Integrand size = 35, antiderivative size = 594 \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {(c f-2 b g) x}{c^3}+\frac {g x^3}{3 c^2}+\frac {x \left (a \left (2 c^3 d-c^2 (b e+2 a f)-b^3 g+b c (b f+3 a g)\right )+\left (b^3 c f+b c^2 (c d-3 a f)-b^4 g-b^2 c (c e-4 a g)+2 a c^2 (c e-a g)\right ) x^2\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)-\frac {3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3 c f-b c^2 (c d+13 a f)-5 b^4 g-b^2 c (c e-24 a g)+2 a c^2 (3 c e-7 a g)+\frac {3 b^4 c f-4 a c^3 (c d-5 a f)-b^2 c^2 (c d+19 a f)-5 b^5 g-b^3 c (c e-34 a g)+4 a b c^2 (2 c e-13 a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{7/2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
(-2*b*g+c*f)*x/c^3+1/3*g*x^3/c^2+1/2*x*(a*(2*c^3*d-c^2*(2*a*f+b*e)-b^3*g+b *c*(3*a*g+b*f))+(b^3*c*f+b*c^2*(-3*a*f+c*d)-b^4*g-b^2*c*(-4*a*g+c*e)+2*a*c ^2*(-a*g+c*e))*x^2)/c^3/(-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))*(3*b^3*c*f-b*c^2*(13*a*f+c*d)-5*b^4* g-b^2*c*(-24*a*g+c*e)+2*a*c^2*(-7*a*g+3*c*e)+(-3*b^4*c*f+4*a*c^3*(-5*a*f+c *d)+b^2*c^2*(19*a*f+c*d)+5*b^5*g+b^3*c*(-34*a*g+c*e)-4*a*b*c^2*(-13*a*g+2* c*e))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(-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))*(3*b^ 3*c*f-b*c^2*(13*a*f+c*d)-5*b^4*g-b^2*c*(-24*a*g+c*e)+2*a*c^2*(-7*a*g+3*c*e )+(3*b^4*c*f-4*a*c^3*(-5*a*f+c*d)-b^2*c^2*(19*a*f+c*d)-5*b^5*g-b^3*c*(-34* a*g+c*e)+4*a*b*c^2*(-13*a*g+2*c*e))/(-4*a*c+b^2)^(1/2))/c^(7/2)/(-4*a*c+b^ 2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.56 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {12 \sqrt {c} (c f-2 b g) x+4 c^{3/2} g x^3+\frac {6 \sqrt {c} x \left (b \left (c^3 d-b c^2 e+b^2 c f-b^3 g\right ) x^2+a^2 c \left (3 b g-2 c \left (f+g x^2\right )\right )+a \left (-b^3 g+2 c^3 \left (d+e x^2\right )-b c^2 \left (e+3 f x^2\right )+b^2 c \left (f+4 g x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \left (-5 b^5 g-b^3 c \left (c e+3 \sqrt {b^2-4 a c} f-34 a g\right )+b^4 \left (3 c f+5 \sqrt {b^2-4 a c} g\right )+2 a c^2 \left (-2 c^2 d-3 c \sqrt {b^2-4 a c} e+10 a c f+7 a \sqrt {b^2-4 a c} g\right )-b^2 c \left (c^2 d-c \sqrt {b^2-4 a c} e+19 a c f+24 a \sqrt {b^2-4 a c} g\right )+b c^2 \left (c \left (\sqrt {b^2-4 a c} d+8 a e\right )+13 a \left (\sqrt {b^2-4 a c} f-4 a g\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \left (5 b^5 g+b^3 c \left (c e-3 \sqrt {b^2-4 a c} f-34 a g\right )+b^4 \left (-3 c f+5 \sqrt {b^2-4 a c} g\right )+b^2 c \left (c^2 d+c \sqrt {b^2-4 a c} e+19 a c f-24 a \sqrt {b^2-4 a c} g\right )+2 a c^2 \left (2 c^2 d-3 c \sqrt {b^2-4 a c} e-10 a c f+7 a \sqrt {b^2-4 a c} g\right )+b c^2 \left (c \left (\sqrt {b^2-4 a c} d-8 a e\right )+13 a \left (\sqrt {b^2-4 a c} f+4 a g\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 c^{7/2}} \]
(12*Sqrt[c]*(c*f - 2*b*g)*x + 4*c^(3/2)*g*x^3 + (6*Sqrt[c]*x*(b*(c^3*d - b *c^2*e + b^2*c*f - b^3*g)*x^2 + a^2*c*(3*b*g - 2*c*(f + g*x^2)) + a*(-(b^3 *g) + 2*c^3*(d + e*x^2) - b*c^2*(e + 3*f*x^2) + b^2*c*(f + 4*g*x^2))))/((b ^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*(-5*b^5*g - b^3*c*(c*e + 3*S qrt[b^2 - 4*a*c]*f - 34*a*g) + b^4*(3*c*f + 5*Sqrt[b^2 - 4*a*c]*g) + 2*a*c ^2*(-2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e + 10*a*c*f + 7*a*Sqrt[b^2 - 4*a*c]* g) - b^2*c*(c^2*d - c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f + 24*a*Sqrt[b^2 - 4*a *c]*g) + b*c^2*(c*(Sqrt[b^2 - 4*a*c]*d + 8*a*e) + 13*a*(Sqrt[b^2 - 4*a*c]* f - 4*a*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]]) + (3*Sqrt[2]*(5*b^5*g + b^3* c*(c*e - 3*Sqrt[b^2 - 4*a*c]*f - 34*a*g) + b^4*(-3*c*f + 5*Sqrt[b^2 - 4*a* c]*g) + b^2*c*(c^2*d + c*Sqrt[b^2 - 4*a*c]*e + 19*a*c*f - 24*a*Sqrt[b^2 - 4*a*c]*g) + 2*a*c^2*(2*c^2*d - 3*c*Sqrt[b^2 - 4*a*c]*e - 10*a*c*f + 7*a*Sq rt[b^2 - 4*a*c]*g) + b*c^2*(c*(Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 13*a*(Sqrt[b ^2 - 4*a*c]*f + 4*a*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]]))/(12*c^(7/2))
Time = 8.04 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2197, 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.
| \(\displaystyle \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {x \left (a \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )+x^2 \left (-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^4 (-g)+b^3 c f\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {2 a \left (4 a-\frac {b^2}{c}\right ) g x^6-\frac {2 a \left (b^2-4 a c\right ) (c f-b g) x^4}{c^2}+\frac {a \left (-g b^4+c f b^3-c (c e-6 a g) b^2-c^2 (c d+5 a f) b+6 a c^2 (c e-a g)\right ) x^2}{c^3}+\frac {a^2 \left (-g b^3+c (b f+3 a g) b+2 c^3 d-c^2 (b e+2 a f)\right )}{c^3}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {x \left (a \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )+x^2 \left (-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^4 (-g)+b^3 c f\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \left (-\frac {2 a \left (b^2-4 a c\right ) g x^2}{c^2}-\frac {2 a \left (b^2-4 a c\right ) (c f-2 b g)}{c^3}+\frac {\left (-5 g b^3+c (3 b f+19 a g) b+2 c^3 d-c^2 (b e+10 a f)\right ) a^2+\left (-5 g b^4+3 c f b^3-c (c e-24 a g) b^2-c^2 (c d+13 a f) b+2 a c^2 (3 c e-7 a g)\right ) x^2 a}{c^3 \left (c x^4+b x^2+a\right )}\right )dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )+x^2 \left (-b^2 c (c e-4 a g)+b c^2 (c d-3 a f)+2 a c^2 (c e-a g)+b^4 (-g)+b^3 c f\right )\right )}{2 c^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-b^2 c (c e-24 a g)-\frac {-b^3 c (c e-34 a g)-b^2 c^2 (19 a f+c d)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)-5 b^5 g+3 b^4 c f}{\sqrt {b^2-4 a c}}-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)-5 b^4 g+3 b^3 c f\right )}{\sqrt {2} c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-b^2 c (c e-24 a g)+\frac {-b^3 c (c e-34 a g)-b^2 c^2 (19 a f+c d)+4 a b c^2 (2 c e-13 a g)-4 a c^3 (c d-5 a f)-5 b^5 g+3 b^4 c f}{\sqrt {b^2-4 a c}}-b c^2 (13 a f+c d)+2 a c^2 (3 c e-7 a g)-5 b^4 g+3 b^3 c f\right )}{\sqrt {2} c^{7/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 a x \left (b^2-4 a c\right ) (c f-2 b g)}{c^3}-\frac {2 a g x^3 \left (b^2-4 a c\right )}{3 c^2}}{2 a \left (b^2-4 a c\right )}\) |
(x*(a*(2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g)) + (b^3*c*f + b*c^2*(c*d - 3*a*f) - b^4*g - b^2*c*(c*e - 4*a*g) + 2*a*c^2*(c*e - a*g) )*x^2))/(2*c^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((-2*a*(b^2 - 4*a*c)*( c*f - 2*b*g)*x)/c^3 - (2*a*(b^2 - 4*a*c)*g*x^3)/(3*c^2) + (a*(3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5*b^4*g - b^2*c*(c*e - 24*a*g) + 2*a*c^2*(3*c*e - 7 *a*g) - (3*b^4*c*f - 4*a*c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^ 5*g - b^3*c*(c*e - 34*a*g) + 4*a*b*c^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c] )*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2 )*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (a*(3*b^3*c*f - b*c^2*(c*d + 13*a*f) - 5* b^4*g - b^2*c*(c*e - 24*a*g) + 2*a*c^2*(3*c*e - 7*a*g) + (3*b^4*c*f - 4*a* c^3*(c*d - 5*a*f) - b^2*c^2*(c*d + 19*a*f) - 5*b^5*g - b^3*c*(c*e - 34*a*g ) + 4*a*b*c^2*(2*c*e - 13*a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c] *x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a *c]]))/(2*a*(b^2 - 4*a*c))
3.2.26.3.1 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 = 0.18 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {g \,x^{3}}{3 c^{2}}-\frac {2 b g x}{c^{3}}+\frac {f x}{c^{2}}+\frac {\frac {\left (2 g \,a^{2} c^{2}-4 a \,b^{2} c g +3 a b \,c^{2} f -2 a \,c^{3} e +b^{4} g -b^{3} c f +b^{2} c^{2} e -b \,c^{3} d \right ) x^{3}}{8 a c -2 b^{2}}-\frac {a \left (3 a b g c -2 a \,c^{2} f -b^{3} g +b^{2} c f -b \,c^{2} e +2 c^{3} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c^{3} \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {\left (14 g \,a^{2} c^{2}-24 a \,b^{2} c g +13 a b \,c^{2} f -6 a \,c^{3} e +5 b^{4} g -3 b^{3} c f +b^{2} c^{2} e +b \,c^{3} d \right ) \textit {\_R}^{2}}{4 a c -b^{2}}+\frac {a \left (19 a b g c -10 a \,c^{2} f -5 b^{3} g +3 b^{2} c f -b \,c^{2} e +2 c^{3} d \right )}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 c^{3}}\) | \(348\) |
| default | \(-\frac {-\frac {1}{3} g \,x^{3} c +2 b g x -c f x}{c^{3}}+\frac {\frac {\frac {\left (2 g \,a^{2} c^{2}-4 a \,b^{2} c g +3 a b \,c^{2} f -2 a \,c^{3} e +b^{4} g -b^{3} c f +b^{2} c^{2} e -b \,c^{3} d \right ) x^{3}}{8 a c -2 b^{2}}-\frac {a \left (3 a b g c -2 a \,c^{2} f -b^{3} g +b^{2} c f -b \,c^{2} e +2 c^{3} d \right ) x}{2 \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\left (-14 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2} g +24 \sqrt {-4 a c +b^{2}}\, a \,b^{2} c g -13 \sqrt {-4 a c +b^{2}}\, a b \,c^{2} f +6 \sqrt {-4 a c +b^{2}}\, a \,c^{3} e -5 \sqrt {-4 a c +b^{2}}\, b^{4} g +3 b^{3} c f \sqrt {-4 a c +b^{2}}-b^{2} c^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{3} d \sqrt {-4 a c +b^{2}}-52 a^{2} c^{2} b g +20 a^{2} c^{3} f +34 a \,b^{3} c g -19 a \,b^{2} c^{2} f +8 b \,c^{3} a e -4 a \,c^{4} d -5 b^{5} g +3 b^{4} c f -b^{3} c^{2} e -b^{2} c^{3} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (-14 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2} g +24 \sqrt {-4 a c +b^{2}}\, a \,b^{2} c g -13 \sqrt {-4 a c +b^{2}}\, a b \,c^{2} f +6 \sqrt {-4 a c +b^{2}}\, a \,c^{3} e -5 \sqrt {-4 a c +b^{2}}\, b^{4} g +3 b^{3} c f \sqrt {-4 a c +b^{2}}-b^{2} c^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{3} d \sqrt {-4 a c +b^{2}}+52 a^{2} c^{2} b g -20 a^{2} c^{3} f -34 a \,b^{3} c g +19 a \,b^{2} c^{2} f -8 b \,c^{3} a e +4 a \,c^{4} d +5 b^{5} g -3 b^{4} c f +b^{3} c^{2} e +b^{2} c^{3} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{c^{3}}\) | \(760\) |
1/3*g*x^3/c^2-2/c^3*b*g*x+f*x/c^2+(1/2*(2*a^2*c^2*g-4*a*b^2*c*g+3*a*b*c^2* f-2*a*c^3*e+b^4*g-b^3*c*f+b^2*c^2*e-b*c^3*d)/(4*a*c-b^2)*x^3-1/2*a*(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)/c^3/(c*x^4+b*x ^2+a)+1/4/c^3*sum((-(14*a^2*c^2*g-24*a*b^2*c*g+13*a*b*c^2*f-6*a*c^3*e+5*b^ 4*g-3*b^3*c*f+b^2*c^2*e+b*c^3*d)/(4*a*c-b^2)*_R^2+a*(19*a*b*c*g-10*a*c^2*f -5*b^3*g+3*b^2*c*f-b*c^2*e+2*c^3*d)/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_R), _R=RootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (g x^{6} + f x^{4} + e x^{2} + d\right )} x^{4}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
1/2*((b*c^3*d - (b^2*c^2 - 2*a*c^3)*e + (b^3*c - 3*a*b*c^2)*f - (b^4 - 4*a *b^2*c + 2*a^2*c^2)*g)*x^3 + (2*a*c^3*d - a*b*c^2*e + (a*b^2*c - 2*a^2*c^2 )*f - (a*b^3 - 3*a^2*b*c)*g)*x)/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^ 5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2) + 1/2*integrate(-(2*a*c^3*d - a*b*c^2* e - (b*c^3*d + (b^2*c^2 - 6*a*c^3)*e - (3*b^3*c - 13*a*b*c^2)*f + (5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*g)*x^2 + (3*a*b^2*c - 10*a^2*c^2)*f - (5*a*b^3 - 19*a^2*b*c)*g)/(c*x^4 + b*x^2 + a), x)/(b^2*c^3 - 4*a*c^4) + 1/3*(c*g*x^3 + 3*(c*f - 2*b*g)*x)/c^3
Leaf count of result is larger than twice the leaf count of optimal. 10752 vs. \(2 (550) = 1100\).
Time = 2.34 (sec) , antiderivative size = 10752, normalized size of antiderivative = 18.10 \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
1/2*(b*c^3*d*x^3 - b^2*c^2*e*x^3 + 2*a*c^3*e*x^3 + b^3*c*f*x^3 - 3*a*b*c^2 *f*x^3 - b^4*g*x^3 + 4*a*b^2*c*g*x^3 - 2*a^2*c^2*g*x^3 + 2*a*c^3*d*x - a*b *c^2*e*x + a*b^2*c*f*x - 2*a^2*c^2*f*x - a*b^3*g*x + 3*a^2*b*c*g*x)/((b^2* c^3 - 4*a*c^4)*(c*x^4 + b*x^2 + a)) + 1/16*((2*b^3*c^5 - 8*a*b*c^6 - sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 + 4*sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 + 2*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^5 - 2*(b^2 - 4*a*c)*b*c^5)*(b^2 *c^3 - 4*a*c^4)^2*d + (2*b^4*c^4 - 20*a*b^2*c^5 + 48*a^2*c^6 - sqrt(2)*sqr t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 10*sqrt(2)*sqrt(b ^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 12*sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a*c^5 - 2*(b^2 - 4*a*c)*b^2*c^4 + 12*(b^2 - 4*a* c)*a*c^5)*(b^2*c^3 - 4*a*c^4)^2*e - (6*b^5*c^3 - 50*a*b^3*c^4 + 104*a^2*b* c^5 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 25*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 6 *sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 52...
Time = 10.71 (sec) , antiderivative size = 47339, normalized size of antiderivative = 79.70 \[ \int \frac {x^4 \left (d+e x^2+f x^4+g x^6\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
((x^3*(b^4*g + b^2*c^2*e + 2*a^2*c^2*g - 2*a*c^3*e - b*c^3*d - b^3*c*f + 3 *a*b*c^2*f - 4*a*b^2*c*g))/(2*(4*a*c - b^2)) + (x*(2*a^2*c^2*f - 2*a*c^3*d + a*b^3*g + a*b*c^2*e - a*b^2*c*f - 3*a^2*b*c*g))/(2*(4*a*c - b^2)))/(a*c ^3 + c^4*x^4 + b*c^3*x^2) + x*(f/c^2 - (2*b*g)/c^3) + atan(((((2048*a^4*c^ 10*d - 10240*a^5*c^9*f + 384*a^2*b^4*c^8*d - 1536*a^3*b^2*c^9*d - 192*a^2* b^5*c^7*e + 768*a^3*b^3*c^8*e + 736*a^2*b^6*c^6*f - 4224*a^3*b^4*c^7*f + 1 0752*a^4*b^2*c^8*f - 1264*a^2*b^7*c^5*g + 7488*a^3*b^5*c^6*g - 19712*a^4*b ^3*c^7*g - 32*a*b^6*c^7*d + 16*a*b^7*c^6*e - 1024*a^4*b*c^9*e - 48*a*b^8*c ^5*f + 80*a*b^9*c^4*g + 19456*a^5*b*c^8*g)/(8*(64*a^3*c^8 - b^6*c^5 + 12*a *b^4*c^6 - 48*a^2*b^2*c^7)) - (x*(-(25*b^15*g^2 + b^9*c^6*d^2 + c^6*d^2*(- (4*a*c - b^2)^9)^(1/2) + b^11*c^4*e^2 + 9*b^13*c^2*f^2 + 25*b^6*g^2*(-(4*a *c - b^2)^9)^(1/2) - 768*a^4*b*c^10*d^2 - 27*a*b^9*c^5*e^2 - 3840*a^5*b*c^ 9*e^2 - 9*a*c^5*e^2*(-(4*a*c - b^2)^9)^(1/2) - 213*a*b^11*c^3*f^2 + 26880* a^6*b*c^8*f^2 - 80640*a^7*b*c^7*g^2 - 30*b^14*c*f*g - 96*a^2*b^5*c^8*d^2 + 512*a^3*b^3*c^9*d^2 + 288*a^2*b^7*c^6*e^2 - 1504*a^3*b^5*c^7*e^2 + 3840*a ^4*b^3*c^8*e^2 + 2077*a^2*b^9*c^4*f^2 - 10656*a^3*b^7*c^5*f^2 + 30240*a^4* b^5*c^6*f^2 - 44800*a^5*b^3*c^7*f^2 + 25*a^2*c^4*f^2*(-(4*a*c - b^2)^9)^(1 /2) + b^2*c^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 6366*a^2*b^11*c^2*g^2 - 35767 *a^3*b^9*c^3*g^2 + 116928*a^4*b^7*c^4*g^2 - 219744*a^5*b^5*c^5*g^2 + 21504 0*a^6*b^3*c^6*g^2 - 49*a^3*c^3*g^2*(-(4*a*c - b^2)^9)^(1/2) + 9*b^4*c^2...