Integrand size = 30, antiderivative size = 244 \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4} \]
-1/6*d/a/x^6+1/4*(-a*e+b*d)/a^2/x^4+1/2*(-b^2*d+a*b*e+a*(-a*f+c*d))/a^3/x^ 2-(b^3*d-a*b^2*e+a^2*c*e-a*b*(-a*f+2*c*d))*ln(x)/a^4+1/4*(b^3*d-a*b^2*e+a^ 2*c*e-a*b*(-a*f+2*c*d))*ln(c*x^4+b*x^2+a)/a^4-1/2*(b^4*d-a*b^3*e+3*a^2*b*c *e+2*a^2*c*(-a*f+c*d)-a*b^2*(-a*f+4*c*d))*arctanh((2*c*x^2+b)/(-4*a*c+b^2) ^(1/2))/a^4/(-4*a*c+b^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.70 \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {2 a^3 d}{x^6}+\frac {3 a^2 (b d-a e)}{x^4}+\frac {6 a \left (-b^2 d+a b e+a (c d-a f)\right )}{x^2}-12 \left (b^3 d-a b^2 e+a^2 c e+a b (-2 c d+a f)\right ) \log (x)+\frac {3 \left (b^4 d+b^3 \left (\sqrt {b^2-4 a c} d-a e\right )+a^2 c \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )+a b^2 \left (-4 c d-\sqrt {b^2-4 a c} e+a f\right )+a b \left (-2 c \sqrt {b^2-4 a c} d+3 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {3 \left (-b^4 d+b^3 \left (\sqrt {b^2-4 a c} d+a e\right )-a b^2 \left (-4 c d+\sqrt {b^2-4 a c} e+a f\right )+a^2 c \left (-2 c d+\sqrt {b^2-4 a c} e+2 a f\right )+a b \left (-2 c \sqrt {b^2-4 a c} d-3 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{12 a^4} \]
((-2*a^3*d)/x^6 + (3*a^2*(b*d - a*e))/x^4 + (6*a*(-(b^2*d) + a*b*e + a*(c* d - a*f)))/x^2 - 12*(b^3*d - a*b^2*e + a^2*c*e + a*b*(-2*c*d + a*f))*Log[x ] + (3*(b^4*d + b^3*(Sqrt[b^2 - 4*a*c]*d - a*e) + a^2*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f) + a*b^2*(-4*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-2 *c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^ 2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (3*(-(b^4*d) + b^3*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*b^2*(-4*c*d + Sqrt[b^2 - 4*a*c]*e + a*f) + a^2*c*(-2*c *d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + a*b*(-2*c*Sqrt[b^2 - 4*a*c]*d - 3*a*c* e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(12*a^4)
Time = 0.66 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2194, 2159, 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.
| \(\displaystyle \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \frac {1}{2} \int \frac {f x^4+e x^2+d}{x^8 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {d}{a x^8}+\frac {d b^4-a e b^3-a (3 c d-a f) b^2+2 a^2 c e b+c \left (d b^3-a e b^2-a (2 c d-a f) b+a^2 c e\right ) x^2+a^2 c (c d-a f)}{a^4 \left (c x^4+b x^2+a\right )}+\frac {-d b^3+a e b^2+a (2 c d-a f) b-a^2 c e}{a^4 x^2}+\frac {d b^2-a e b-a (c d-a f)}{a^3 x^4}+\frac {a e-b d}{a^2 x^6}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-a b e-a (c d-a f)+b^2 d}{a^3 x^2}+\frac {b d-a e}{2 a^2 x^4}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right )}{a^4 \sqrt {b^2-4 a c}}-\frac {\log \left (x^2\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}+\frac {\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{2 a^4}-\frac {d}{3 a x^6}\right )\) |
(-1/3*d/(a*x^6) + (b*d - a*e)/(2*a^2*x^4) - (b^2*d - a*b*e - a*(c*d - a*f) )/(a^3*x^2) - ((b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^ 2*(4*c*d - a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - ((b^3*d - a*b^2*e + a^2*c*e - a*b*(2*c*d - a*f))*Log[x^2])/a^4 + ((b^3*d - a*b^2*e + a^2*c*e - a*b*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4]) /(2*a^4))/2
3.1.54.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Time = 0.15 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {d}{6 a \,x^{6}}-\frac {a e -b d}{4 a^{2} x^{4}}-\frac {f \,a^{2}-a b e -a c d +b^{2} d}{2 a^{3} x^{2}}+\frac {\left (-a^{2} b f -a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) \ln \left (x \right )}{a^{4}}-\frac {\frac {\left (-a^{2} b c f -a^{2} c^{2} e +a \,b^{2} c e +2 a b \,c^{2} d -b^{3} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{3} c f -a^{2} b^{2} f -2 a^{2} b c e -a^{2} c^{2} d +a \,b^{3} e +3 a \,b^{2} c d -d \,b^{4}-\frac {\left (-a^{2} b c f -a^{2} c^{2} e +a \,b^{2} c e +2 a b \,c^{2} d -b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a^{4}}\) | \(294\) |
| risch | \(\frac {-\frac {\left (f \,a^{2}-a b e -a c d +b^{2} d \right ) x^{4}}{2 a^{3}}-\frac {\left (a e -b d \right ) x^{2}}{4 a^{2}}-\frac {d}{6 a}}{x^{6}}-\frac {\ln \left (x \right ) b f}{a^{2}}-\frac {\ln \left (x \right ) c e}{a^{2}}+\frac {\ln \left (x \right ) b^{2} e}{a^{3}}+\frac {2 \ln \left (x \right ) b c d}{a^{3}}-\frac {\ln \left (x \right ) b^{3} d}{a^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{5}-b^{2} a^{4}\right ) \textit {\_Z}^{2}+\left (-4 a^{3} b c f -4 a^{3} c^{2} e +a^{2} b^{3} f +5 a^{2} b^{2} c e +8 a^{2} b \,c^{2} d -a \,b^{4} e -6 a \,b^{3} c d +b^{5} d \right ) \textit {\_Z} +a^{2} c^{2} f^{2}-a b \,c^{2} e f -2 a \,c^{3} d f +a \,c^{3} e^{2}+b^{2} c^{2} d f -b \,c^{3} d e +d^{2} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a^{7} c -3 b^{2} a^{6}\right ) \textit {\_R}^{2}+\left (-4 a^{5} b c f -5 a^{5} c^{2} e +4 a^{4} b^{2} c e +9 a^{4} b \,c^{2} d -4 a^{3} b^{3} c d \right ) \textit {\_R} +2 a^{4} c^{2} f^{2}-4 a^{3} b \,c^{2} e f -4 a^{3} c^{3} d f +4 a^{2} b^{2} c^{2} d f +2 a^{2} b^{2} c^{2} e^{2}+4 a^{2} b \,c^{3} d e +2 a^{2} c^{4} d^{2}-4 a \,b^{3} c^{2} d e -4 a \,b^{2} c^{3} d^{2}+2 b^{4} c^{2} d^{2}\right ) x^{2}-a^{7} b \,\textit {\_R}^{2}+\left (a^{6} c f -2 a^{5} b^{2} f -3 a^{5} b c e -a^{5} c^{2} d +2 a^{4} b^{3} e +5 a^{4} b^{2} c d -2 a^{3} b^{4} d \right ) \textit {\_R} +2 a^{4} b c \,f^{2}+2 a^{4} c^{2} e f -4 a^{3} b^{2} c e f -6 a^{3} b \,c^{2} d f -2 a^{3} b \,c^{2} e^{2}-2 a^{3} c^{3} d e +4 a^{2} b^{3} c d f +2 a^{2} b^{3} c \,e^{2}+8 a^{2} b^{2} c^{2} d e +4 a^{2} b \,c^{3} d^{2}-4 a \,b^{4} c d e -6 a \,b^{3} c^{2} d^{2}+2 b^{5} c \,d^{2}\right )\right )}{2}\) | \(667\) |
-1/6*d/a/x^6-1/4*(a*e-b*d)/a^2/x^4-1/2*(a^2*f-a*b*e-a*c*d+b^2*d)/a^3/x^2+1 /a^4*(-a^2*b*f-a^2*c*e+a*b^2*e+2*a*b*c*d-b^3*d)*ln(x)-1/2/a^4*(1/2*(-a^2*b *c*f-a^2*c^2*e+a*b^2*c*e+2*a*b*c^2*d-b^3*c*d)/c*ln(c*x^4+b*x^2+a)+2*(a^3*c *f-a^2*b^2*f-2*a^2*b*c*e-a^2*c^2*d+a*b^3*e+3*a*b^2*c*d-d*b^4-1/2*(-a^2*b*c *f-a^2*c^2*e+a*b^2*c*e+2*a*b*c^2*d-b^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan( (2*c*x^2+b)/(4*a*c-b^2)^(1/2)))
Time = 1.67 (sec) , antiderivative size = 834, normalized size of antiderivative = 3.42 \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\left [-\frac {3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (x\right ) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}, -\frac {6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (x\right ) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}\right ] \]
[-1/12*(3*sqrt(b^2 - 4*a*c)*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3* a^2*b*c)*e + (a^2*b^2 - 2*a^3*c)*f)*x^6*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - 3*((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d - (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e + (a^2*b ^3 - 4*a^3*b*c)*f)*x^6*log(c*x^4 + b*x^2 + a) + 12*((b^5 - 6*a*b^3*c + 8*a ^2*b*c^2)*d - (a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*e + (a^2*b^3 - 4*a^3*b*c)* f)*x^6*log(x) + 6*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d - (a^2*b^3 - 4*a^3* b*c)*e + (a^3*b^2 - 4*a^4*c)*f)*x^4 - 3*((a^2*b^3 - 4*a^3*b*c)*d - (a^3*b^ 2 - 4*a^4*c)*e)*x^2 + 2*(a^3*b^2 - 4*a^4*c)*d)/((a^4*b^2 - 4*a^5*c)*x^6), -1/12*(6*sqrt(-b^2 + 4*a*c)*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3* a^2*b*c)*e + (a^2*b^2 - 2*a^3*c)*f)*x^6*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 3*((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d - (a*b^4 - 5* a^2*b^2*c + 4*a^3*c^2)*e + (a^2*b^3 - 4*a^3*b*c)*f)*x^6*log(c*x^4 + b*x^2 + a) + 12*((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d - (a*b^4 - 5*a^2*b^2*c + 4*a^ 3*c^2)*e + (a^2*b^3 - 4*a^3*b*c)*f)*x^6*log(x) + 6*((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d - (a^2*b^3 - 4*a^3*b*c)*e + (a^3*b^2 - 4*a^4*c)*f)*x^4 - 3*( (a^2*b^3 - 4*a^3*b*c)*d - (a^3*b^2 - 4*a^4*c)*e)*x^2 + 2*(a^3*b^2 - 4*a^4* c)*d)/((a^4*b^2 - 4*a^5*c)*x^6)]
Timed out. \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.61 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e + a^{2} b f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e + a^{2} b f\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e + a^{2} b^{2} f - 2 \, a^{3} c f\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{4}} + \frac {11 \, b^{3} d x^{6} - 22 \, a b c d x^{6} - 11 \, a b^{2} e x^{6} + 11 \, a^{2} c e x^{6} + 11 \, a^{2} b f x^{6} - 6 \, a b^{2} d x^{4} + 6 \, a^{2} c d x^{4} + 6 \, a^{2} b e x^{4} - 6 \, a^{3} f x^{4} + 3 \, a^{2} b d x^{2} - 3 \, a^{3} e x^{2} - 2 \, a^{3} d}{12 \, a^{4} x^{6}} \]
1/4*(b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e + a^2*b*f)*log(c*x^4 + b*x^2 + a)/a^4 - 1/2*(b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e + a^2*b*f)*log(x^2)/a^ 4 + 1/2*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e + a^2*b ^2*f - 2*a^3*c*f)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4* a*c)*a^4) + 1/12*(11*b^3*d*x^6 - 22*a*b*c*d*x^6 - 11*a*b^2*e*x^6 + 11*a^2* c*e*x^6 + 11*a^2*b*f*x^6 - 6*a*b^2*d*x^4 + 6*a^2*c*d*x^4 + 6*a^2*b*e*x^4 - 6*a^3*f*x^4 + 3*a^2*b*d*x^2 - 3*a^3*e*x^2 - 2*a^3*d)/(a^4*x^6)
Time = 17.78 (sec) , antiderivative size = 9141, normalized size of antiderivative = 37.46 \[ \int \frac {d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(atan((16*a^12*(4*a*c - b^2)^(3/2)*(x^2*((((a^3*c^8*d^3 - b^6*c^5*d^3 - a^ 6*c^5*f^3 + 3*a*b^4*c^6*d^3 - 3*a^4*c^7*d^2*f + 3*a^5*c^6*d*f^2 - 3*a^2*b^ 2*c^7*d^3 + a^3*b^3*c^5*e^3 + 3*a*b^5*c^5*d^2*e + 3*a^3*b*c^7*d^2*e + 3*a^ 5*b*c^5*e*f^2 - 6*a^2*b^3*c^6*d^2*e - 3*a^2*b^4*c^5*d*e^2 + 3*a^3*b^2*c^6* d*e^2 - 3*a^2*b^4*c^5*d^2*f + 6*a^3*b^2*c^6*d^2*f - 3*a^4*b^2*c^5*d*f^2 - 3*a^4*b^2*c^5*e^2*f - 6*a^4*b*c^6*d*e*f + 6*a^3*b^3*c^5*d*e*f)/a^9 - (((11 *a^5*b*c^6*d^2 - 5*a^6*b*c^5*e^2 + 6*a^7*b*c^4*f^2 + 6*a^3*b^5*c^4*d^2 - 1 7*a^4*b^3*c^5*d^2 + 6*a^5*b^3*c^4*e^2 - 5*a^6*c^6*d*e + 5*a^7*c^5*e*f - 17 *a^6*b*c^5*d*f - 12*a^4*b^4*c^4*d*e + 22*a^5*b^2*c^5*d*e + 12*a^5*b^3*c^4* d*f - 12*a^6*b^2*c^4*e*f)/a^9 + (((20*a^9*c^4*f - 20*a^8*c^5*d + 2*a^6*b^4 *c^3*d + 8*a^7*b^2*c^4*d - 2*a^7*b^3*c^3*e + 2*a^8*b^2*c^3*f - 10*a^8*b*c^ 4*e)/a^9 + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*f - 8*a^ 3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2 *b^2*c*e))/(2*a^9*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3* c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b ^2*c*e))/(2*(16*a^5*c - 4*a^4*b^2)))*(2*b^5*d + 2*a^2*b^3*f - 8*a^3*c^2*e - 2*a*b^4*e - 12*a*b^3*c*d - 8*a^3*b*c*f + 16*a^2*b*c^2*d + 10*a^2*b^2*c*e ))/(2*(16*a^5*c - 4*a^4*b^2)) + (((((20*a^9*c^4*f - 20*a^8*c^5*d + 2*a^6*b ^4*c^3*d + 8*a^7*b^2*c^4*d - 2*a^7*b^3*c^3*e + 2*a^8*b^2*c^3*f - 10*a^8*b* c^4*e)/a^9 + ((40*a^10*b*c^3 - 12*a^9*b^3*c^2)*(2*b^5*d + 2*a^2*b^3*f -...