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

Optimal. Leaf size=403 \[ \frac {\left (4 a^2 c C+A \left (b^3-6 a b c\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {A \log (x)}{a^2}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {B x \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {B \sqrt {c} \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {B \sqrt {c} \left (-b \sqrt {b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.93, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1662, 1251, 822, 800, 634, 618, 206, 628, 12, 1092, 1166, 205} \begin {gather*} \frac {\left (4 a^2 c C+A \left (b^3-6 a b c\right )\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 {A \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac {A \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 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {B x \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 )}+\frac {B \sqrt {c} \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {B \sqrt {c} \left (-b \sqrt {b^2-4 a c}-12 a c+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 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x*(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)*(a + b*x^2 + c*x^4)) + (B*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) -
(B*Sqrt[c]*(b^2 - 12*a*c - b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*S
qrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((A*(b^3 - 6*a*b*c) + 4*a^2*c*C)*ArcTanh[(b + 2*c*
x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*(b^2 - 4*a*c)^(3/2)) + (A*Log[x])/a^2 - (A*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 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 822

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

Rule 1092

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 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 1251

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

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

________________________________________________________________________________________

Mathematica [A]  time = 1.47, size = 458, normalized size = 1.14 \begin {gather*} \frac {-\frac {\left (4 a^2 c C+A \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 )\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 {\left (A \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 )-4 a^2 c C\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 {2 a \left (-A \left (-2 a c+b^2+b c x^2\right )+a b C+2 a c x (B+C x)-b B x \left (b+c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} a B \sqrt {c} \left (b \sqrt {b^2-4 a c}-12 a c+b^2\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} a B \sqrt {c} \left (b \sqrt {b^2-4 a c}+12 a c-b^2\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}}+4 A \log (x)}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [B]  time = 6.55, size = 6022, normalized size = 14.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*A*log(abs(c*x^4 + b*x^2 + a))/a^2 + A*log(abs(x))/a^2 + 1/16*((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)^2*
(2*b^3*c^2 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2
*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*B + 2*(sqrt(2)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c - 18*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^2 - 2*sqrt(2)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^2 - 2*a^4*b^8*c^2 + 120*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*
b^4*c^3 + 28*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4
*b^6*c^3 + 36*a^5*b^6*c^3 - 352*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^4 - 128*sqrt(2)*sqrt(b*c + s
qrt(b^2 - 4*a*c)*c)*a^6*b^3*c^4 - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^4 - 240*a^6*b^4*c^4 + 3
84*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*c^5 + 192*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b*c^5 + 6
4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 + 704*a^7*b^2*c^5 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^7*c^6 - 768*a^8*c^6 + 2*(b^2 - 4*a*c)*a^4*b^6*c^2 - 28*(b^2 - 4*a*c)*a^5*b^4*c^3 + 128*(b^2 - 4*a*c)*
a^6*b^2*c^4 - 192*(b^2 - 4*a*c)*a^7*c^5)*B*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + (2*a^8*b^11*c^4 - 56*
a^9*b^9*c^5 + 576*a^10*b^7*c^6 - 2816*a^11*b^5*c^7 + 6656*a^12*b^3*c^8 - 6144*a^13*b*c^9 - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^11*c^2 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^9*b^9*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c^3 - 288*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^4 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^9*b^8*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^4 + 1408*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^5 + 384*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*a^10*b^6*c^5 + 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^7*c^5 - 33
28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^6 - 1280*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^6 - 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10
*b^5*c^6 + 3072*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^13*b*c^7 + 1536*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^7 + 640*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^11*b^3*c^7 - 768*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b*c^8 - 2*(b^2 - 4*a*
c)*a^8*b^9*c^4 + 48*(b^2 - 4*a*c)*a^9*b^7*c^5 - 384*(b^2 - 4*a*c)*a^10*b^5*c^6 + 1280*(b^2 - 4*a*c)*a^11*b^3*c
^7 - 1536*(b^2 - 4*a*c)*a^12*b*c^8)*B)*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 + s
qrt((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^6*b^8*c - 16*a^7*b^6*c^2 - 2*a^6
*b^7*c^2 + 96*a^8*b^4*c^3 + 24*a^7*b^5*c^3 + a^6*b^6*c^3 - 256*a^9*b^2*c^4 - 96*a^8*b^3*c^4 - 12*a^7*b^4*c^4 +
256*a^10*c^5 + 128*a^9*b*c^5 + 48*a^8*b^2*c^5 - 64*a^9*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*(2*b^3*c^2 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c - sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*B - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c - 18*sqrt(2)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^2 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^2 + 2*a^4*b^8*
c^2 + 120*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^3 + 28*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5
*b^5*c^3 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^3 - 36*a^5*b^6*c^3 - 352*sqrt(2)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a^7*b^2*c^4 - 128*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^4 - 14*sqrt(2)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^4 + 240*a^6*b^4*c^4 + 384*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*c^5 + 192
*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b*c^5 + 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 -
704*a^7*b^2*c^5 - 96*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*c^6 + 768*a^8*c^6 - 2*(b^2 - 4*a*c)*a^4*b^6*c
^2 + 28*(b^2 - 4*a*c)*a^5*b^4*c^3 - 128*(b^2 - 4*a*c)*a^6*b^2*c^4 + 192*(b^2 - 4*a*c)*a^7*c^5)*B*abs(a^4*b^4*c
- 8*a^5*b^2*c^2 + 16*a^6*c^3) + (2*a^8*b^11*c^4 - 56*a^9*b^9*c^5 + 576*a^10*b^7*c^6 - 2816*a^11*b^5*c^7 + 665
6*a^12*b^3*c^8 - 6144*a^13*b*c^9 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^11*c^2 + 28
*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a^8*b^10*c^3 - 288*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c
^4 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^8*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^4 + 1408*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*
b^5*c^5 + 384*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^6*c^5 + 24*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^7*c^5 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a^12*b^3*c^6 - 1280*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^6 - 192*sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^5*c^6 + 3072*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^13*b*c^7 + 1536*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^7 + 6
40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^3*c^7 - 768*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b*c^8 - 2*(b^2 - 4*a*c)*a^8*b^9*c^4 + 48*(b^2 - 4*a*c)*a^9*b^7*c^5 - 384*(b^2
- 4*a*c)*a^10*b^5*c^6 + 1280*(b^2 - 4*a*c)*a^11*b^3*c^7 - 1536*(b^2 - 4*a*c)*a^12*b*c^8)*B)*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^6*b^8*c - 16*a^7*b^6*c^2 - 2*a^6*b^7*c^2 + 96*a^8*b^4*c^3 + 24*a^7*b^5*c^3 + a^6*b^6*c^3
- 256*a^9*b^2*c^4 - 96*a^8*b^3*c^4 - 12*a^7*b^4*c^4 + 256*a^10*c^5 + 128*a^9*b*c^5 + 48*a^8*b^2*c^5 - 64*a^9*
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))*A*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + 4*(a^2*b^3*c^2 - 4*
a^3*b*c^3 - 2*a^2*b^2*c^3 + a^2*b*c^4 + (a^2*b^2*c^2 - 4*a^3*c^3 - 2*a^2*b*c^3 + a^2*c^4)*sqrt(b^2 - 4*a*c))*C
*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))*A + 4*(a^6*b^7*c^3 - 12*a^7*b^5*c^4 - 2*a^6*b^6*c^4 + 48*a^8*b^3*c^5 + 16*a^7*b^4*c^5 + a^6*b^5*c^5
- 64*a^9*b*c^6 - 32*a^8*b^2*c^6 - 8*a^7*b^3*c^6 + 16*a^8*b*c^7 + (a^6*b^6*c^3 - 8*a^7*b^4*c^4 - 2*a^6*b^5*c^4
+ 16*a^8*b^2*c^5 + 8*a^7*b^3*c^5 + a^6*b^4*c^5 - 4*a^7*b^2*c^6)*sqrt(b^2 - 4*a*c))*C)*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))*A*abs(a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3) + 4*(a^2*b^3*c^2 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 +
a^2*b*c^4 - (a^2*b^2*c^2 - 4*a^3*c^3 - 2*a^2*b*c^3 + a^2*c^4)*sqrt(b^2 - 4*a*c))*C*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))*A - 4*(a^6*b^7*c^3
- 12*a^7*b^5*c^4 - 2*a^6*b^6*c^4 + 48*a^8*b^3*c^5 + 16*a^7*b^4*c^5 + a^6*b^5*c^5 - 64*a^9*b*c^6 - 32*a^8*b^2*
c^6 - 8*a^7*b^3*c^6 + 16*a^8*b*c^7 - (a^6*b^6*c^3 - 8*a^7*b^4*c^4 - 2*a^6*b^5*c^4 + 16*a^8*b^2*c^5 + 8*a^7*b^3
*c^5 + a^6*b^4*c^5 - 4*a^7*b^2*c^6)*sqrt(b^2 - 4*a*c))*C)*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))
+ 1/2*(B*a*b*c*x^3 - C*a^2*b + A*a*b^2 - 2*A*a^2*c - (2*C*a^2*c - A*a*b*c)*x^2 + (B*a*b^2 - 2*B*a^2*c)*x)/((c
*x^4 + b*x^2 + a)*(b^2 - 4*a*c)*a^2)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 1603, normalized size = 3.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*B*(-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)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*B*(-4*a*c+b^2)^(1/2)*b^2-1/2/(c*x^4+b*x^2+a)/(4
*a*c-b^2)*B/a*b^2*x-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B/a*b*c*x^3+A/a^2*ln(x)+1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*C*
b+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A*c-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A/a*b*c*x^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))*A*b^4-1/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))*A
*b^4-4*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))*C*(-4*a*c+b^2)^(1/2)+4*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))*C*(-4*a*c+b^2)^(1/2)-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))*A*(-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)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*B*(-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)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*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))*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))*A-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))*A*(-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))*A*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))*A*(-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))*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)*B*b^3+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)*B*b^3-1/2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A
/a*b^2+1/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*C+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)*B*(-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)*b*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))*A*(-4*a*c+b^2)^(1/2)*b+1/(c*x^4+b*x^2+a)/(4*a*c-
b^2)*B*c*x

________________________________________________________________________________________

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

[Out]

1/2*(B*b*c*x^3 - (2*C*a - A*b)*c*x^2 - C*a*b + A*b^2 - 2*A*a*c + (B*b^2 - 2*B*a*c)*x)/((a*b^2*c - 4*a^2*c^2)*x
^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*x^2) + 1/2*integrate((B*a*b*c*x^2 + B*a*b^2 - 6*B*a^2*c - 2*(A*b^
2*c - 4*A*a*c^2)*x^3 - 2*(A*b^3 + (2*C*a^2 - 5*A*a*b)*c)*x)/(c*x^4 + b*x^2 + a), x)/(a^2*b^2 - 4*a^3*c) + A*lo
g(x)/a^2

________________________________________________________________________________________

mupad [B]  time = 1.84, size = 8129, normalized size = 20.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*A*a*c - A*b^2 + C*a*b)/(2*a*(4*a*c - b^2)) + (B*x*(2*a*c - b^2))/(2*a*(4*a*c - b^2)) - (c*x^2*(A*b - 2*C*a
))/(2*a*(4*a*c - b^2)) - (B*b*c*x^3)/(2*a*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) + symsum(log(root(1572864*a^9*b^
2*c^5*z^4 - 983040*a^8*b^4*c^4*z^4 + 327680*a^7*b^6*c^3*z^4 - 61440*a^6*b^8*c^2*z^4 + 6144*a^5*b^10*c*z^4 - 10
48576*a^10*c^6*z^4 - 256*a^4*b^12*z^4 + 1572864*A*a^7*b^2*c^5*z^3 - 983040*A*a^6*b^4*c^4*z^3 + 327680*A*a^5*b^
6*c^3*z^3 - 61440*A*a^4*b^8*c^2*z^3 + 6144*A*a^3*b^10*c*z^3 - 1048576*A*a^8*c^6*z^3 - 256*A*a^2*b^12*z^3 + 983
04*A*C*a^6*b*c^5*z^2 + 256*A*C*a^2*b^9*c*z^2 - 90112*A*C*a^5*b^3*c^4*z^2 + 30720*A*C*a^4*b^5*c^3*z^2 - 4608*A*
C*a^3*b^7*c^2*z^2 + 61440*B^2*a^6*b*c^5*z^2 + 432*B^2*a^2*b^9*c*z^2 + 1536*A^2*a*b^10*c*z^2 + 24576*C^2*a^6*b^
2*c^4*z^2 - 6144*C^2*a^5*b^4*c^3*z^2 + 512*C^2*a^4*b^6*c^2*z^2 - 61440*B^2*a^5*b^3*c^4*z^2 + 24064*B^2*a^4*b^5
*c^3*z^2 - 4608*B^2*a^3*b^7*c^2*z^2 + 516096*A^2*a^5*b^2*c^5*z^2 - 288768*A^2*a^4*b^4*c^4*z^2 + 88576*A^2*a^3*
b^6*c^3*z^2 - 15744*A^2*a^2*b^8*c^2*z^2 - 16*B^2*a*b^11*z^2 - 32768*C^2*a^7*c^5*z^2 - 393216*A^2*a^6*c^6*z^2 -
64*A^2*b^12*z^2 + 49152*A^2*C*a^4*b*c^5*z - 2304*A^2*C*a*b^7*c^2*z + 3072*A*B^2*a^4*b*c^5*z - 48*A*B^2*a*b^7*
c^2*z + 32*B^2*C*a*b^8*c*z - 15872*B^2*C*a^4*b^2*c^4*z + 4992*B^2*C*a^3*b^4*c^3*z - 672*B^2*C*a^2*b^6*c^2*z -
45056*A^2*C*a^3*b^3*c^4*z + 15360*A^2*C*a^2*b^5*c^3*z + 12288*A*C^2*a^4*b^2*c^4*z - 3072*A*C^2*a^3*b^4*c^3*z +
256*A*C^2*a^2*b^6*c^2*z - 2304*A*B^2*a^3*b^3*c^4*z + 576*A*B^2*a^2*b^5*c^3*z + 128*A^2*C*b^9*c*z + 61440*A^3*
a^3*b^2*c^5*z - 21504*A^3*a^2*b^4*c^4*z + 3328*A^3*a*b^6*c^3*z + 18432*B^2*C*a^5*c^5*z - 16384*A*C^2*a^5*c^5*z
- 192*A^3*b^8*c^2*z - 65536*A^3*a^4*c^6*z - 1088*A*B^2*C*a^2*b^2*c^4 + 48*A*B^2*C*a*b^4*c^3 + 240*B^2*C^2*a^2
*b^3*c^3 - 1920*A^2*C^2*a^2*b^2*c^4 - 960*B^2*C^2*a^3*b*c^4 - 16*B^2*C^2*a*b^5*c^2 + 768*A^2*C^2*a*b^4*c^3 - 2
56*A*C^3*a^2*b^3*c^3 - 3072*A^2*B^2*a^2*b*c^5 + 1104*A^2*B^2*a*b^3*c^4 + 6144*A^3*C*a^2*b*c^5 - 2176*A^3*C*a*b
^3*c^4 + 1536*A*C^3*a^3*b*c^4 + 4608*A*B^2*C*a^3*c^5 - 25*B^4*a*b^4*c^3 + 1536*A^4*a*b^2*c^5 + 192*A^3*C*b^5*c
^3 + 360*B^4*a^2*b^2*c^4 - 64*A^2*C^2*b^6*c^2 - 2048*A^2*C^2*a^3*c^5 - 100*A^2*B^2*b^5*c^3 - 256*C^4*a^4*c^4 -
1296*B^4*a^3*c^5 - 144*A^4*b^4*c^4 - 4096*A^4*a^2*c^6, z, k)*(root(1572864*a^9*b^2*c^5*z^4 - 983040*a^8*b^4*c
^4*z^4 + 327680*a^7*b^6*c^3*z^4 - 61440*a^6*b^8*c^2*z^4 + 6144*a^5*b^10*c*z^4 - 1048576*a^10*c^6*z^4 - 256*a^4
*b^12*z^4 + 1572864*A*a^7*b^2*c^5*z^3 - 983040*A*a^6*b^4*c^4*z^3 + 327680*A*a^5*b^6*c^3*z^3 - 61440*A*a^4*b^8*
c^2*z^3 + 6144*A*a^3*b^10*c*z^3 - 1048576*A*a^8*c^6*z^3 - 256*A*a^2*b^12*z^3 + 98304*A*C*a^6*b*c^5*z^2 + 256*A
*C*a^2*b^9*c*z^2 - 90112*A*C*a^5*b^3*c^4*z^2 + 30720*A*C*a^4*b^5*c^3*z^2 - 4608*A*C*a^3*b^7*c^2*z^2 + 61440*B^
2*a^6*b*c^5*z^2 + 432*B^2*a^2*b^9*c*z^2 + 1536*A^2*a*b^10*c*z^2 + 24576*C^2*a^6*b^2*c^4*z^2 - 6144*C^2*a^5*b^4
*c^3*z^2 + 512*C^2*a^4*b^6*c^2*z^2 - 61440*B^2*a^5*b^3*c^4*z^2 + 24064*B^2*a^4*b^5*c^3*z^2 - 4608*B^2*a^3*b^7*
c^2*z^2 + 516096*A^2*a^5*b^2*c^5*z^2 - 288768*A^2*a^4*b^4*c^4*z^2 + 88576*A^2*a^3*b^6*c^3*z^2 - 15744*A^2*a^2*
b^8*c^2*z^2 - 16*B^2*a*b^11*z^2 - 32768*C^2*a^7*c^5*z^2 - 393216*A^2*a^6*c^6*z^2 - 64*A^2*b^12*z^2 + 49152*A^2
*C*a^4*b*c^5*z - 2304*A^2*C*a*b^7*c^2*z + 3072*A*B^2*a^4*b*c^5*z - 48*A*B^2*a*b^7*c^2*z + 32*B^2*C*a*b^8*c*z -
15872*B^2*C*a^4*b^2*c^4*z + 4992*B^2*C*a^3*b^4*c^3*z - 672*B^2*C*a^2*b^6*c^2*z - 45056*A^2*C*a^3*b^3*c^4*z +
15360*A^2*C*a^2*b^5*c^3*z + 12288*A*C^2*a^4*b^2*c^4*z - 3072*A*C^2*a^3*b^4*c^3*z + 256*A*C^2*a^2*b^6*c^2*z - 2
304*A*B^2*a^3*b^3*c^4*z + 576*A*B^2*a^2*b^5*c^3*z + 128*A^2*C*b^9*c*z + 61440*A^3*a^3*b^2*c^5*z - 21504*A^3*a^
2*b^4*c^4*z + 3328*A^3*a*b^6*c^3*z + 18432*B^2*C*a^5*c^5*z - 16384*A*C^2*a^5*c^5*z - 192*A^3*b^8*c^2*z - 65536
*A^3*a^4*c^6*z - 1088*A*B^2*C*a^2*b^2*c^4 + 48*A*B^2*C*a*b^4*c^3 + 240*B^2*C^2*a^2*b^3*c^3 - 1920*A^2*C^2*a^2*
b^2*c^4 - 960*B^2*C^2*a^3*b*c^4 - 16*B^2*C^2*a*b^5*c^2 + 768*A^2*C^2*a*b^4*c^3 - 256*A*C^3*a^2*b^3*c^3 - 3072*
A^2*B^2*a^2*b*c^5 + 1104*A^2*B^2*a*b^3*c^4 + 6144*A^3*C*a^2*b*c^5 - 2176*A^3*C*a*b^3*c^4 + 1536*A*C^3*a^3*b*c^
4 + 4608*A*B^2*C*a^3*c^5 - 25*B^4*a*b^4*c^3 + 1536*A^4*a*b^2*c^5 + 192*A^3*C*b^5*c^3 + 360*B^4*a^2*b^2*c^4 - 6
4*A^2*C^2*b^6*c^2 - 2048*A^2*C^2*a^3*c^5 - 100*A^2*B^2*b^5*c^3 - 256*C^4*a^4*c^4 - 1296*B^4*a^3*c^5 - 144*A^4*
b^4*c^4 - 4096*A^4*a^2*c^6, z, k)*((1032*A*B*a^3*b^5*c^4 - 152*A*B*a^2*b^7*c^3 - 768*B*C*a^6*c^6 - 2944*A*B*a^
4*b^3*c^5 + 16*B*C*a^3*b^6*c^3 - 208*B*C*a^4*b^4*c^4 + 768*B*C*a^5*b^2*c^5 + 8*A*B*a*b^9*c^2 + 2944*A*B*a^5*b*
c^6)/(4*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)) + root(1572864*a^9*b^2*c^5*z^4 - 983040*a^8*b^
4*c^4*z^4 + 327680*a^7*b^6*c^3*z^4 - 61440*a^6*b^8*c^2*z^4 + 6144*a^5*b^10*c*z^4 - 1048576*a^10*c^6*z^4 - 256*
a^4*b^12*z^4 + 1572864*A*a^7*b^2*c^5*z^3 - 983040*A*a^6*b^4*c^4*z^3 + 327680*A*a^5*b^6*c^3*z^3 - 61440*A*a^4*b
^8*c^2*z^3 + 6144*A*a^3*b^10*c*z^3 - 1048576*A*a^8*c^6*z^3 - 256*A*a^2*b^12*z^3 + 98304*A*C*a^6*b*c^5*z^2 + 25
6*A*C*a^2*b^9*c*z^2 - 90112*A*C*a^5*b^3*c^4*z^2 + 30720*A*C*a^4*b^5*c^3*z^2 - 4608*A*C*a^3*b^7*c^2*z^2 + 61440
*B^2*a^6*b*c^5*z^2 + 432*B^2*a^2*b^9*c*z^2 + 1536*A^2*a*b^10*c*z^2 + 24576*C^2*a^6*b^2*c^4*z^2 - 6144*C^2*a^5*
b^4*c^3*z^2 + 512*C^2*a^4*b^6*c^2*z^2 - 61440*B^2*a^5*b^3*c^4*z^2 + 24064*B^2*a^4*b^5*c^3*z^2 - 4608*B^2*a^3*b
^7*c^2*z^2 + 516096*A^2*a^5*b^2*c^5*z^2 - 288768*A^2*a^4*b^4*c^4*z^2 + 88576*A^2*a^3*b^6*c^3*z^2 - 15744*A^2*a
^2*b^8*c^2*z^2 - 16*B^2*a*b^11*z^2 - 32768*C^2*a^7*c^5*z^2 - 393216*A^2*a^6*c^6*z^2 - 64*A^2*b^12*z^2 + 49152*
A^2*C*a^4*b*c^5*z - 2304*A^2*C*a*b^7*c^2*z + 3072*A*B^2*a^4*b*c^5*z - 48*A*B^2*a*b^7*c^2*z + 32*B^2*C*a*b^8*c*
z - 15872*B^2*C*a^4*b^2*c^4*z + 4992*B^2*C*a^3*b^4*c^3*z - 672*B^2*C*a^2*b^6*c^2*z - 45056*A^2*C*a^3*b^3*c^4*z
+ 15360*A^2*C*a^2*b^5*c^3*z + 12288*A*C^2*a^4*b^2*c^4*z - 3072*A*C^2*a^3*b^4*c^3*z + 256*A*C^2*a^2*b^6*c^2*z
- 2304*A*B^2*a^3*b^3*c^4*z + 576*A*B^2*a^2*b^5*c^3*z + 128*A^2*C*b^9*c*z + 61440*A^3*a^3*b^2*c^5*z - 21504*A^3
*a^2*b^4*c^4*z + 3328*A^3*a*b^6*c^3*z + 18432*B^2*C*a^5*c^5*z - 16384*A*C^2*a^5*c^5*z - 192*A^3*b^8*c^2*z - 65
536*A^3*a^4*c^6*z - 1088*A*B^2*C*a^2*b^2*c^4 + 48*A*B^2*C*a*b^4*c^3 + 240*B^2*C^2*a^2*b^3*c^3 - 1920*A^2*C^2*a
^2*b^2*c^4 - 960*B^2*C^2*a^3*b*c^4 - 16*B^2*C^2*a*b^5*c^2 + 768*A^2*C^2*a*b^4*c^3 - 256*A*C^3*a^2*b^3*c^3 - 30
72*A^2*B^2*a^2*b*c^5 + 1104*A^2*B^2*a*b^3*c^4 + 6144*A^3*C*a^2*b*c^5 - 2176*A^3*C*a*b^3*c^4 + 1536*A*C^3*a^3*b
*c^4 + 4608*A*B^2*C*a^3*c^5 - 25*B^4*a*b^4*c^3 + 1536*A^4*a*b^2*c^5 + 192*A^3*C*b^5*c^3 + 360*B^4*a^2*b^2*c^4
- 64*A^2*C^2*b^6*c^2 - 2048*A^2*C^2*a^3*c^5 - 100*A^2*B^2*b^5*c^3 - 256*C^4*a^4*c^4 - 1296*B^4*a^3*c^5 - 144*A
^4*b^4*c^4 - 4096*A^4*a^2*c^6, z, k)*((x*(983040*A*a^8*c^8 - 32768*C*a^8*b*c^7 + 192*A*a^2*b^12*c^2 - 4736*A*a
^3*b^10*c^3 + 48896*A*a^4*b^8*c^4 - 270336*A*a^5*b^6*c^5 + 843776*A*a^6*b^4*c^6 - 1409024*A*a^7*b^2*c^7 - 128*
C*a^4*b^9*c^3 + 2048*C*a^5*b^7*c^4 - 12288*C*a^6*b^5*c^5 + 32768*C*a^7*b^3*c^6))/(16*(a^3*b^8 + 256*a^7*c^4 -
16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3)) - (3584*B*a^7*b*c^6 + 8*B*a^3*b^9*c^2 - 152*B*a^4*b^7*c^3 +
1056*B*a^5*b^5*c^4 - 3200*B*a^6*b^3*c^5)/(4*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)) + (root(15
72864*a^9*b^2*c^5*z^4 - 983040*a^8*b^4*c^4*z^4 + 327680*a^7*b^6*c^3*z^4 - 61440*a^6*b^8*c^2*z^4 + 6144*a^5*b^1
0*c*z^4 - 1048576*a^10*c^6*z^4 - 256*a^4*b^12*z^4 + 1572864*A*a^7*b^2*c^5*z^3 - 983040*A*a^6*b^4*c^4*z^3 + 327
680*A*a^5*b^6*c^3*z^3 - 61440*A*a^4*b^8*c^2*z^3 + 6144*A*a^3*b^10*c*z^3 - 1048576*A*a^8*c^6*z^3 - 256*A*a^2*b^
12*z^3 + 98304*A*C*a^6*b*c^5*z^2 + 256*A*C*a^2*b^9*c*z^2 - 90112*A*C*a^5*b^3*c^4*z^2 + 30720*A*C*a^4*b^5*c^3*z
^2 - 4608*A*C*a^3*b^7*c^2*z^2 + 61440*B^2*a^6*b*c^5*z^2 + 432*B^2*a^2*b^9*c*z^2 + 1536*A^2*a*b^10*c*z^2 + 2457
6*C^2*a^6*b^2*c^4*z^2 - 6144*C^2*a^5*b^4*c^3*z^2 + 512*C^2*a^4*b^6*c^2*z^2 - 61440*B^2*a^5*b^3*c^4*z^2 + 24064
*B^2*a^4*b^5*c^3*z^2 - 4608*B^2*a^3*b^7*c^2*z^2 + 516096*A^2*a^5*b^2*c^5*z^2 - 288768*A^2*a^4*b^4*c^4*z^2 + 88
576*A^2*a^3*b^6*c^3*z^2 - 15744*A^2*a^2*b^8*c^2*z^2 - 16*B^2*a*b^11*z^2 - 32768*C^2*a^7*c^5*z^2 - 393216*A^2*a
^6*c^6*z^2 - 64*A^2*b^12*z^2 + 49152*A^2*C*a^4*b*c^5*z - 2304*A^2*C*a*b^7*c^2*z + 3072*A*B^2*a^4*b*c^5*z - 48*
A*B^2*a*b^7*c^2*z + 32*B^2*C*a*b^8*c*z - 15872*B^2*C*a^4*b^2*c^4*z + 4992*B^2*C*a^3*b^4*c^3*z - 672*B^2*C*a^2*
b^6*c^2*z - 45056*A^2*C*a^3*b^3*c^4*z + 15360*A^2*C*a^2*b^5*c^3*z + 12288*A*C^2*a^4*b^2*c^4*z - 3072*A*C^2*a^3
*b^4*c^3*z + 256*A*C^2*a^2*b^6*c^2*z - 2304*A*B^2*a^3*b^3*c^4*z + 576*A*B^2*a^2*b^5*c^3*z + 128*A^2*C*b^9*c*z
+ 61440*A^3*a^3*b^2*c^5*z - 21504*A^3*a^2*b^4*c^4*z + 3328*A^3*a*b^6*c^3*z + 18432*B^2*C*a^5*c^5*z - 16384*A*C
^2*a^5*c^5*z - 192*A^3*b^8*c^2*z - 65536*A^3*a^4*c^6*z - 1088*A*B^2*C*a^2*b^2*c^4 + 48*A*B^2*C*a*b^4*c^3 + 240
*B^2*C^2*a^2*b^3*c^3 - 1920*A^2*C^2*a^2*b^2*c^4 - 960*B^2*C^2*a^3*b*c^4 - 16*B^2*C^2*a*b^5*c^2 + 768*A^2*C^2*a
*b^4*c^3 - 256*A*C^3*a^2*b^3*c^3 - 3072*A^2*B^2*a^2*b*c^5 + 1104*A^2*B^2*a*b^3*c^4 + 6144*A^3*C*a^2*b*c^5 - 21
76*A^3*C*a*b^3*c^4 + 1536*A*C^3*a^3*b*c^4 + 4608*A*B^2*C*a^3*c^5 - 25*B^4*a*b^4*c^3 + 1536*A^4*a*b^2*c^5 + 192
*A^3*C*b^5*c^3 + 360*B^4*a^2*b^2*c^4 - 64*A^2*C^2*b^6*c^2 - 2048*A^2*C^2*a^3*c^5 - 100*A^2*B^2*b^5*c^3 - 256*C
^4*a^4*c^4 - 1296*B^4*a^3*c^5 - 144*A^4*b^4*c^4 - 4096*A^4*a^2*c^6, z, k)*x*(1310720*a^10*c^8 + 384*a^4*b^12*c
^2 - 8960*a^5*b^10*c^3 + 87040*a^6*b^8*c^4 - 450560*a^7*b^6*c^5 + 1310720*a^8*b^4*c^6 - 2031616*a^9*b^2*c^7))/
(16*(a^3*b^8 + 256*a^7*c^4 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3))) - (x*(26560*A^2*a^3*b^6*c^5 -
36864*C^2*a^7*c^7 - 2912*A^2*a^2*b^8*c^4 - 245760*A^2*a^6*c^8 - 120832*A^2*a^4*b^4*c^6 + 273408*A^2*a^5*b^2*c^
7 + 432*B^2*a^2*b^9*c^3 - 4616*B^2*a^3*b^7*c^4 + 24032*B^2*a^4*b^5*c^5 - 60800*B^2*a^5*b^3*c^6 + 640*C^2*a^4*b
^6*c^4 - 7424*C^2*a^5*b^4*c^5 + 28672*C^2*a^6*b^2*c^6 + 128*A^2*a*b^10*c^3 - 16*B^2*a*b^11*c^2 + 59904*B^2*a^6
*b*c^7 + 256*A*C*a^2*b^9*c^3 - 4608*A*C*a^3*b^7*c^4 + 30464*A*C*a^4*b^5*c^5 - 88064*A*C*a^5*b^3*c^6 + 94208*A*
C*a^6*b*c^7))/(16*(a^3*b^8 + 256*a^7*c^4 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3))) + (108*B^3*a^4*c
^6 - 15*B^3*a^3*b^2*c^5 + 24*A^2*B*a*b^5*c^4 + 704*A^2*B*a^3*b*c^6 + 56*B*C^2*a^4*b*c^5 - 266*A^2*B*a^2*b^3*c^
5 - 8*B*C^2*a^3*b^3*c^4 + 576*A*B*C*a^4*c^6 - 16*A*B*C*a*b^6*c^3 + 208*A*B*C*a^2*b^4*c^4 - 744*A*B*C*a^3*b^2*c
^5)/(4*(a^3*b^6 - 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)) + (x*(20480*A^3*a^4*c^8 - 32*A^3*b^8*c^4 + 1216
*A^3*a^2*b^4*c^6 - 11008*A^3*a^3*b^2*c^7 + 128*C^3*a^4*b^3*c^5 + 13312*A*C^2*a^5*c^7 - 19584*B^2*C*a^5*c^7 + 1
92*A^3*a*b^6*c^5 - 512*C^3*a^5*b*c^6 + 40*A*B^2*a*b^7*c^4 - 2496*A*B^2*a^4*b*c^7 + 256*A^2*C*a*b^7*c^4 - 25600
*A^2*C*a^4*b*c^7 - 32*B^2*C*a*b^8*c^3 - 508*A*B^2*a^2*b^5*c^5 + 2016*A*B^2*a^3*b^3*c^6 - 64*A*C^2*a^2*b^6*c^4
+ 1152*A*C^2*a^3*b^4*c^5 - 6912*A*C^2*a^4*b^2*c^6 - 3552*A^2*C*a^2*b^5*c^5 + 16512*A^2*C*a^3*b^3*c^6 + 672*B^2
*C*a^2*b^6*c^4 - 5000*B^2*C*a^3*b^4*c^5 + 16192*B^2*C*a^4*b^2*c^6))/(16*(a^3*b^8 + 256*a^7*c^4 - 16*a^4*b^6*c
+ 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3))) - (108*A*B^3*a^2*c^6 - 10*A^3*B*b^3*c^5 - 192*A^2*B*C*a^2*c^6 - 15*A*B^3
*a*b^2*c^5 + 64*A^3*B*a*b*c^6 - 8*A*B*C^2*a*b^3*c^4 + 56*A*B*C^2*a^2*b*c^5 + 24*A^2*B*C*a*b^2*c^5)/(4*(a^3*b^6
- 64*a^6*c^3 - 12*a^4*b^4*c + 48*a^5*b^2*c^2)) + (x*(1296*B^4*a^3*c^7 - 48*A^4*b^4*c^6 + 256*C^4*a^4*c^6 + 10
24*A^2*C^2*a^3*c^7 - 360*B^4*a^2*b^2*c^6 + 32*A^3*C*b^5*c^5 + 256*A^4*a*b^2*c^7 + 25*B^4*a*b^4*c^5 - 3456*A*B^
2*C*a^3*c^7 - 1024*A*C^3*a^3*b*c^6 - 1024*A^3*C*a^2*b*c^7 - 176*A^2*B^2*a*b^3*c^6 + 960*A^2*B^2*a^2*b*c^7 + 12
8*A*C^3*a^2*b^3*c^5 - 128*A^2*C^2*a*b^4*c^5 + 16*B^2*C^2*a*b^5*c^4 + 960*B^2*C^2*a^3*b*c^6 + 640*A^2*C^2*a^2*b
^2*c^6 - 240*B^2*C^2*a^2*b^3*c^5 - 40*A*B^2*C*a*b^4*c^5 + 768*A*B^2*C*a^2*b^2*c^6))/(16*(a^3*b^8 + 256*a^7*c^4
- 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3)))*root(1572864*a^9*b^2*c^5*z^4 - 983040*a^8*b^4*c^4*z^4 +
327680*a^7*b^6*c^3*z^4 - 61440*a^6*b^8*c^2*z^4 + 6144*a^5*b^10*c*z^4 - 1048576*a^10*c^6*z^4 - 256*a^4*b^12*z^4
+ 1572864*A*a^7*b^2*c^5*z^3 - 983040*A*a^6*b^4*c^4*z^3 + 327680*A*a^5*b^6*c^3*z^3 - 61440*A*a^4*b^8*c^2*z^3 +
6144*A*a^3*b^10*c*z^3 - 1048576*A*a^8*c^6*z^3 - 256*A*a^2*b^12*z^3 + 98304*A*C*a^6*b*c^5*z^2 + 256*A*C*a^2*b^
9*c*z^2 - 90112*A*C*a^5*b^3*c^4*z^2 + 30720*A*C*a^4*b^5*c^3*z^2 - 4608*A*C*a^3*b^7*c^2*z^2 + 61440*B^2*a^6*b*c
^5*z^2 + 432*B^2*a^2*b^9*c*z^2 + 1536*A^2*a*b^10*c*z^2 + 24576*C^2*a^6*b^2*c^4*z^2 - 6144*C^2*a^5*b^4*c^3*z^2
+ 512*C^2*a^4*b^6*c^2*z^2 - 61440*B^2*a^5*b^3*c^4*z^2 + 24064*B^2*a^4*b^5*c^3*z^2 - 4608*B^2*a^3*b^7*c^2*z^2 +
516096*A^2*a^5*b^2*c^5*z^2 - 288768*A^2*a^4*b^4*c^4*z^2 + 88576*A^2*a^3*b^6*c^3*z^2 - 15744*A^2*a^2*b^8*c^2*z
^2 - 16*B^2*a*b^11*z^2 - 32768*C^2*a^7*c^5*z^2 - 393216*A^2*a^6*c^6*z^2 - 64*A^2*b^12*z^2 + 49152*A^2*C*a^4*b*
c^5*z - 2304*A^2*C*a*b^7*c^2*z + 3072*A*B^2*a^4*b*c^5*z - 48*A*B^2*a*b^7*c^2*z + 32*B^2*C*a*b^8*c*z - 15872*B^
2*C*a^4*b^2*c^4*z + 4992*B^2*C*a^3*b^4*c^3*z - 672*B^2*C*a^2*b^6*c^2*z - 45056*A^2*C*a^3*b^3*c^4*z + 15360*A^2
*C*a^2*b^5*c^3*z + 12288*A*C^2*a^4*b^2*c^4*z - 3072*A*C^2*a^3*b^4*c^3*z + 256*A*C^2*a^2*b^6*c^2*z - 2304*A*B^2
*a^3*b^3*c^4*z + 576*A*B^2*a^2*b^5*c^3*z + 128*A^2*C*b^9*c*z + 61440*A^3*a^3*b^2*c^5*z - 21504*A^3*a^2*b^4*c^4
*z + 3328*A^3*a*b^6*c^3*z + 18432*B^2*C*a^5*c^5*z - 16384*A*C^2*a^5*c^5*z - 192*A^3*b^8*c^2*z - 65536*A^3*a^4*
c^6*z - 1088*A*B^2*C*a^2*b^2*c^4 + 48*A*B^2*C*a*b^4*c^3 + 240*B^2*C^2*a^2*b^3*c^3 - 1920*A^2*C^2*a^2*b^2*c^4 -
960*B^2*C^2*a^3*b*c^4 - 16*B^2*C^2*a*b^5*c^2 + 768*A^2*C^2*a*b^4*c^3 - 256*A*C^3*a^2*b^3*c^3 - 3072*A^2*B^2*a
^2*b*c^5 + 1104*A^2*B^2*a*b^3*c^4 + 6144*A^3*C*a^2*b*c^5 - 2176*A^3*C*a*b^3*c^4 + 1536*A*C^3*a^3*b*c^4 + 4608*
A*B^2*C*a^3*c^5 - 25*B^4*a*b^4*c^3 + 1536*A^4*a*b^2*c^5 + 192*A^3*C*b^5*c^3 + 360*B^4*a^2*b^2*c^4 - 64*A^2*C^2
*b^6*c^2 - 2048*A^2*C^2*a^3*c^5 - 100*A^2*B^2*b^5*c^3 - 256*C^4*a^4*c^4 - 1296*B^4*a^3*c^5 - 144*A^4*b^4*c^4 -
4096*A^4*a^2*c^6, z, k), k, 1, 4) + (A*log(x))/a^2

________________________________________________________________________________________

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/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________