Integrand size = 35, antiderivative size = 471 \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {g x}{c^2}-\frac {x \left (b c (c d+a f)-a b^2 g-2 a c (c e-a g)+\left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right ) x^2\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)+\frac {b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (2 c^3 d-c^2 (b e-6 a f)+3 b^3 g-b c (b f+13 a g)-\frac {b^3 c f-4 b c^2 (c d+2 a f)-3 b^4 g+4 a c^2 (c e-5 a g)+b^2 c (c e+19 a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
g*x/c^2-1/2*x*(b*c*(a*f+c*d)-a*b^2*g-2*a*c*(-a*g+c*e)+(2*c^3*d-c^2*(2*a*f+ b*e)-b^3*g+b*c*(3*a*g+b*f))*x^2)/c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-1/4*arct an(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(2*c^3*d-c^2*(-6*a*f+b* e)+3*b^3*g-b*c*(13*a*g+b*f)+(b^3*c*f-4*b*c^2*(2*a*f+c*d)-3*b^4*g+4*a*c^2*( -5*a*g+c*e)+b^2*c*(19*a*g+c*e))/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^2)*2 ^(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))*(2*c^3*d-c^2*(-6*a*f+b*e)+3*b^3*g-b*c*(13*a*g+b*f)+(- b^3*c*f+4*b*c^2*(2*a*f+c*d)+3*b^4*g-4*a*c^2*(-5*a*g+c*e)-b^2*c*(19*a*g+c*e ))/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4*a*c+b^2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2)) ^(1/2)
Time = 1.16 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {4 \sqrt {c} g x-\frac {2 \sqrt {c} x \left (-b^3 g x^2+b^2 \left (-a g+c f x^2\right )+2 c \left (a^2 g+c^2 d x^2-a c \left (e+f x^2\right )\right )+b c \left (c \left (d-e x^2\right )+a \left (f+3 g x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\sqrt {2} \left (-3 b^4 g+b^2 c \left (c e-\sqrt {b^2-4 a c} f+19 a g\right )+2 c^2 \left (c \sqrt {b^2-4 a c} d+2 a c e+3 a \sqrt {b^2-4 a c} f-10 a^2 g\right )+b^3 \left (c f+3 \sqrt {b^2-4 a c} g\right )-b c \left (4 c^2 d+c \sqrt {b^2-4 a c} e+8 a c f+13 a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (3 b^4 g-b^2 c \left (c e+\sqrt {b^2-4 a c} f+19 a g\right )+2 c^2 \left (c \sqrt {b^2-4 a c} d-2 a c e+3 a \sqrt {b^2-4 a c} f+10 a^2 g\right )+b^3 \left (-c f+3 \sqrt {b^2-4 a c} g\right )+b c \left (4 c^2 d-c \sqrt {b^2-4 a c} e+8 a c f-13 a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 c^{5/2}} \]
(4*Sqrt[c]*g*x - (2*Sqrt[c]*x*(-(b^3*g*x^2) + b^2*(-(a*g) + c*f*x^2) + 2*c *(a^2*g + c^2*d*x^2 - a*c*(e + f*x^2)) + b*c*(c*(d - e*x^2) + a*(f + 3*g*x ^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*(-3*b^4*g + b^2*c*(c *e - Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d + 2*a*c* e + 3*a*Sqrt[b^2 - 4*a*c]*f - 10*a^2*g) + b^3*(c*f + 3*Sqrt[b^2 - 4*a*c]*g ) - b*c*(4*c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 8*a*c*f + 13*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]*(3*b^4*g - b^2*c*(c*e + Sqrt[b^2 - 4*a*c]*f + 19*a*g) + 2*c^2*(c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + 3 *a*Sqrt[b^2 - 4*a*c]*f + 10*a^2*g) + b^3*(-(c*f) + 3*Sqrt[b^2 - 4*a*c]*g) + b*c*(4*c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 8*a*c*f - 13*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*c^(5/2))
Time = 3.93 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2197, 25, 2205, 2009}
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.
| \(\displaystyle \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle -\frac {\int -\frac {-2 a \left (4 a-\frac {b^2}{c}\right ) g x^4-\frac {a \left (g b^3-c (b f+5 a g) b+2 c^3 d-c^2 (b e-6 a f)\right ) x^2}{c^2}+\frac {a \left (-a g b^2+c (c d+a f) b-2 a c (c e-a g)\right )}{c^2}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-2 a \left (4 a-\frac {b^2}{c}\right ) g x^4-\frac {a \left (g b^3-c (b f+5 a g) b+2 c^3 d-c^2 (b e-6 a f)\right ) x^2}{c^2}+\frac {a \left (-a g b^2+c (c d+a f) b-2 a c (c e-a g)\right )}{c^2}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {\int \left (\frac {2 a \left (b^2-4 a c\right ) g}{c^2}+\frac {a \left (-3 a g b^2+c (c d+a f) b-2 a c (c e-5 a g)\right )-a \left (3 g b^3-c (b f+13 a g) b+2 c^3 d-c^2 (b e-6 a f)\right ) x^2}{c^2 \left (c x^4+b x^2+a\right )}\right )dx}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)-3 b^4 g+b^3 c f}{\sqrt {b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{\sqrt {2} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {b^2 c (19 a g+c e)-4 b c^2 (2 a f+c d)+4 a c^2 (c e-5 a g)-3 b^4 g+b^3 c f}{\sqrt {b^2-4 a c}}-c^2 (b e-6 a f)-b c (13 a g+b f)+3 b^3 g+2 c^3 d\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 a g x \left (b^2-4 a c\right )}{c^2}}{2 a \left (b^2-4 a c\right )}-\frac {x \left (x^2 \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )-a b^2 g+b c (a f+c d)-2 a c (c e-a g)\right )}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
-1/2*(x*(b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g) + (2*c^3*d - c^2*(b *e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g))*x^2))/(c^2*(b^2 - 4*a*c)*(a + b*x ^2 + c*x^4)) + ((2*a*(b^2 - 4*a*c)*g*x)/c^2 - (a*(2*c^3*d - c^2*(b*e - 6*a *f) + 3*b^3*g - b*c*(b*f + 13*a*g) + (b^3*c*f - 4*b*c^2*(c*d + 2*a*f) - 3* b^4*g + 4*a*c^2*(c*e - 5*a*g) + b^2*c*(c*e + 19*a*g))/Sqrt[b^2 - 4*a*c])*A rcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*S qrt[b - Sqrt[b^2 - 4*a*c]]) - (a*(2*c^3*d - c^2*(b*e - 6*a*f) + 3*b^3*g - b*c*(b*f + 13*a*g) - (b^3*c*f - 4*b*c^2*(c*d + 2*a*f) - 3*b^4*g + 4*a*c^2* (c*e - 5*a*g) + b^2*c*(c*e + 19*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*S qrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c))
3.2.27.3.1 Defintions of rubi rules used
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, 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)*Qx + 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]] /; Fre eQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {g x}{c^{2}}+\frac {\frac {\left (3 a b g c -2 a \,c^{2} f -b^{3} g +b^{2} c f -b \,c^{2} e +2 c^{3} d \right ) x^{3}}{8 a c -2 b^{2}}+\frac {\left (2 a^{2} c g -a \,b^{2} g +a b c f -2 a \,c^{2} e +b \,c^{2} d \right ) x}{8 a c -2 b^{2}}}{c^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (13 a b g c -6 a \,c^{2} f -3 b^{3} g +b^{2} c f +b \,c^{2} e -2 c^{3} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {10 a^{2} c g -3 a \,b^{2} g +a b c f -2 a \,c^{2} e +b \,c^{2} d}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 c^{2}}\) | \(274\) |
| default | \(\frac {g x}{c^{2}}-\frac {\frac {-\frac {\left (3 a b g c -2 a \,c^{2} f -b^{3} g +b^{2} c f -b \,c^{2} e +2 c^{3} d \right ) x^{3}}{2 \left (4 a c -b^{2}\right )}-\frac {\left (2 a^{2} c g -a \,b^{2} g +a b c f -2 a \,c^{2} e +b \,c^{2} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\left (13 \sqrt {-4 a c +b^{2}}\, a b c g -6 a \,c^{2} f \sqrt {-4 a c +b^{2}}-3 b^{3} g \sqrt {-4 a c +b^{2}}+b^{2} c f \sqrt {-4 a c +b^{2}}+b \,c^{2} e \sqrt {-4 a c +b^{2}}-2 c^{3} d \sqrt {-4 a c +b^{2}}-20 g \,a^{2} c^{2}+19 a \,b^{2} c g -8 a b \,c^{2} f +4 a \,c^{3} e -3 b^{4} g +b^{3} c f +b^{2} c^{2} e -4 b \,c^{3} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (13 \sqrt {-4 a c +b^{2}}\, a b c g -6 a \,c^{2} f \sqrt {-4 a c +b^{2}}-3 b^{3} g \sqrt {-4 a c +b^{2}}+b^{2} c f \sqrt {-4 a c +b^{2}}+b \,c^{2} e \sqrt {-4 a c +b^{2}}-2 c^{3} d \sqrt {-4 a c +b^{2}}+20 g \,a^{2} c^{2}-19 a \,b^{2} c g +8 a b \,c^{2} f -4 a \,c^{3} e +3 b^{4} g -b^{3} c f -b^{2} c^{2} e +4 b \,c^{3} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{c^{2}}\) | \(584\) |
g*x/c^2+(1/2*(3*a*b*c*g-2*a*c^2*f-b^3*g+b^2*c*f-b*c^2*e+2*c^3*d)/(4*a*c-b^ 2)*x^3+1/2*(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)*x)/c^ 2/(c*x^4+b*x^2+a)+1/4/c^2*sum((-(13*a*b*c*g-6*a*c^2*f-3*b^3*g+b^2*c*f+b*c^ 2*e-2*c^3*d)/(4*a*c-b^2)*_R^2-(10*a^2*c*g-3*a*b^2*g+a*b*c*f-2*a*c^2*e+b*c^ 2*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. 23774 vs. \(2 (430) = 860\).
Time = 181.70 (sec) , antiderivative size = 23774, normalized size of antiderivative = 50.48 \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (g x^{6} + f x^{4} + e x^{2} + d\right )} x^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
-1/2*((2*c^3*d - b*c^2*e + (b^2*c - 2*a*c^2)*f - (b^3 - 3*a*b*c)*g)*x^3 + (b*c^2*d - 2*a*c^2*e + a*b*c*f - (a*b^2 - 2*a^2*c)*g)*x)/(a*b^2*c^2 - 4*a^ 2*c^3 + (b^2*c^3 - 4*a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2) + g*x/c^2 + 1 /2*integrate((b*c^2*d - 2*a*c^2*e + a*b*c*f - (2*c^3*d - b*c^2*e - (b^2*c - 6*a*c^2)*f + (3*b^3 - 13*a*b*c)*g)*x^2 - (3*a*b^2 - 10*a^2*c)*g)/(c*x^4 + b*x^2 + a), x)/(b^2*c^2 - 4*a*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 9152 vs. \(2 (430) = 860\).
Time = 2.04 (sec) , antiderivative size = 9152, normalized size of antiderivative = 19.43 \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
g*x/c^2 - 1/2*(2*c^3*d*x^3 - b*c^2*e*x^3 + b^2*c*f*x^3 - 2*a*c^2*f*x^3 - b ^3*g*x^3 + 3*a*b*c*g*x^3 + b*c^2*d*x - 2*a*c^2*e*x + a*b*c*f*x - a*b^2*g*x + 2*a^2*c*g*x)/((c*x^4 + b*x^2 + a)*(b^2*c^2 - 4*a*c^3)) - 1/16*(2*(2*b^2 *c^5 - 8*a*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c ^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - s qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*c^5 - 2*(b^2 - 4* a*c)*c^5)*(b^2*c^2 - 4*a*c^3)^2*d - (2*b^3*c^4 - 8*a*b*c^5 - sqrt(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)*sqrt(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)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(b^2*c^2 - 4 *a*c^3)^2*e - (2*b^4*c^3 - 20*a*b^2*c^4 + 48*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 12*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 + s qrt(b^2 - 4*a*c)*c)*b^2*c^3 + 6*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)*b^2*c^3 + 12*(b^2 - 4*a*c)*a*c^...
Time = 10.39 (sec) , antiderivative size = 36589, normalized size of antiderivative = 77.68 \[ \int \frac {x^2 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
((x^3*(2*c^3*d - b^3*g - 2*a*c^2*f - b*c^2*e + b^2*c*f + 3*a*b*c*g))/(2*(4 *a*c - b^2)) + (x*(b*c^2*d - 2*a*c^2*e - a*b^2*g + 2*a^2*c*g + a*b*c*f))/( 2*(4*a*c - b^2)))/(a*c^2 + c^3*x^4 + b*c^2*x^2) - atan(((((10240*a^5*c^7*g - 16*b^7*c^5*d - 2048*a^4*c^8*e - 768*a^2*b^3*c^7*d - 384*a^2*b^4*c^6*e + 1536*a^3*b^2*c^7*e + 192*a^2*b^5*c^5*f - 768*a^3*b^3*c^6*f - 736*a^2*b^6* c^4*g + 4224*a^3*b^4*c^5*g - 10752*a^4*b^2*c^6*g + 192*a*b^5*c^6*d + 1024* a^3*b*c^8*d + 32*a*b^6*c^5*e - 16*a*b^7*c^4*f + 1024*a^4*b*c^7*f + 48*a*b^ 8*c^3*g)/(8*(64*a^3*c^6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) - (x*( (c^5*d^2*(-(4*a*c - b^2)^9)^(1/2) - b^9*c^5*d^2 - 9*a*b^13*g^2 + 768*a^4*b *c^9*d^2 - a*b^9*c^4*e^2 + 768*a^5*b*c^8*e^2 - a*c^4*e^2*(-(4*a*c - b^2)^9 )^(1/2) - a*b^11*c^2*f^2 + 3840*a^6*b*c^7*f^2 - 9*a*b^4*g^2*(-(4*a*c - b^2 )^9)^(1/2) + 213*a^2*b^11*c*g^2 - 26880*a^7*b*c^6*g^2 + 96*a^2*b^5*c^7*d^2 - 512*a^3*b^3*c^8*d^2 + 96*a^3*b^5*c^6*e^2 - 512*a^4*b^3*c^7*e^2 + 27*a^2 *b^9*c^3*f^2 - 288*a^3*b^7*c^4*f^2 + 1504*a^4*b^5*c^5*f^2 - 3840*a^5*b^3*c ^6*f^2 + 9*a^2*c^3*f^2*(-(4*a*c - b^2)^9)^(1/2) - 2077*a^3*b^9*c^2*g^2 + 1 0656*a^4*b^7*c^3*g^2 - 30240*a^5*b^5*c^4*g^2 + 44800*a^6*b^3*c^5*g^2 - 25* a^3*c^2*g^2*(-(4*a*c - b^2)^9)^(1/2) - 1024*a^5*c^9*d*e + 5120*a^6*c^8*d*g - 3072*a^6*c^8*e*f + 15360*a^7*c^7*f*g + 12*a*b^8*c^5*d*e + 6*a*b^9*c^4*d *f + 3584*a^5*b*c^8*d*f + 6*a*c^4*d*f*(-(4*a*c - b^2)^9)^(1/2) - 18*a*b^10 *c^3*d*g - 2*a*b^10*c^3*e*f + 6*a*b^11*c^2*e*g + 1536*a^6*b*c^7*e*g - 1...