Integrand size = 32, antiderivative size = 449 \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (c \left (b^2 d-2 a (c d-a f)-\frac {a b (c e+a g)}{c}\right )+\left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b (c d+a f)+\frac {a b^2 g}{c}-2 a (c e+3 a g)+\frac {b^2 c (c d-a f)-4 a c^2 (3 c d+a f)-a b^3 g+4 a b c (c e+2 a g)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b (c d+a f)+\frac {a b^2 g}{c}-2 a (c e+3 a g)-\frac {b^2 c (c d-a f)-4 a c^2 (3 c d+a f)-a b^3 g+4 a b c (c e+2 a g)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
1/2*x*(c*(b^2*d-2*a*(-a*f+c*d)-a*b*(a*g+c*e)/c)+(b*c*(a*f+c*d)-a*b^2*g-2*a *c*(-a*g+c*e))*x^2)/a/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*arctan(x*2^(1/2)* c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b*(a*f+c*d)+a*b^2*g/c-2*a*(3*a*g+c* e)+(b^2*c*(-a*f+c*d)-4*a*c^2*(a*f+3*c*d)-a*b^3*g+4*a*b*c*(2*a*g+c*e))/c/(- 4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)*2^(1/2)/c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1 /2)+1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b*(a*f+c*d )+a*b^2*g/c-2*a*(3*a*g+c*e)+(-b^2*c*(-a*f+c*d)+4*a*c^2*(a*f+3*c*d)+a*b^3*g -4*a*b*c*(2*a*g+c*e))/c/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)*2^(1/2)/c^(1/2) /(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.01 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 \sqrt {c} x \left (b \left (-a c e-a^2 g+c^2 d x^2+a c f x^2\right )+b^2 \left (c d-a g x^2\right )+2 a c \left (-c \left (d+e x^2\right )+a \left (f+g x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (-a b^3 g+b c \left (c \sqrt {b^2-4 a c} d+4 a c e+a \sqrt {b^2-4 a c} f+8 a^2 g\right )+b^2 \left (c^2 d-a c f+a \sqrt {b^2-4 a c} g\right )-2 a c \left (6 c^2 d+c \sqrt {b^2-4 a c} e+2 a c f+3 a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (a b^3 g+b c \left (c \sqrt {b^2-4 a c} d-4 a c e+a \sqrt {b^2-4 a c} f-8 a^2 g\right )+2 a c \left (6 c^2 d-c \sqrt {b^2-4 a c} e+2 a c f-3 a \sqrt {b^2-4 a c} g\right )+b^2 \left (-c^2 d+a c f+a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a c^{3/2}} \]
((2*Sqrt[c]*x*(b*(-(a*c*e) - a^2*g + c^2*d*x^2 + a*c*f*x^2) + b^2*(c*d - a *g*x^2) + 2*a*c*(-(c*(d + e*x^2)) + a*(f + g*x^2))))/((b^2 - 4*a*c)*(a + b *x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*g) + b*c*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c *e + a*Sqrt[b^2 - 4*a*c]*f + 8*a^2*g) + b^2*(c^2*d - a*c*f + a*Sqrt[b^2 - 4*a*c]*g) - 2*a*c*(6*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 2*a*c*f + 3*a*Sqrt[b^ 2 - 4*a*c]*g))*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^3*g + b*c* (c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*e + a*Sqrt[b^2 - 4*a*c]*f - 8*a^2*g) + 2*a* c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 2*a*c*f - 3*a*Sqrt[b^2 - 4*a*c]*g) + b^2*(-(c^2*d) + a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*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*c^(3/2))
Time = 0.95 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2206, 25, 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.
| \(\displaystyle \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac {a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {d b^2+\frac {a (c e+a g) b}{c}+\left (\frac {a g b^2}{c}+(c d+a f) b-2 a (c e+3 a g)\right ) x^2-2 a (3 c d+a f)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d b^2+\frac {a (c e+a g) b}{c}+\left (\frac {a g b^2}{c}+(c d+a f) b-2 a (c e+3 a g)\right ) x^2-2 a (3 c d+a f)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac {a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {a b^2 g}{c}+\frac {-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt {b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {a b^2 g}{c}-\frac {-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt {b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac {a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {a b^2 g}{c}+\frac {-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt {b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {a b^2 g}{c}-\frac {-a b^3 g+b^2 c (c d-a f)+4 a b c (2 a g+c e)-4 a c^2 (a f+3 c d)}{c \sqrt {b^2-4 a c}}+b (a f+c d)-2 a (3 a g+c e)\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (x^2 \left (-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )+c \left (-\frac {a b (a g+c e)}{c}-2 a (c d-a f)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
(x*(c*(b^2*d - 2*a*(c*d - a*f) - (a*b*(c*e + a*g))/c) + (b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^ 4)) + (((b*(c*d + a*f) + (a*b^2*g)/c - 2*a*(c*e + 3*a*g) + (b^2*c*(c*d - a *f) - 4*a*c^2*(3*c*d + a*f) - a*b^3*g + 4*a*b*c*(c*e + 2*a*g))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt [2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*(c*d + a*f) + (a*b^2*g)/c - 2*a*(c*e + 3*a*g) - (b^2*c*(c*d - a*f) - 4*a*c^2*(3*c*d + a*f) - a*b^3*g + 4*a*b*c*(c*e + 2*a*g))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x) /Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c] ]))/(2*a*(b^2 - 4*a*c))
3.2.28.3.1 Defintions of rubi rules used
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[((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[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {-\frac {\left (2 a^{2} c g -a \,b^{2} g +a b c f -2 a \,c^{2} e +b \,c^{2} d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right ) c}+\frac {\left (a^{2} b g -2 a^{2} c f +a b c e +2 a \,c^{2} d -b^{2} c d \right ) x}{2 a c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (6 a^{2} c g -a \,b^{2} g -a b c f +2 a \,c^{2} e -b \,c^{2} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {a^{2} b g -2 a^{2} c f +a b c e -6 a \,c^{2} d +b^{2} c d}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 a c}\) | \(269\) |
| default | \(\frac {-\frac {\left (2 a^{2} c g -a \,b^{2} g +a b c f -2 a \,c^{2} e +b \,c^{2} d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right ) c}+\frac {\left (a^{2} b g -2 a^{2} c f +a b c e +2 a \,c^{2} d -b^{2} c d \right ) x}{2 a c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (6 a^{2} c g \sqrt {-4 a c +b^{2}}-a \,b^{2} g \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, a b c f +2 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e -b \,c^{2} d \sqrt {-4 a c +b^{2}}+8 a^{2} b g c -4 a^{2} c^{2} f -a \,b^{3} g -a \,b^{2} c f +4 a b \,c^{2} e -12 a \,c^{3} d +b^{2} c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (6 a^{2} c g \sqrt {-4 a c +b^{2}}-a \,b^{2} g \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, a b c f +2 \sqrt {-4 a c +b^{2}}\, a \,c^{2} e -b \,c^{2} d \sqrt {-4 a c +b^{2}}-8 a^{2} b g c +4 a^{2} c^{2} f +a \,b^{3} g +a \,b^{2} c f -4 a b \,c^{2} e +12 a \,c^{3} d -b^{2} c^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a \left (4 a c -b^{2}\right )}\) | \(543\) |
(-1/2/a*(2*a^2*c*g-a*b^2*g+a*b*c*f-2*a*c^2*e+b*c^2*d)/(4*a*c-b^2)/c*x^3+1/ 2*(a^2*b*g-2*a^2*c*f+a*b*c*e+2*a*c^2*d-b^2*c*d)/a/c/(4*a*c-b^2)*x)/(c*x^4+ b*x^2+a)+1/4/a/c*sum(((6*a^2*c*g-a*b^2*g-a*b*c*f+2*a*c^2*e-b*c^2*d)/(4*a*c -b^2)*_R^2-(a^2*b*g-2*a^2*c*f+a*b*c*e-6*a*c^2*d+b^2*c*d)/(4*a*c-b^2))/(2*_ R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 19375 vs. \(2 (408) = 816\).
Time = 151.26 (sec) , antiderivative size = 19375, normalized size of antiderivative = 43.15 \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {g x^{6} + f x^{4} + e x^{2} + d}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
1/2*((b*c^2*d - 2*a*c^2*e + a*b*c*f - (a*b^2 - 2*a^2*c)*g)*x^3 - (a*b*c*e - 2*a^2*c*f + a^2*b*g - (b^2*c - 2*a*c^2)*d)*x)/(a^2*b^2*c - 4*a^3*c^2 + ( a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2) - 1/2*integrate( -(a*b*c*e - 2*a^2*c*f + a^2*b*g + (b*c^2*d - 2*a*c^2*e + a*b*c*f + (a*b^2 - 6*a^2*c)*g)*x^2 + (b^2*c - 6*a*c^2)*d)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 8905 vs. \(2 (408) = 816\).
Time = 1.89 (sec) , antiderivative size = 8905, normalized size of antiderivative = 19.83 \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
1/2*(b*c^2*d*x^3 - 2*a*c^2*e*x^3 + a*b*c*f*x^3 - a*b^2*g*x^3 + 2*a^2*c*g*x ^3 + b^2*c*d*x - 2*a*c^2*d*x - a*b*c*e*x + 2*a^2*c*f*x - a^2*b*g*x)/((c*x^ 4 + b*x^2 + a)*(a*b^2*c - 4*a^2*c^2)) + 1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(a *b^2*c - 4*a^2*c^2)^2*d - 2*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2 *c^2)^2*e + (2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(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)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*(a*b^2*c - 4*a^2*c^2)^2* f + (2*a*b^4*c^2 - 20*a^2*b^2*c^3 + 48*a^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr...
Time = 11.16 (sec) , antiderivative size = 32587, normalized size of antiderivative = 72.58 \[ \int \frac {d+e x^2+f x^4+g x^6}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
((x*(2*a*c^2*d - b^2*c*d + a^2*b*g - 2*a^2*c*f + a*b*c*e))/(2*a*c*(4*a*c - b^2)) - (x^3*(b*c^2*d - 2*a*c^2*e - a*b^2*g + 2*a^2*c*g + a*b*c*f))/(2*a* c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - atan(((((6144*a^5*c^7*d + 2048*a^6 *c^6*f - 288*a^2*b^6*c^4*d + 1920*a^3*b^4*c^5*d - 5632*a^4*b^2*c^6*d + 16* a^2*b^7*c^3*e - 192*a^3*b^5*c^4*e + 768*a^4*b^3*c^5*e - 32*a^3*b^6*c^3*f + 384*a^4*b^4*c^4*f - 1536*a^5*b^2*c^5*f + 16*a^3*b^7*c^2*g - 192*a^4*b^5*c ^3*g + 768*a^5*b^3*c^4*g + 16*a*b^8*c^3*d - 1024*a^5*b*c^6*e - 1024*a^6*b* c^5*g)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3)) - (x *((27*a*b^9*c^4*d^2 - a^3*b^11*g^2 - b^11*c^3*d^2 + 3840*a^5*b*c^8*d^2 - 9 *a*c^4*d^2*(-(4*a*c - b^2)^9)^(1/2) + 768*a^6*b*c^7*e^2 + 768*a^7*b*c^6*f^ 2 + 27*a^4*b^9*c*g^2 + 3840*a^8*b*c^5*g^2 + 9*a^4*c*g^2*(-(4*a*c - b^2)^9) ^(1/2) - 288*a^2*b^7*c^5*d^2 + 1504*a^3*b^5*c^6*d^2 - 3840*a^4*b^3*c^7*d^2 - a^2*b^9*c^3*e^2 + 96*a^4*b^5*c^5*e^2 - 512*a^5*b^3*c^6*e^2 + a^2*c^3*e^ 2*(-(4*a*c - b^2)^9)^(1/2) + b^2*c^3*d^2*(-(4*a*c - b^2)^9)^(1/2) - a^3*b^ 9*c^2*f^2 + 96*a^5*b^5*c^4*f^2 - 512*a^6*b^3*c^5*f^2 - a^3*b^2*g^2*(-(4*a* c - b^2)^9)^(1/2) - a^3*c^2*f^2*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2 *g^2 + 1504*a^6*b^5*c^3*g^2 - 3840*a^7*b^3*c^4*g^2 - 3072*a^6*c^8*d*e - 92 16*a^7*c^7*d*g - 1024*a^7*c^7*e*f - 3072*a^8*c^6*f*g - 2*a*b^10*c^3*d*e + 3584*a^6*b*c^7*d*f + 3584*a^7*b*c^6*e*g - 2*a^3*b^10*c*f*g + 36*a^2*b^8*c^ 4*d*e - 192*a^3*b^6*c^5*d*e + 128*a^4*b^4*c^6*d*e + 1536*a^5*b^2*c^7*d*...