2*a^3*b^
3*c^5 - 45*A^2*C*a^2*b^4*c^5 + 102*A^2*C*a^3*b^2*c^6 + 48*B^2*C*a^2*b^5*c^4 - 532*B^2*C*a^3*b^3*c^5)/(8*(a^4*b
^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)) + (x*(20480*B^3*a^5*c^8 + 192*B^3*a^2*b^6*c^5 + 1216*B^3*a^3
*b^4*c^6 - 11008*B^3*a^4*b^2*c^7 + 360*A^2*B*b^9*c^4 - 32*B^3*a*b^8*c^4 - 6072*A^2*B*a*b^7*c^5 + 112320*A^2*B*
a^4*b*c^8 - 2496*B*C^2*a^5*b*c^7 + 38284*A^2*B*a^2*b^5*c^6 - 107104*A^2*B*a^3*b^3*c^7 + 40*B*C^2*a^2*b^7*c^4 -
508*B*C^2*a^3*b^5*c^5 + 2016*B*C^2*a^4*b^3*c^6 - 99840*A*B*C*a^5*c^8 - 240*A*B*C*a*b^8*c^4 + 4448*A*B*C*a^2*b
^6*c^5 - 30176*A*B*C*a^3*b^4*c^6 + 89856*A*B*C*a^4*b^2*c^7))/(16*(a^4*b^8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^
6*b^4*c^2 - 256*a^7*b^2*c^3))) - (63*A^3*B*b^3*c^6 - 640*A*B^3*a^2*c^7 + 216*B*C^3*a^3*c^6 + 600*A^2*B*C*a^2*c
^7 - 45*A^2*B*C*b^4*c^5 + 136*A*B^3*a*b^2*c^6 - 20*B^3*C*a*b^3*c^5 + 128*B^3*C*a^2*b*c^6 - 30*B*C^3*a^2*b^2*c^
5 - 300*A^3*B*a*b*c^7 + 105*A*B*C^2*a*b^3*c^5 - 564*A*B*C^2*a^2*b*c^6 + 102*A^2*B*C*a*b^2*c^6)/(8*(a^4*b^6 - 6
4*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)) + (x*(10000*A^4*a^2*c^9 + 441*A^4*b^4*c^7 + 1296*C^4*a^4*c^7 + 216
*A^2*B^2*b^5*c^6 + 7200*A^2*C^2*a^3*c^8 + 225*A^2*C^2*b^6*c^5 + 256*B^4*a^2*b^2*c^7 + 25*C^4*a^2*b^4*c^5 - 360
*C^4*a^3*b^2*c^6 - 630*A^3*C*b^5*c^6 - 4200*A^4*a*b^2*c^8 - 48*B^4*a*b^4*c^6 - 7680*A*B^2*C*a^3*c^8 - 150*A*C^
3*a*b^5*c^5 - 5472*A*C^3*a^3*b*c^7 + 6192*A^3*C*a*b^3*c^7 - 15200*A^3*C*a^2*b*c^8 - 2160*A^2*B^2*a*b^3*c^7 + 5
440*A^2*B^2*a^2*b*c^8 + 1840*A*C^3*a^2*b^3*c^6 - 2070*A^2*C^2*a*b^4*c^6 + 960*B^2*C^2*a^3*b*c^7 + 3264*A^2*C^2
*a^2*b^2*c^7 - 176*B^2*C^2*a^2*b^3*c^6 - 144*A*B^2*C*a*b^4*c^6 + 2240*A*B^2*C*a^2*b^2*c^7))/(16*(a^4*b^8 + 256
*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3)))*root(1572864*a^10*b^2*c^5*z^4 - 983040*a^9*b^4*c
^4*z^4 + 327680*a^8*b^6*c^3*z^4 - 61440*a^7*b^8*c^2*z^4 + 6144*a^6*b^10*c*z^4 - 1048576*a^11*c^6*z^4 - 256*a^5
*b^12*z^4 + 1572864*B*a^8*b^2*c^5*z^3 - 983040*B*a^7*b^4*c^4*z^3 + 327680*B*a^6*b^6*c^3*z^3 - 61440*B*a^5*b^8*
c^2*z^3 + 6144*B*a^4*b^10*c*z^3 - 1048576*B*a^9*c^6*z^3 - 256*B*a^3*b^12*z^3 - 2432*A*C*a^2*b^10*c*z^2 - 49152
0*A*C*a^6*b^2*c^5*z^2 + 358400*A*C*a^5*b^4*c^4*z^2 - 129024*A*C*a^4*b^6*c^3*z^2 + 24768*A*C*a^3*b^8*c^2*z^2 +
96*A*C*a*b^12*z^2 + 61440*C^2*a^7*b*c^5*z^2 + 432*C^2*a^3*b^9*c*z^2 + 1536*B^2*a^2*b^10*c*z^2 - 430080*A^2*a^6
*b*c^6*z^2 + 3408*A^2*a*b^11*c*z^2 + 245760*A*C*a^7*c^6*z^2 - 61440*C^2*a^6*b^3*c^4*z^2 + 24064*C^2*a^5*b^5*c^
3*z^2 - 4608*C^2*a^4*b^7*c^2*z^2 + 516096*B^2*a^6*b^2*c^5*z^2 - 288768*B^2*a^5*b^4*c^4*z^2 + 88576*B^2*a^4*b^6
*c^3*z^2 - 15744*B^2*a^3*b^8*c^2*z^2 + 716800*A^2*a^5*b^3*c^5*z^2 - 483840*A^2*a^4*b^5*c^4*z^2 + 170496*A^2*a^
3*b^7*c^3*z^2 - 33232*A^2*a^2*b^9*c^2*z^2 - 64*B^2*a*b^12*z^2 - 393216*B^2*a^7*c^6*z^2 - 16*C^2*a^2*b^11*z^2 -
144*A^2*b^13*z^2 - 110592*A*B*C*a^4*b^2*c^5*z + 36864*A*B*C*a^3*b^4*c^4*z - 5376*A*B*C*a^2*b^6*c^3*z + 288*A*
B*C*a*b^8*c^2*z + 3072*B*C^2*a^5*b*c^5*z - 138240*A^2*B*a^4*b*c^6*z + 7344*A^2*B*a*b^7*c^3*z + 122880*A*B*C*a^
5*c^6*z - 2304*B*C^2*a^4*b^3*c^4*z + 576*B*C^2*a^3*b^5*c^3*z - 48*B*C^2*a^2*b^7*c^2*z + 131328*A^2*B*a^3*b^3*c
^5*z - 46656*A^2*B*a^2*b^5*c^4*z + 61440*B^3*a^4*b^2*c^5*z - 21504*B^3*a^3*b^4*c^4*z + 3328*B^3*a^2*b^6*c^3*z
- 192*B^3*a*b^8*c^2*z - 432*A^2*B*b^9*c^2*z - 65536*B^3*a^5*c^6*z - 5568*A*B^2*C*a^2*b^2*c^5 + 496*A*B^2*C*a*b
^4*c^4 + 1104*B^2*C^2*a^2*b^3*c^4 - 3264*A^2*C^2*a^2*b^2*c^5 - 3072*B^2*C^2*a^3*b*c^5 - 100*B^2*C^2*a*b^5*c^3
+ 2070*A^2*C^2*a*b^4*c^4 - 1840*A*C^3*a^2*b^3*c^4 - 7680*A^2*B^2*a^2*b*c^6 + 3152*A^2*B^2*a*b^3*c^5 + 15200*A^
3*C*a^2*b*c^6 - 6192*A^3*C*a*b^3*c^5 + 5472*A*C^3*a^3*b*c^5 + 150*A*C^3*a*b^5*c^3 + 15360*A*B^2*C*a^3*c^6 - 14
4*B^4*a*b^4*c^4 + 4200*A^4*a*b^2*c^6 + 630*A^3*C*b^5*c^4 + 360*C^4*a^3*b^2*c^4 - 25*C^4*a^2*b^4*c^3 + 1536*B^4
*a^2*b^2*c^5 - 225*A^2*C^2*b^6*c^3 - 7200*A^2*C^2*a^3*c^6 - 324*A^2*B^2*b^5*c^4 - 1296*C^4*a^4*c^5 - 4096*B^4*
a^3*c^6 - 441*A^4*b^4*c^5 - 10000*A^4*a^2*c^7, z, k), k, 1, 4) - (A/a - (x^2*(3*A*b^3 - C*a*b^2 + 2*C*a^2*c -
11*A*a*b*c))/(2*a^2*(4*a*c - b^2)) + (x^4*(10*A*a*c^2 - 3*A*b^2*c + C*a*b*c))/(2*a^2*(4*a*c - b^2)) - (B*x*(2*
a*c - b^2))/(2*a*(4*a*c - b^2)) + (B*b*c*x^3)/(2*a*(4*a*c - b^2)))/(a*x + b*x^3 + c*x^5) + (B*log(x))/a^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/x**2/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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