Integrand size = 35, antiderivative size = 542 \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {d}{3 a^2 x^3}+\frac {2 b d-a e}{a^3 x}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}+2 c^2 d+3 b c e-\frac {b^2 (4 c d+b e)}{a}+b^2 f-a (2 c f+b g)\right )+c \left (b^3 d-a b^2 e-a b (3 c d-a f)+2 a^2 (c e-a g)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)+\frac {5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (5 b^3 d-3 a b^2 e-a b (19 c d-a f)+2 a^2 (5 c e-a g)-\frac {5 b^4 d-3 a b^3 e+4 a^2 c (7 c d-3 a f)-a b^2 (29 c d-a f)+4 a^2 b (4 c e+a g)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \]
-1/3*d/a^2/x^3+(-a*e+2*b*d)/a^3/x+1/2*x*(a^2*(b^4*d/a^2+2*c^2*d+3*b*c*e-b^ 2*(b*e+4*c*d)/a+b^2*f-a*(b*g+2*c*f))+c*(b^3*d-a*b^2*e-a*b*(-a*f+3*c*d)+2*a ^2*(-a*g+c*e))*x^2)/a^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))*c^(1/2)*(5*b^3*d-3*a*b^2*e-a*b*(-a*f +19*c*d)+2*a^2*(-a*g+5*c*e)+(5*b^4*d-3*a*b^3*e+4*a^2*c*(-3*a*f+7*c*d)-a*b^ 2*(-a*f+29*c*d)+4*a^2*b*(a*g+4*c*e))/(-4*a*c+b^2)^(1/2))/a^3/(-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))*c^(1/2)*(5*b^3*d-3*a*b^2*e-a*b*(-a*f+19*c*d)+2*a^2*( -a*g+5*c*e)+(-5*b^4*d+3*a*b^3*e-4*a^2*c*(-3*a*f+7*c*d)+a*b^2*(-a*f+29*c*d) -4*a^2*b*(a*g+4*c*e))/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)*2^(1/2)/(b+(-4* a*c+b^2)^(1/2))^(1/2)
Time = 1.36 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.13 \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {4 a d}{x^3}+\frac {24 b d-12 a e}{x}+\frac {6 x \left (b^4 d+b^3 \left (-a e+c d x^2\right )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )+a b \left (-a^2 g-3 c^2 d x^2+a c \left (3 e+f x^2\right )\right )+2 a^2 c \left (c \left (d+e x^2\right )-a \left (f+g x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4 d+b^3 \left (5 \sqrt {b^2-4 a c} d-3 a e\right )+a b^2 \left (-29 c d-3 \sqrt {b^2-4 a c} e+a f\right )+a b \left (-19 c \sqrt {b^2-4 a c} d+16 a c e+a \sqrt {b^2-4 a c} f+4 a^2 g\right )-2 a^2 \left (-14 c^2 d-5 c \sqrt {b^2-4 a c} e+6 a c f+a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4 d-b^3 \left (5 \sqrt {b^2-4 a c} d+3 a e\right )+a b^2 \left (-29 c d+3 \sqrt {b^2-4 a c} e+a f\right )+a b \left (19 c \sqrt {b^2-4 a c} d+16 a c e-a \sqrt {b^2-4 a c} f+4 a^2 g\right )+2 a^2 \left (14 c^2 d-5 c \sqrt {b^2-4 a c} e-6 a c f+a \sqrt {b^2-4 a c} g\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 a^3} \]
((-4*a*d)/x^3 + (24*b*d - 12*a*e)/x + (6*x*(b^4*d + b^3*(-(a*e) + c*d*x^2) + a*b^2*(a*f - c*(4*d + e*x^2)) + a*b*(-(a^2*g) - 3*c^2*d*x^2 + a*c*(3*e + f*x^2)) + 2*a^2*c*(c*(d + e*x^2) - a*(f + g*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4*d + b^3*(5*Sqrt[b^2 - 4*a*c]*d - 3*a*e) + a*b^2*(-29*c*d - 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-19*c*Sqr t[b^2 - 4*a*c]*d + 16*a*c*e + a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) - 2*a^2*(-1 4*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e + 6*a*c*f + a*Sqrt[b^2 - 4*a*c]*g))*ArcT an[(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]*Sqrt[c]*(5*b^4*d - b^3*(5*Sqrt[b ^2 - 4*a*c]*d + 3*a*e) + a*b^2*(-29*c*d + 3*Sqrt[b^2 - 4*a*c]*e + a*f) + a *b*(19*c*Sqrt[b^2 - 4*a*c]*d + 16*a*c*e - a*Sqrt[b^2 - 4*a*c]*f + 4*a^2*g) + 2*a^2*(14*c^2*d - 5*c*Sqrt[b^2 - 4*a*c]*e - 6*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]]))/(12*a^3)
Time = 3.95 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2198, 25, 2195, 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.
| \(\displaystyle \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {x \left (a^2 \left (\frac {b^4 d}{a^2}-\frac {b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {c \left (\frac {d b^3}{a^2}-\frac {(3 c d+b e) b}{a}+f b+2 c e-2 a g\right ) x^6+\frac {\left (d b^4-a e b^3-a (6 c d-a f) b^2+a^2 (5 c e+a g) b+6 a^2 c (c d-a f)\right ) x^4}{a^2}-\frac {2 \left (b^2-4 a c\right ) (b d-a e) x^2}{a}+2 \left (b^2-4 a c\right ) d}{x^4 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c \left (\frac {d b^3}{a^2}-\frac {(3 c d+b e) b}{a}+f b+2 c e-2 a g\right ) x^6+\frac {\left (d b^4-a e b^3-a (6 c d-a f) b^2+a^2 (5 c e+a g) b+6 a^2 c (c d-a f)\right ) x^4}{a^2}-\frac {2 \left (b^2-4 a c\right ) (b d-a e) x^2}{a}+2 \left (b^2-4 a c\right ) d}{x^4 \left (c x^4+b x^2+a\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}-\frac {b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {\int \left (-\frac {2 \left (4 a c-b^2\right ) d}{a x^4}+\frac {5 d b^4-3 a e b^3-a (24 c d-a f) b^2+a^2 (13 c e+a g) b+c \left (5 d b^3-3 a e b^2-a (19 c d-a f) b+2 a^2 (5 c e-a g)\right ) x^2+2 a^2 c (7 c d-3 a f)}{a^2 \left (c x^4+b x^2+a\right )}-\frac {2 \left (4 a c-b^2\right ) (a e-2 b d)}{a^2 x^2}\right )dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}-\frac {b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt {b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{\sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {4 a^2 b (a g+4 c e)+4 a^2 c (7 c d-3 a f)-3 a b^3 e-a b^2 (29 c d-a f)+5 b^4 d}{\sqrt {b^2-4 a c}}+2 a^2 (5 c e-a g)-3 a b^2 e-a b (19 c d-a f)+5 b^3 d\right )}{\sqrt {2} a^2 \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {2 \left (b^2-4 a c\right ) (2 b d-a e)}{a^2 x}-\frac {2 d \left (b^2-4 a c\right )}{3 a x^3}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (a^2 \left (\frac {b^4 d}{a^2}-\frac {b^2 (b e+4 c d)}{a}-a (b g+2 c f)+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 (c e-a g)-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
(x*(a^2*((b^4*d)/a^2 + 2*c^2*d + 3*b*c*e - (b^2*(4*c*d + b*e))/a + b^2*f - a*(2*c*f + b*g)) + c*(b^3*d - a*b^2*e - a*b*(3*c*d - a*f) + 2*a^2*(c*e - a*g))*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((-2*(b^2 - 4*a*c) *d)/(3*a*x^3) + (2*(b^2 - 4*a*c)*(2*b*d - a*e))/(a^2*x) + (Sqrt[c]*(5*b^3* d - 3*a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) + (5*b^4*d - 3*a* b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4* a*c]]])/(Sqrt[2]*a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(5*b^3*d - 3* a*b^2*e - a*b*(19*c*d - a*f) + 2*a^2*(5*c*e - a*g) - (5*b^4*d - 3*a*b^3*e + 4*a^2*c*(7*c*d - 3*a*f) - a*b^2*(29*c*d - a*f) + 4*a^2*b*(4*c*e + a*g))/ Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]] )/(Sqrt[2]*a^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c))
3.2.30.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
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[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x ^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
Time = 0.27 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(-\frac {d}{3 a^{2} x^{3}}-\frac {a e -2 b d}{a^{3} x}+\frac {\frac {\frac {c \left (2 a^{3} g -a^{2} b f -2 a^{2} c e +a \,b^{2} e +3 a b c d -b^{3} d \right ) x^{3}}{8 a c -2 b^{2}}+\frac {\left (a^{3} b g +2 a^{3} c f -a^{2} b^{2} f -3 a^{2} b c e -2 a^{2} c^{2} d +a \,b^{3} e +4 a \,b^{2} c d -d \,b^{4}\right ) x}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {2 c \left (\frac {\left (2 a^{3} g \sqrt {-4 a c +b^{2}}-a^{2} b f \sqrt {-4 a c +b^{2}}-10 a^{2} c e \sqrt {-4 a c +b^{2}}+3 a \,b^{2} e \sqrt {-4 a c +b^{2}}+19 a b c d \sqrt {-4 a c +b^{2}}-5 b^{3} d \sqrt {-4 a c +b^{2}}+4 a^{3} b g -12 a^{3} c f +a^{2} b^{2} f +16 a^{2} b c e +28 a^{2} c^{2} d -3 a \,b^{3} e -29 a \,b^{2} c d +5 d \,b^{4}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (2 a^{3} g \sqrt {-4 a c +b^{2}}-a^{2} b f \sqrt {-4 a c +b^{2}}-10 a^{2} c e \sqrt {-4 a c +b^{2}}+3 a \,b^{2} e \sqrt {-4 a c +b^{2}}+19 a b c d \sqrt {-4 a c +b^{2}}-5 b^{3} d \sqrt {-4 a c +b^{2}}-4 a^{3} b g +12 a^{3} c f -a^{2} b^{2} f -16 a^{2} b c e -28 a^{2} c^{2} d +3 a \,b^{3} e +29 a \,b^{2} c d -5 d \,b^{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{3}}\) | \(629\) |
| risch | \(\text {Expression too large to display}\) | \(7383\) |
-1/3*d/a^2/x^3-(a*e-2*b*d)/a^3/x+1/a^3*((1/2*c*(2*a^3*g-a^2*b*f-2*a^2*c*e+ a*b^2*e+3*a*b*c*d-b^3*d)/(4*a*c-b^2)*x^3+1/2*(a^3*b*g+2*a^3*c*f-a^2*b^2*f- 3*a^2*b*c*e-2*a^2*c^2*d+a*b^3*e+4*a*b^2*c*d-b^4*d)/(4*a*c-b^2)*x)/(c*x^4+b *x^2+a)+2/(4*a*c-b^2)*c*(1/8*(2*a^3*g*(-4*a*c+b^2)^(1/2)-a^2*b*f*(-4*a*c+b ^2)^(1/2)-10*a^2*c*e*(-4*a*c+b^2)^(1/2)+3*a*b^2*e*(-4*a*c+b^2)^(1/2)+19*a* b*c*d*(-4*a*c+b^2)^(1/2)-5*b^3*d*(-4*a*c+b^2)^(1/2)+4*a^3*b*g-12*a^3*c*f+a ^2*b^2*f+16*a^2*b*c*e+28*a^2*c^2*d-3*a*b^3*e-29*a*b^2*c*d+5*d*b^4)/(-4*a*c +b^2)^(1/2)*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))-1/8*(2*a^3*g*(-4*a*c+b^2)^(1/2)-a^2*b*f*(- 4*a*c+b^2)^(1/2)-10*a^2*c*e*(-4*a*c+b^2)^(1/2)+3*a*b^2*e*(-4*a*c+b^2)^(1/2 )+19*a*b*c*d*(-4*a*c+b^2)^(1/2)-5*b^3*d*(-4*a*c+b^2)^(1/2)-4*a^3*b*g+12*a^ 3*c*f-a^2*b^2*f-16*a^2*b*c*e-28*a^2*c^2*d+3*a*b^3*e+29*a*b^2*c*d-5*d*b^4)/ (-4*a*c+b^2)^(1/2)*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))))
Leaf count of result is larger than twice the leaf count of optimal. 33432 vs. \(2 (498) = 996\).
Time = 297.37 (sec) , antiderivative size = 33432, normalized size of antiderivative = 61.68 \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {g x^{6} + f x^{4} + e x^{2} + d}{{\left (c x^{4} + b x^{2} + a\right )}^{2} x^{4}} \,d x } \]
1/6*(3*(a^2*b*c*f - 2*a^3*c*g + (5*b^3*c - 19*a*b*c^2)*d - (3*a*b^2*c - 10 *a^2*c^2)*e)*x^6 - (3*a^3*b*g - (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d + 3*( 3*a*b^3 - 11*a^2*b*c)*e - 3*(a^2*b^2 - 2*a^3*c)*f)*x^4 + 2*(5*(a*b^3 - 4*a ^2*b*c)*d - 3*(a^2*b^2 - 4*a^3*c)*e)*x^2 - 2*(a^2*b^2 - 4*a^3*c)*d)/((a^3* b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x ^3) - 1/2*integrate(-(a^3*b*g + (a^2*b*c*f - 2*a^3*c*g + (5*b^3*c - 19*a*b *c^2)*d - (3*a*b^2*c - 10*a^2*c^2)*e)*x^2 + (5*b^4 - 24*a*b^2*c + 14*a^2*c ^2)*d - (3*a*b^3 - 13*a^2*b*c)*e + (a^2*b^2 - 6*a^3*c)*f)/(c*x^4 + b*x^2 + a), x)/(a^3*b^2 - 4*a^4*c)
Leaf count of result is larger than twice the leaf count of optimal. 10411 vs. \(2 (498) = 996\).
Time = 1.80 (sec) , antiderivative size = 10411, normalized size of antiderivative = 19.21 \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
1/2*(b^3*c*d*x^3 - 3*a*b*c^2*d*x^3 - a*b^2*c*e*x^3 + 2*a^2*c^2*e*x^3 + a^2 *b*c*f*x^3 - 2*a^3*c*g*x^3 + b^4*d*x - 4*a*b^2*c*d*x + 2*a^2*c^2*d*x - a*b ^3*e*x + 3*a^2*b*c*e*x + a^2*b^2*f*x - 2*a^3*c*f*x - a^3*b*g*x)/((a^3*b^2 - 4*a^4*c)*(c*x^4 + b*x^2 + a)) + 1/16*((10*b^5*c^2 - 78*a*b^3*c^3 + 152*a ^2*b*c^4 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 39*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 76*sq rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 38*sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 5*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 19*sqrt(2)* sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 10*(b^2 - 4*a* c)*b^3*c^2 + 38*(b^2 - 4*a*c)*a*b*c^3)*(a^3*b^2 - 4*a^4*c)^2*d - (6*a*b^4* c^2 - 44*a^2*b^2*c^3 + 80*a^3*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*a^2*b^2*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b ^2 - 4*a*c)*c)*a*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a^2*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*a^2*c^3 - 6*(b^2 - 4*a*c)*a*b^2*c^2 + 20*(b^2 - 4*a*c)*a^2*c^3)*(...
Time = 12.57 (sec) , antiderivative size = 51386, normalized size of antiderivative = 94.81 \[ \int \frac {d+e x^2+f x^4+g x^6}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
atan((((-(25*b^15*d^2 + 9*a^2*b^13*e^2 + 25*b^6*d^2*(-(4*a*c - b^2)^9)^(1/ 2) + a^4*b^11*f^2 + a^6*b^9*g^2 + a^6*g^2*(-(4*a*c - b^2)^9)^(1/2) - 80640 *a^7*b*c^7*d^2 - 213*a^3*b^11*c*e^2 + 26880*a^8*b*c^6*e^2 - 27*a^5*b^9*c*f ^2 - 3840*a^9*b*c^5*f^2 - 9*a^5*c*f^2*(-(4*a*c - b^2)^9)^(1/2) - 768*a^10* b*c^4*g^2 - 30*a*b^14*d*e + 6366*a^2*b^11*c^2*d^2 - 35767*a^3*b^9*c^3*d^2 + 116928*a^4*b^7*c^4*d^2 - 219744*a^5*b^5*c^5*d^2 + 215040*a^6*b^3*c^6*d^2 + 9*a^2*b^4*e^2*(-(4*a*c - b^2)^9)^(1/2) - 49*a^3*c^3*d^2*(-(4*a*c - b^2) ^9)^(1/2) + 2077*a^4*b^9*c^2*e^2 - 10656*a^5*b^7*c^3*e^2 + 30240*a^6*b^5*c ^4*e^2 - 44800*a^7*b^3*c^5*e^2 + a^4*b^2*f^2*(-(4*a*c - b^2)^9)^(1/2) + 25 *a^4*c^2*e^2*(-(4*a*c - b^2)^9)^(1/2) + 288*a^6*b^7*c^2*f^2 - 1504*a^7*b^5 *c^3*f^2 + 3840*a^8*b^3*c^4*f^2 - 96*a^8*b^5*c^2*g^2 + 512*a^9*b^3*c^3*g^2 - 615*a*b^13*c*d^2 + 10*a^2*b^13*d*f + 35840*a^8*c^7*d*e + 10*a^3*b^12*d* g - 6*a^3*b^12*e*f - 6*a^4*b^11*e*g - 7168*a^9*c^6*d*g - 15360*a^9*c^6*e*f + 2*a^5*b^10*f*g + 3072*a^10*c^5*f*g - 30*a*b^5*d*e*(-(4*a*c - b^2)^9)^(1 /2) + 724*a^2*b^12*c*d*e - 258*a^3*b^11*c*d*f + 43520*a^8*b*c^6*d*f - 168* a^4*b^10*c*d*g + 152*a^4*b^10*c*e*f + 98*a^5*b^9*c*e*g - 1536*a^9*b*c^5*e* g + 2*a^5*b*f*g*(-(4*a*c - b^2)^9)^(1/2) - 10*a^5*c*e*g*(-(4*a*c - b^2)^9) ^(1/2) - 36*a^6*b^8*c*f*g + 246*a^2*b^2*c^2*d^2*(-(4*a*c - b^2)^9)^(1/2) - 165*a*b^4*c*d^2*(-(4*a*c - b^2)^9)^(1/2) - 7278*a^3*b^10*c^2*d*e + 39132* a^4*b^8*c^3*d*e - 119616*a^5*b^6*c^4*d*e + 201600*a^6*b^4*c^5*d*e - 161...