3.1.36 \(\int \frac {A+B x+C x^2}{x^3 (a+b x^2+c x^4)^2} \, dx\) [36]

Optimal. Leaf size=534 \[ -\frac {2 A b^2-6 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac {B \left (3 b^2-10 a c\right )}{2 a^2 \left (b^2-4 a c\right ) x}+\frac {B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}+\frac {A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {B \sqrt {c} \left (3 b^3-16 a b c+\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {B \sqrt {c} \left (3 b^3-16 a b c-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {\left (2 A \left (b^4-6 a b^2 c+6 a^2 c^2\right )-a b \left (b^2-6 a c\right ) C\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 A b-a C) \log (x)}{a^3}+\frac {(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3} \]

[Out]

1/2*(6*A*a*c-2*A*b^2+C*a*b)/a^2/(-4*a*c+b^2)/x^2-1/2*B*(-10*a*c+3*b^2)/a^2/(-4*a*c+b^2)/x+1/2*B*(b*c*x^2-2*a*c
+b^2)/a/(-4*a*c+b^2)/x/(c*x^4+b*x^2+a)+1/2*(A*(-2*a*c+b^2)-a*b*C+c*(A*b-2*C*a)*x^2)/a/(-4*a*c+b^2)/x^2/(c*x^4+
b*x^2+a)-1/2*(2*A*(6*a^2*c^2-6*a*b^2*c+b^4)-a*b*(-6*a*c+b^2)*C)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^3/(-
4*a*c+b^2)^(3/2)-(2*A*b-C*a)*ln(x)/a^3+1/4*(2*A*b-C*a)*ln(c*x^4+b*x^2+a)/a^3-1/4*B*arctan(x*2^(1/2)*c^(1/2)/(b
-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*b^3-16*a*b*c+(-10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)
*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*B*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(3*
b^3-16*a*b*c-(-10*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 1.36, antiderivative size = 534, 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 = {1676, 1265, 836, 814, 648, 632, 212, 642, 12, 1135, 1295, 1180, 211} \begin {gather*} \frac {(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac {\log (x) (2 A b-a C)}{a^3}-\frac {-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac {B \sqrt {c} \left (\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \text {ArcTan}\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 {B \sqrt {c} \left (-\left (3 b^2-10 a c\right ) \sqrt {b^2-4 a c}-16 a b c+3 b^3\right ) \text {ArcTan}\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 )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {B \left (3 b^2-10 a c\right )}{2 a^2 x \left (b^2-4 a c\right )}-\frac {\left (2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )-a b C \left (b^2-6 a c\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac {A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x^2 \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 x \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^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/2*(2*A*b^2 - 6*a*A*c - a*b*C)/(a^2*(b^2 - 4*a*c)*x^2) - (B*(3*b^2 - 10*a*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)*x*(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^2*(a + b*x^2 + c*x^4)) - (B*Sqrt[c]*(3*b^3 - 16*a*b*c + (3*b^2 - 10*a*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)^(3/2)*Sqrt[b -
Sqrt[b^2 - 4*a*c]]) + (B*Sqrt[c]*(3*b^3 - 16*a*b*c - (3*b^2 - 10*a*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)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - ((2*A*(b
^4 - 6*a*b^2*c + 6*a^2*c^2) - a*b*(b^2 - 6*a*c)*C)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a
*c)^(3/2)) - ((2*A*b - a*C)*Log[x])/a^3 + ((2*A*b - a*C)*Log[a + b*x^2 + c*x^4])/(4*a^3)

Rule 12

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

Rule 211

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

Rule 212

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

Rule 632

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 814

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 836

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 1135

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*
a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*
a*c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7
)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integ
erQ[p] || IntegerQ[m])

Rule 1180

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 1265

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 1295

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 1676

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

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Mathematica [A]
time = 1.54, size = 655, normalized size = 1.23 \begin {gather*} \frac {-\frac {2 a A}{x^2}-\frac {4 a B}{x}-\frac {2 a \left (2 a^2 c C+b^2 B x \left (b+c x^2\right )+A \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )-a \left (b^2 C+2 B c^2 x^3+b c x (3 B+C x)\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 (-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 ) \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 (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 ) \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}}}+4 (-2 A b+a C) \log (x)+\frac {\left (2 A \left (b^4-6 a b^2 c+6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right )+a \left (-b^3+6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) C\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (2 A \left (-b^4+6 a b^2 c-6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right )+a \left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) C\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x]

[Out]

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

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Maple [A]
time = 0.14, size = 778, normalized size = 1.46

method	result	size
default	\(-\frac {\frac {\frac {a c B \left (2 a c -b^{2}\right ) x^{3}}{8 a c -2 b^{2}}+\frac {a c \left (2 a c A -A \,b^{2}+a b C \right ) x^{2}}{8 a c -2 b^{2}}+\frac {a b B \left (3 a c -b^{2}\right ) x}{8 a c -2 b^{2}}+\frac {a \left (3 A a b c -A \,b^{3}-2 a^{2} c C +C a \,b^{2}\right )}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {-\frac {\left (24 A \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}-24 A \sqrt {-4 a c +b^{2}}\, a \,b^{2} c +4 A \sqrt {-4 a c +b^{2}}\, b^{4}+64 A \,a^{2} b \,c^{2}-32 A a \,b^{3} c +4 A \,b^{5}+12 C \sqrt {-4 a c +b^{2}}\, a^{2} b c -2 C \sqrt {-4 a c +b^{2}}\, a \,b^{3}-32 C \,a^{3} c^{2}+16 C \,a^{2} b^{2} c -2 C a \,b^{4}\right ) \ln \left (-b -2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {\left (16 B \sqrt {-4 a c +b^{2}}\, a^{2} b c -3 B \sqrt {-4 a c +b^{2}}\, a \,b^{3}-40 a^{3} B \,c^{2}+22 B \,a^{2} b^{2} c -3 B a \,b^{4}\right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}+\frac {\frac {\left (24 A \sqrt {-4 a c +b^{2}}\, a^{2} c^{2}-24 A \sqrt {-4 a c +b^{2}}\, a \,b^{2} c +4 A \sqrt {-4 a c +b^{2}}\, b^{4}-64 A \,a^{2} b \,c^{2}+32 A a \,b^{3} c -4 A \,b^{5}+12 C \sqrt {-4 a c +b^{2}}\, a^{2} b c -2 C \sqrt {-4 a c +b^{2}}\, a \,b^{3}+32 C \,a^{3} c^{2}-16 C \,a^{2} b^{2} c +2 C a \,b^{4}\right ) \ln \left (b +2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {\left (16 B \sqrt {-4 a c +b^{2}}\, a^{2} b c -3 B \sqrt {-4 a c +b^{2}}\, a \,b^{3}+40 a^{3} B \,c^{2}-22 B \,a^{2} b^{2} c +3 B a \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}\right )}{4 a c -b^{2}}}{a^{3}}-\frac {A}{2 a^{2} x^{2}}-\frac {B}{a^{2} x}+\frac {\left (-2 A b +a C \right ) \ln \left (x \right )}{a^{3}}\)	\(778\)
risch	\(\text {Expression too large to display}\)	\(3877\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/x^3/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/a^3*((1/2*a*c*B*(2*a*c-b^2)/(4*a*c-b^2)*x^3+1/2*a*c*(2*A*a*c-A*b^2+C*a*b)/(4*a*c-b^2)*x^2+1/2*a*b*B*(3*a*c-
b^2)/(4*a*c-b^2)*x+1/2*a*(3*A*a*b*c-A*b^3-2*C*a^2*c+C*a*b^2)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)*c*(1/(
16*a*c-4*b^2)*(-1/4*(24*A*(-4*a*c+b^2)^(1/2)*a^2*c^2-24*A*(-4*a*c+b^2)^(1/2)*a*b^2*c+4*A*(-4*a*c+b^2)^(1/2)*b^
4+64*A*a^2*b*c^2-32*A*a*b^3*c+4*A*b^5+12*C*(-4*a*c+b^2)^(1/2)*a^2*b*c-2*C*(-4*a*c+b^2)^(1/2)*a*b^3-32*C*a^3*c^
2+16*C*a^2*b^2*c-2*C*a*b^4)/c*ln(-b-2*c*x^2+(-4*a*c+b^2)^(1/2))+1/2*(16*B*(-4*a*c+b^2)^(1/2)*a^2*b*c-3*B*(-4*a
*c+b^2)^(1/2)*a*b^3-40*a^3*B*c^2+22*B*a^2*b^2*c-3*B*a*b^4)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c
*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))+1/(16*a*c-4*b^2)*(1/4*(24*A*(-4*a*c+b^2)^(1/2)*a^2*c^2-24*A*(-4
*a*c+b^2)^(1/2)*a*b^2*c+4*A*(-4*a*c+b^2)^(1/2)*b^4-64*A*a^2*b*c^2+32*A*a*b^3*c-4*A*b^5+12*C*(-4*a*c+b^2)^(1/2)
*a^2*b*c-2*C*(-4*a*c+b^2)^(1/2)*a*b^3+32*C*a^3*c^2-16*C*a^2*b^2*c+2*C*a*b^4)/c*ln(b+2*c*x^2+(-4*a*c+b^2)^(1/2)
)+1/2*(16*B*(-4*a*c+b^2)^(1/2)*a^2*b*c-3*B*(-4*a*c+b^2)^(1/2)*a*b^3+40*a^3*B*c^2-22*B*a^2*b^2*c+3*B*a*b^4)*2^(
1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))))-1/2/a^2*A/x^2-1/
a^2*B/x+(-2*A*b+C*a)/a^3*ln(x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*((3*B*b^2*c - 10*B*a*c^2)*x^5 - (6*A*a*c^2 + (C*a*b - 2*A*b^2)*c)*x^4 + A*a*b^2 - 4*A*a^2*c + (3*B*b^3 -
11*B*a*b*c)*x^3 - (C*a*b^2 - 2*A*b^3 - (2*C*a^2 - 7*A*a*b)*c)*x^2 + 2*(B*a*b^2 - 4*B*a^2*c)*x)/((a^2*b^2*c - 4
*a^3*c^2)*x^6 + (a^2*b^3 - 4*a^3*b*c)*x^4 + (a^3*b^2 - 4*a^4*c)*x^2) - 1/2*integrate((3*B*a*b^3 - 13*B*a^2*b*c
- 2*(4*(C*a^2 - 2*A*a*b)*c^2 - (C*a*b^2 - 2*A*b^3)*c)*x^3 + (3*B*a*b^2*c - 10*B*a^2*c^2)*x^2 + 2*(C*a*b^3 - 2
*A*b^4 - 6*A*a^2*c^2 - 5*(C*a^2*b - 2*A*a*b^2)*c)*x)/(c*x^4 + b*x^2 + a), x)/(a^3*b^2 - 4*a^4*c) + (C*a - 2*A*
b)*log(x)/a^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/x**3/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6939 vs. \(2 (470) = 940\).
time = 8.19, size = 6939, normalized size = 12.99 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x^3/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/16*((a^6*b^4*c - 8*a^7*b^2*c^2 + 16*a^8*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*s
qrt(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 + sqr
t(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)*B - 2*(3*sqrt(2)*sqrt(b*c + sqrt(b
^2 - 4*a*c)*c)*a^6*b^9*c - 49*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^7*c^2 - 6*sqrt(2)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^6*b^8*c^2 - 6*a^6*b^9*c^2 + 300*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^3 + 74*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^3 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^3 + 98*
a^7*b^7*c^3 - 816*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^3*c^4 - 304*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*a^8*b^4*c^4 - 37*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^5*c^4 - 600*a^8*b^5*c^4 + 832*sqrt(2)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b*c^5 + 416*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^2*c^5 + 152*sqrt(2
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^3*c^5 + 1632*a^9*b^3*c^5 - 208*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*a^9*b*c^6 - 1664*a^10*b*c^6 + 6*(b^2 - 4*a*c)*a^6*b^7*c^2 - 74*(b^2 - 4*a*c)*a^7*b^5*c^3 + 304*(b^2 - 4*a*c)*
a^8*b^3*c^4 - 416*(b^2 - 4*a*c)*a^9*b*c^5)*B*abs(a^6*b^4*c - 8*a^7*b^2*c^2 + 16*a^8*c^3) + (6*a^12*b^12*c^4 -
128*a^13*b^10*c^5 + 1088*a^14*b^8*c^6 - 4608*a^15*b^6*c^7 + 9728*a^16*b^4*c^8 - 8192*a^17*b^2*c^9 - 3*sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^12*c^2 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a^13*b^10*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^11*c^3 - 5
44*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^14*b^8*c^4 - 104*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^13*b^9*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^
10*c^4 + 2304*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^15*b^6*c^5 + 672*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^14*b^7*c^5 + 52*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^13*b^8*c^5 - 4864*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^16*b^4*c^6 - 1920*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^15*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^14*b^6*c^6 + 4096*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^17*b^2*c^7 +
2048*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^16*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^15*b^4*c^7 - 1024*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^16*b^2*c^8 - 6*(b^2 - 4*a*c)*a^12*b^10*c^4 + 104*(b^2 - 4*a*c)*a^13*b^8*c^5 - 672*(b^2 - 4*a*c)*a^14*b^6*c^6
+ 1920*(b^2 - 4*a*c)*a^15*b^4*c^7 - 2048*(b^2 - 4*a*c)*a^16*b^2*c^8)*B)*arctan(2*sqrt(1/2)*x/sqrt((a^6*b^5*c -
8*a^7*b^3*c^2 + 16*a^8*b*c^3 + sqrt((a^6*b^5*c - 8*a^7*b^3*c^2 + 16*a^8*b*c^3)^2 - 4*(a^7*b^4*c - 8*a^8*b^2*c
^2 + 16*a^9*c^3)*(a^6*b^4*c^2 - 8*a^7*b^2*c^3 + 16*a^8*c^4)))/(a^6*b^4*c^2 - 8*a^7*b^2*c^3 + 16*a^8*c^4)))/((a
^9*b^8*c - 16*a^10*b^6*c^2 - 2*a^9*b^7*c^2 + 96*a^11*b^4*c^3 + 24*a^10*b^5*c^3 + a^9*b^6*c^3 - 256*a^12*b^2*c^
4 - 96*a^11*b^3*c^4 - 12*a^10*b^4*c^4 + 256*a^13*c^5 + 128*a^12*b*c^5 + 48*a^11*b^2*c^5 - 64*a^12*c^6)*abs(a^6
*b^4*c - 8*a^7*b^2*c^2 + 16*a^8*c^3)*abs(c)) + 1/16*((a^6*b^4*c - 8*a^7*b^2*c^2 + 16*a^8*c^3)^2*(6*b^4*c^2 - 4
4*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)*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)*B - 2*(3*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^9*c - 49*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^7*b^7*c^2 - 6*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^8*c^2 + 6*a^6*b^9*c^2 + 300*sqrt(2)*sqrt(
b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^5*c^3 + 74*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^6*c^3 + 3*sqrt(2)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^3 - 98*a^7*b^7*c^3 - 816*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b
^3*c^4 - 304*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^4*c^4 - 37*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*
a^7*b^5*c^4 + 600*a^8*b^5*c^4 + 832*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b*c^5 + 416*sqrt(2)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a^9*b^2*c^5 + 152*sqrt(2...

________________________________________________________________________________________

Mupad [B]
time = 2.77, size = 2500, normalized size = 4.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/(x^3*(a + b*x^2 + c*x^4)^2),x)

[Out]

symsum(log(root(1572864*a^11*b^2*c^5*z^4 - 983040*a^10*b^4*c^4*z^4 + 327680*a^9*b^6*c^3*z^4 - 61440*a^8*b^8*c^
2*z^4 + 6144*a^7*b^10*c*z^4 - 1048576*a^12*c^6*z^4 - 256*a^6*b^12*z^4 + 1572864*C*a^9*b^2*c^5*z^3 - 983040*C*a
^8*b^4*c^4*z^3 + 327680*C*a^7*b^6*c^3*z^3 - 61440*C*a^6*b^8*c^2*z^3 - 3145728*A*a^8*b^3*c^5*z^3 + 1966080*A*a^
7*b^5*c^4*z^3 - 655360*A*a^6*b^7*c^3*z^3 + 122880*A*a^5*b^9*c^2*z^3 + 6144*C*a^5*b^10*c*z^3 + 2097152*A*a^9*b*
c^6*z^3 - 12288*A*a^4*b^11*c*z^3 - 1048576*C*a^10*c^6*z^3 - 256*C*a^4*b^12*z^3 + 512*A*a^3*b^13*z^3 + 1277952*
A*C*a^7*b*c^6*z^2 - 6144*A*C*a^2*b^11*c*z^2 - 1794048*A*C*a^6*b^3*c^5*z^2 + 1062912*A*C*a^5*b^5*c^4*z^2 - 3404
80*A*C*a^4*b^7*c^3*z^2 + 62208*A*C*a^3*b^9*c^2*z^2 + 256*A*C*a*b^13*z^2 + 1536*C^2*a^3*b^10*c*z^2 - 430080*B^2
*a^7*b*c^6*z^2 + 3408*B^2*a^2*b^11*c*z^2 + 6144*A^2*a*b^12*c*z^2 + 516096*C^2*a^7*b^2*c^5*z^2 - 288768*C^2*a^6
*b^4*c^4*z^2 + 88576*C^2*a^5*b^6*c^3*z^2 - 15744*C^2*a^4*b^8*c^2*z^2 + 716800*B^2*a^6*b^3*c^5*z^2 - 483840*B^2
*a^5*b^5*c^4*z^2 + 170496*B^2*a^4*b^7*c^3*z^2 - 33232*B^2*a^3*b^9*c^2*z^2 + 1468416*A^2*a^5*b^4*c^5*z^2 - 9661
44*A^2*a^4*b^6*c^4*z^2 - 761856*A^2*a^6*b^2*c^6*z^2 + 326656*A^2*a^3*b^8*c^3*z^2 - 61440*A^2*a^2*b^10*c^2*z^2
- 144*B^2*a*b^13*z^2 - 393216*C^2*a^8*c^6*z^2 - 64*C^2*a^2*b^12*z^2 - 294912*A^2*a^7*c^7*z^2 - 256*A^2*b^14*z^
2 - 138240*B^2*C*a^5*b*c^6*z - 432*B^2*C*a*b^9*c^2*z + 245760*A*C^2*a^5*b*c^6*z + 12288*A^2*C*a*b^8*c^3*z + 76
8*A*C^2*a*b^9*c^2*z + 576*A*B^2*a*b^8*c^3*z + 131328*B^2*C*a^4*b^3*c^5*z - 46656*B^2*C*a^3*b^5*c^4*z + 7344*B^
2*C*a^2*b^7*c^3*z - 233472*A*C^2*a^4*b^3*c^5*z + 168960*A^2*C*a^3*b^4*c^5*z - 86016*A^2*C*a^4*b^2*c^6*z + 8294
4*A*C^2*a^3*b^5*c^4*z - 71424*A^2*C*a^2*b^6*c^4*z - 13056*A*C^2*a^2*b^7*c^3*z - 152064*A*B^2*a^4*b^2*c^6*z + 5
6448*A*B^2*a^3*b^4*c^5*z - 9312*A*B^2*a^2*b^6*c^4*z + 61440*C^3*a^5*b^2*c^5*z - 21504*C^3*a^4*b^4*c^4*z + 3328
*C^3*a^3*b^6*c^3*z - 192*C^3*a^2*b^8*c^2*z - 286720*A^3*a^3*b^3*c^6*z + 104448*A^3*a^2*b^5*c^5*z + 294912*A^3*
a^4*b*c^7*z - 16896*A^3*a*b^7*c^4*z - 768*A^2*C*b^10*c^2*z - 147456*A^2*C*a^5*c^7*z + 153600*A*B^2*a^5*c^7*z -
65536*C^3*a^6*c^6*z + 1024*A^3*b^9*c^3*z - 15936*A*B^2*C*a^2*b^2*c^6 + 1648*A*B^2*C*a*b^4*c^5 + 3152*B^2*C^2*
a^2*b^3*c^5 - 4992*A^2*C^2*a^2*b^2*c^6 - 7680*B^2*C^2*a^3*b*c^6 - 324*B^2*C^2*a*b^5*c^4 - 5760*A*C^3*a^2*b^3*c
^5 + 4608*A^2*C^2*a*b^4*c^5 - 16320*A^2*B^2*a^2*b*c^7 + 7152*A^2*B^2*a*b^3*c^6 + 32256*A^3*C*a^2*b*c^7 + 14336
*A*C^3*a^3*b*c^6 - 14080*A^3*C*a*b^3*c^6 + 576*A*C^3*a*b^5*c^4 + 38400*A*B^2*C*a^3*c^7 - 441*B^4*a*b^4*c^5 + 9
216*A^4*a*b^2*c^7 + 1536*A^3*C*b^5*c^5 + 1536*C^4*a^3*b^2*c^5 - 144*C^4*a^2*b^4*c^4 + 4200*B^4*a^2*b^2*c^6 - 5
76*A^2*C^2*b^6*c^4 - 18432*A^2*C^2*a^3*c^7 - 784*A^2*B^2*b^5*c^5 - 4096*C^4*a^4*c^6 - 10000*B^4*a^3*c^7 - 1024
*A^4*b^4*c^6 - 20736*A^4*a^2*c^8, z, k)*(root(1572864*a^11*b^2*c^5*z^4 - 983040*a^10*b^4*c^4*z^4 + 327680*a^9*
b^6*c^3*z^4 - 61440*a^8*b^8*c^2*z^4 + 6144*a^7*b^10*c*z^4 - 1048576*a^12*c^6*z^4 - 256*a^6*b^12*z^4 + 1572864*
C*a^9*b^2*c^5*z^3 - 983040*C*a^8*b^4*c^4*z^3 + 327680*C*a^7*b^6*c^3*z^3 - 61440*C*a^6*b^8*c^2*z^3 - 3145728*A*
a^8*b^3*c^5*z^3 + 1966080*A*a^7*b^5*c^4*z^3 - 655360*A*a^6*b^7*c^3*z^3 + 122880*A*a^5*b^9*c^2*z^3 + 6144*C*a^5
*b^10*c*z^3 + 2097152*A*a^9*b*c^6*z^3 - 12288*A*a^4*b^11*c*z^3 - 1048576*C*a^10*c^6*z^3 - 256*C*a^4*b^12*z^3 +
512*A*a^3*b^13*z^3 + 1277952*A*C*a^7*b*c^6*z^2 - 6144*A*C*a^2*b^11*c*z^2 - 1794048*A*C*a^6*b^3*c^5*z^2 + 1062
912*A*C*a^5*b^5*c^4*z^2 - 340480*A*C*a^4*b^7*c^3*z^2 + 62208*A*C*a^3*b^9*c^2*z^2 + 256*A*C*a*b^13*z^2 + 1536*C
^2*a^3*b^10*c*z^2 - 430080*B^2*a^7*b*c^6*z^2 + 3408*B^2*a^2*b^11*c*z^2 + 6144*A^2*a*b^12*c*z^2 + 516096*C^2*a^
7*b^2*c^5*z^2 - 288768*C^2*a^6*b^4*c^4*z^2 + 88576*C^2*a^5*b^6*c^3*z^2 - 15744*C^2*a^4*b^8*c^2*z^2 + 716800*B^
2*a^6*b^3*c^5*z^2 - 483840*B^2*a^5*b^5*c^4*z^2 + 170496*B^2*a^4*b^7*c^3*z^2 - 33232*B^2*a^3*b^9*c^2*z^2 + 1468
416*A^2*a^5*b^4*c^5*z^2 - 966144*A^2*a^4*b^6*c^4*z^2 - 761856*A^2*a^6*b^2*c^6*z^2 + 326656*A^2*a^3*b^8*c^3*z^2
- 61440*A^2*a^2*b^10*c^2*z^2 - 144*B^2*a*b^13*z^2 - 393216*C^2*a^8*c^6*z^2 - 64*C^2*a^2*b^12*z^2 - 294912*A^2
*a^7*c^7*z^2 - 256*A^2*b^14*z^2 - 138240*B^2*C*a^5*b*c^6*z - 432*B^2*C*a*b^9*c^2*z + 245760*A*C^2*a^5*b*c^6*z
+ 12288*A^2*C*a*b^8*c^3*z + 768*A*C^2*a*b^9*c^2*z + 576*A*B^2*a*b^8*c^3*z + 131328*B^2*C*a^4*b^3*c^5*z - 46656
*B^2*C*a^3*b^5*c^4*z + 7344*B^2*C*a^2*b^7*c^3*z - 233472*A*C^2*a^4*b^3*c^5*z + 168960*A^2*C*a^3*b^4*c^5*z - 86
016*A^2*C*a^4*b^2*c^6*z + 82944*A*C^2*a^3*b^5*c^4*z - 71424*A^2*C*a^2*b^6*c^4*z - 13056*A*C^2*a^2*b^7*c^3*z -
152064*A*B^2*a^4*b^2*c^6*z + 56448*A*B^2*a^3*b^4*c^5*z - 9312*A*B^2*a^2*b^6*c^4*z + 61440*C^3*a^5*b^2*c^5*z -
21504*C^3*a^4*b^4*c^4*z + 3328*C^3*a^3*b^6*c^3*z - 192*C^3*a^2*b^8*c^2*z - 286720*A^3*a^3*b^3*c^6*z + 104448*A
^3*a^2*b^5*c^5*z + 294912*A^3*a^4*b*c^7*z - 16896*A^3*a*b^7*c^4*z - 768*A^2*C*b^10*c^2*z - 147456*A^2*C*a^5*c^
7*z + 153600*A*B^2*a^5*c^7*z - 65536*C^3*a^6*c^6*z + 1024*A^3*b^9*c^3*z - 15936*A*B^2*C*a^2*b^2*c^6 + 1648*A*B
^2*C*a*b^4*c^5 + 3152*B^2*C^2*a^2*b^3*c^5 - 499...

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