Integrand size = 28, antiderivative size = 260 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=-\frac {A}{a x}-\frac {\sqrt {c} \left (A+\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (A-\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {b B \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}+\frac {B \log (x)}{a}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a} \]
-A/a/x+B*ln(x)/a-1/4*B*ln(c*x^4+b*x^2+a)/a+1/2*b*B*arctanh((2*c*x^2+b)/(-4 *a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^(1/2)-1/2*arctan(x*2^(1/2)*c^(1/2)/(b-(-4* a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(A+(A*b-2*C*a)/(-4*a*c+b^2)^(1/2))/a*2^(1/2 )/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2 )^(1/2))^(1/2))*c^(1/2)*(A+(-A*b+2*C*a)/(-4*a*c+b^2)^(1/2))/a*2^(1/2)/(b+( -4*a*c+b^2)^(1/2))^(1/2)
Time = 0.69 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\frac {4 A}{x}+\frac {2 \sqrt {2} \sqrt {c} \left (A \left (b+\sqrt {b^2-4 a c}\right )-2 a C\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {2 \sqrt {2} \sqrt {c} \left (A \left (-b+\sqrt {b^2-4 a c}\right )+2 a C\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}-4 B \log (x)+\frac {B \left (b+\sqrt {b^2-4 a c}\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {B \left (-b+\sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{4 a} \]
-1/4*((4*A)/x + (2*Sqrt[2]*Sqrt[c]*(A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*Arc Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*S qrt[b - Sqrt[b^2 - 4*a*c]]) + (2*Sqrt[2]*Sqrt[c]*(A*(-b + Sqrt[b^2 - 4*a*c ]) + 2*a*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt [b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - 4*B*Log[x] + (B*(b + Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (B*( -b + Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4 *a*c])/a
Time = 0.67 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2193, 27, 1434, 1144, 25, 1142, 1083, 219, 1103, 1604, 1480, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2193 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\int \frac {B}{x \left (c x^4+b x^2+a\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+B \int \frac {1}{x \left (c x^4+b x^2+a\right )}dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \int \frac {1}{x^2 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1144 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\int -\frac {c x^2+b}{c x^4+b x^2+a}dx^2}{a}+\frac {\log \left (x^2\right )}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\int \frac {c x^2+b}{c x^4+b x^2+a}dx^2}{a}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} b \int \frac {1}{c x^4+b x^2+a}dx^2+\frac {1}{2} \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{a}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2-b \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{a}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \int \frac {C x^2+A}{x^2 \left (c x^4+b x^2+a\right )}dx+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \log \left (a+b x^2+c x^4\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\right )\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle -\frac {\int \frac {A c x^2+A b-a C}{c x^4+b x^2+a}dx}{a}-\frac {A}{a x}+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \log \left (a+b x^2+c x^4\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {A b-2 a C}{\sqrt {b^2-4 a c}}+A\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (A-\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{a}-\frac {A}{a x}+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \log \left (a+b x^2+c x^4\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\sqrt {c} \left (\frac {A b-2 a C}{\sqrt {b^2-4 a c}}+A\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (A-\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {A}{a x}+\frac {1}{2} B \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \log \left (a+b x^2+c x^4\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\right )\) |
-(A/(a*x)) - ((Sqrt[c]*(A + (A*b - 2*a*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[ 2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4 *a*c]]) + (Sqrt[c]*(A - (A*b - 2*a*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*S qrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c ]]))/a + (B*(Log[x^2]/a - (-((b*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/ Sqrt[b^2 - 4*a*c]) + Log[a + b*x^2 + c*x^4]/2)/a))/2
3.1.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S imp[1/(c*d^2 - b*d*e + a*e^2) Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
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] + Simp[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])
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_S ymbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(d*x)^m*(a + b*x^2 + c*x^4)^p, x] + Simp[1/d Int[Sum[Coe ff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q + 1)/2}]*(d*x)^(m + 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]
Time = 0.11 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {A}{a x}+\frac {B \ln \left (x \right )}{a}+\frac {4 c \left (\frac {\frac {\left (-B b \sqrt {-4 a c +b^{2}}-4 B a c +B \,b^{2}\right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (-A b \sqrt {-4 a c +b^{2}}-4 A a c +A \,b^{2}+2 C \sqrt {-4 a c +b^{2}}\, a \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}+\frac {-\frac {\left (-B b \sqrt {-4 a c +b^{2}}+4 B a c -B \,b^{2}\right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (-A b \sqrt {-4 a c +b^{2}}+4 A a c -A \,b^{2}+2 C \sqrt {-4 a c +b^{2}}\, a \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}\right )}{a}\) | \(325\) |
| risch | \(-\frac {A}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{5} c^{2}-8 a^{4} b^{2} c +b^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (32 B \,a^{4} c^{2}-16 B \,a^{3} b^{2} c +2 B \,a^{2} b^{4}\right ) \textit {\_Z}^{3}+\left (12 A^{2} a^{2} b \,c^{2}-7 A^{2} a \,b^{3} c +A^{2} b^{5}-16 A C \,a^{3} c^{2}+12 A C \,a^{2} b^{2} c -2 A C a \,b^{4}+24 B^{2} a^{3} c^{2}-10 B^{2} a^{2} b^{2} c +B^{2} a \,b^{4}-4 C^{2} a^{3} b c +C^{2} a^{2} b^{3}\right ) \textit {\_Z}^{2}+\left (8 A^{2} B a b \,c^{2}-2 A^{2} B \,b^{3} c -16 A B C \,a^{2} c^{2}+4 A B C a \,b^{2} c +8 B^{3} a^{2} c^{2}-2 B^{3} a \,b^{2} c \right ) \textit {\_Z} +c^{3} A^{4}-2 A^{3} C b \,c^{2}+A^{2} B^{2} b \,c^{2}+2 A^{2} C^{2} a \,c^{2}+A^{2} C^{2} b^{2} c -4 A \,B^{2} C a \,c^{2}-2 A \,C^{3} a b c +B^{4} a \,c^{2}+B^{2} C^{2} a b c +C^{4} a^{2} c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2}-22 a^{4} b^{2} c +3 b^{4} a^{3}\right ) \textit {\_R}^{4}+\left (60 B \,a^{4} c^{2}-27 B \,a^{3} b^{2} c +3 B \,a^{2} b^{4}\right ) \textit {\_R}^{3}+\left (25 A^{2} a^{2} b \,c^{2}-14 A^{2} a \,b^{3} c +2 A^{2} b^{5}-36 A C \,a^{3} c^{2}+24 A C \,a^{2} b^{2} c -4 A C a \,b^{4}+30 B^{2} a^{3} c^{2}-8 B^{2} a^{2} b^{2} c -7 C^{2} a^{3} b c +2 C^{2} a^{2} b^{3}\right ) \textit {\_R}^{2}+\left (14 A^{2} B a b \,c^{2}-4 A^{2} B \,b^{3} c -26 A B C \,a^{2} c^{2}+8 A B C a \,b^{2} c +5 B^{3} a^{2} c^{2}-B \,C^{2} a^{2} b c \right ) \textit {\_R} +2 c^{3} A^{4}-4 A^{3} C b \,c^{2}+2 A^{2} B^{2} b \,c^{2}+4 A^{2} C^{2} a \,c^{2}+2 A^{2} C^{2} b^{2} c -4 A \,B^{2} C a \,c^{2}-4 A \,C^{3} a b c +2 C^{4} a^{2} c \right ) x +\left (4 A \,a^{4} c^{2}-5 A \,a^{3} b^{2} c +A \,a^{2} b^{4}+4 C \,a^{4} b c -C \,a^{3} b^{3}\right ) \textit {\_R}^{3}+\left (-4 A B \,a^{3} c^{2}+8 A B \,a^{2} b^{2} c -2 A B a \,b^{4}-6 B C \,a^{3} b c +2 B C \,a^{2} b^{3}\right ) \textit {\_R}^{2}+\left (-A^{2} C \,a^{2} c^{2}-7 A \,B^{2} a^{2} c^{2}+4 A \,B^{2} a \,b^{2} c +A \,C^{2} a^{2} b c -4 B^{2} C \,a^{2} b c -C^{3} a^{3} c \right ) \textit {\_R} +2 A^{2} B C a \,c^{2}-2 A \,B^{3} a \,c^{2}-2 A B \,C^{2} a b c +2 B \,C^{3} a^{2} c \right )\right )}{2}+\frac {B \ln \left (x \right )}{a}\) | \(903\) |
-A/a/x+B*ln(x)/a+4/a*c*(1/(16*a*c-4*b^2)*(1/4*(-B*b*(-4*a*c+b^2)^(1/2)-4*B *a*c+B*b^2)/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+1/2*(-A*b*(-4*a*c+b^2)^(1/2 )-4*A*a*c+A*b^2+2*C*(-4*a*c+b^2)^(1/2)*a)*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/(16*a*c-4 *b^2)*(-1/4*(-B*b*(-4*a*c+b^2)^(1/2)+4*B*a*c-B*b^2)/c*ln(-2*c*x^2+(-4*a*c+ b^2)^(1/2)-b)+1/2*(-A*b*(-4*a*c+b^2)^(1/2)+4*A*a*c-A*b^2+2*C*(-4*a*c+b^2)^ (1/2)*a)*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))))
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}} \,d x } \]
B*log(x)/a - integrate((B*c*x^3 + A*c*x^2 + B*b*x - C*a + A*b)/(c*x^4 + b* x^2 + a), x)/a - A/(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 3505 vs. \(2 (218) = 436\).
Time = 1.57 (sec) , antiderivative size = 3505, normalized size of antiderivative = 13.48 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
-1/4*B*log(abs(c*x^4 + b*x^2 + a))/a + B*log(abs(x))/a - A/(a*x) - 1/8*((2 *b^4*c^2 - 16*a*b^2*c^3 + 32*a^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b ^2 - 4*a*c)*c)*a*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*a^2*c^2 - 8*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^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 2*(b^2 - 4*a*c)*b^2*c^2 + 8*(b^2 - 4*a*c)*a*c^3)*A*c^2 + 2*(sqrt(2)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c )*a*b^3*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 2*b^5*c^ 2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 8*sqrt(2)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*b^3*c^3 + 16*a*b^3*c^3 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a* b*c^4 - 32*a^2*b*c^4 + 2*(b^2 - 4*a*c)*b^3*c^2 - 8*(b^2 - 4*a*c)*a*b*c^3)* A*abs(c) - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 8*sqrt(2)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a*b^3*c^2 - 2*a*b^4*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a^3*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + s qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 16*a^2*b^2*c^3 - 4*s...
Time = 8.16 (sec) , antiderivative size = 2588, normalized size of antiderivative = 9.95 \[ \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
symsum(log(root(128*a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128 *B*a^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c *z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16* C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^ 2 - 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4*A^2*b^5*z^2 - 8*A*B*C*a*b^2 *c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a *b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a* b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C^2*b^2*c - A^2*B^2*b*c^2 - C^ 4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)*(root(128*a^4*b^2*c*z^4 - 256*a^5*c^2 *z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2 *b^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a ^3*c^2*z^2 - 4*B^2*a*b^4*z^2 - 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4* A^2*b^5*z^2 - 8*A*B*C*a*b^2*c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a*b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C^2 *b^2*c - A^2*B^2*b*c^2 - C^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)*(root(128* a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b^2*c*z^3 - 2 56*B*a^4*c^2*z^3 - 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z ^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c*z^2 +...