Integrand size = 30, antiderivative size = 329 \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {d}{4 a^2 x^4}+\frac {2 b d-a e}{2 a^3 x^2}+\frac {b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^5 d-2 a b^4 e+12 a^2 b^2 c e-12 a^3 c^2 e+6 a^2 b c (5 c d-a f)-a b^3 (20 c d-a f)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log (x)}{a^4}-\frac {\left (3 b^2 d-2 a b e-a (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4} \]
-1/4*d/a^2/x^4+1/2*(-a*e+2*b*d)/a^3/x^2+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)+c*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(-a*f+3* c*d))*x^2)/a^3/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*(3*b^5*d-2*a*b^4*e+12*a^2* b^2*c*e-12*a^3*c^2*e+6*a^2*b*c*(-a*f+5*c*d)-a*b^3*(-a*f+20*c*d))*arctanh(( 2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^4/(-4*a*c+b^2)^(3/2)+(3*b^2*d-2*a*b*e-a*( -a*f+2*c*d))*ln(x)/a^4-1/4*(3*b^2*d-2*a*b*e-a*(-a*f+2*c*d))*ln(c*x^4+b*x^2 +a)/a^4
Time = 0.77 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.80 \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {\frac {a^2 d}{x^4}+\frac {2 a (-2 b d+a e)}{x^2}+\frac {2 a \left (-b^4 d+b^3 \left (a e-c d x^2\right )+a b^2 \left (4 c d-a f+c e x^2\right )-a b c \left (3 a e-3 c d x^2+a f x^2\right )+2 a^2 c \left (a f-c \left (d+e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-4 \left (3 b^2 d-2 a b e+a (-2 c d+a f)\right ) \log (x)+\frac {\left (3 b^5 d+b^4 \left (3 \sqrt {b^2-4 a c} d-2 a e\right )+2 a^2 b c \left (15 c d+4 \sqrt {b^2-4 a c} e-3 a f\right )+a b^3 \left (-20 c d-2 \sqrt {b^2-4 a c} e+a f\right )-4 a^2 c \left (-2 c \sqrt {b^2-4 a c} d+3 a c e+a \sqrt {b^2-4 a c} f\right )+a b^2 \left (-14 c \sqrt {b^2-4 a c} d+12 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-3 b^5 d+b^4 \left (3 \sqrt {b^2-4 a c} d+2 a e\right )-a b^3 \left (-20 c d+2 \sqrt {b^2-4 a c} e+a f\right )+2 a^2 b c \left (-15 c d+4 \sqrt {b^2-4 a c} e+3 a f\right )+4 a^2 c \left (2 c \sqrt {b^2-4 a c} d+3 a c e-a \sqrt {b^2-4 a c} f\right )+a b^2 \left (-2 c \left (7 \sqrt {b^2-4 a c} d+6 a e\right )+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^4} \]
-1/4*((a^2*d)/x^4 + (2*a*(-2*b*d + a*e))/x^2 + (2*a*(-(b^4*d) + b^3*(a*e - c*d*x^2) + a*b^2*(4*c*d - a*f + c*e*x^2) - a*b*c*(3*a*e - 3*c*d*x^2 + a*f *x^2) + 2*a^2*c*(a*f - c*(d + e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4) ) - 4*(3*b^2*d - 2*a*b*e + a*(-2*c*d + a*f))*Log[x] + ((3*b^5*d + b^4*(3*S qrt[b^2 - 4*a*c]*d - 2*a*e) + 2*a^2*b*c*(15*c*d + 4*Sqrt[b^2 - 4*a*c]*e - 3*a*f) + a*b^3*(-20*c*d - 2*Sqrt[b^2 - 4*a*c]*e + a*f) - 4*a^2*c*(-2*c*Sqr t[b^2 - 4*a*c]*d + 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-14*c*Sqrt[b^ 2 - 4*a*c]*d + 12*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c ] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((-3*b^5*d + b^4*(3*Sqrt[b^2 - 4*a*c]* d + 2*a*e) - a*b^3*(-20*c*d + 2*Sqrt[b^2 - 4*a*c]*e + a*f) + 2*a^2*b*c*(-1 5*c*d + 4*Sqrt[b^2 - 4*a*c]*e + 3*a*f) + 4*a^2*c*(2*c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - a*Sqrt[b^2 - 4*a*c]*f) + a*b^2*(-2*c*(7*Sqrt[b^2 - 4*a*c]*d + 6*a*e) + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^ 2 - 4*a*c)^(3/2))/a^4
Time = 1.17 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2194, 2177, 25, 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.
| \(\displaystyle \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \frac {1}{2} \int \frac {f x^4+e x^2+d}{x^6 \left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 2177 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {\frac {c \left (d b^3-a e b^2-a (3 c d-a f) b+2 a^2 c e\right ) x^6}{a^3}+\frac {\left (b^2-4 a c\right ) \left (d b^2-a e b-a (c d-a f)\right ) x^4}{a^3}-\frac {\left (b^2-4 a c\right ) (b d-a e) x^2}{a^2}+\left (\frac {b^2}{a}-4 c\right ) d}{x^6 \left (c x^4+b x^2+a\right )}dx^2}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\frac {c \left (d b^3-a e b^2-a (3 c d-a f) b+2 a^2 c e\right ) x^6}{a^3}+\frac {\left (b^2-4 a c\right ) \left (d b^2-a e b-a (c d-a f)\right ) x^4}{a^3}-\frac {\left (b^2-4 a c\right ) (b d-a e) x^2}{a^2}+\left (\frac {b^2}{a}-4 c\right ) d}{x^6 \left (c x^4+b x^2+a\right )}dx^2}{b^2-4 a c}+\frac {c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \left (-\frac {\left (4 a c-b^2\right ) d}{a^2 x^6}+\frac {-3 d b^5+2 a e b^4+a (17 c d-a f) b^3-10 a^2 c e b^2-a^2 c (19 c d-5 a f) b-c \left (b^2-4 a c\right ) \left (3 d b^2-2 a e b-a (2 c d-a f)\right ) x^2+6 a^3 c^2 e}{a^4 \left (c x^4+b x^2+a\right )}+\frac {\left (b^2-4 a c\right ) \left (3 d b^2-2 a e b-a (2 c d-a f)\right )}{a^4 x^2}-\frac {\left (4 a c-b^2\right ) (a e-2 b d)}{a^3 x^4}\right )dx^2}{b^2-4 a c}+\frac {c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )+3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d}{a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\frac {\log \left (x^2\right ) \left (b^2-4 a c\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{a^4}-\frac {\left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right ) \left (-2 a b e-a (2 c d-a f)+3 b^2 d\right )}{2 a^4}+\frac {\left (b^2-4 a c\right ) (2 b d-a e)}{a^3 x^2}-\frac {d \left (b^2-4 a c\right )}{2 a^2 x^4}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-12 a^3 c^2 e+12 a^2 b^2 c e+6 a^2 b c (5 c d-a f)-2 a b^4 e-a b^3 (20 c d-a f)+3 b^5 d\right )}{a^4 \sqrt {b^2-4 a c}}}{b^2-4 a c}\right )\) |
((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 ) + c*(b^3*d - a*b^2*e + 2*a^2*c*e - a*b*(3*c*d - a*f))*x^2)/(a^3*(b^2 - 4 *a*c)*(a + b*x^2 + c*x^4)) + (-1/2*((b^2 - 4*a*c)*d)/(a^2*x^4) + ((b^2 - 4 *a*c)*(2*b*d - a*e))/(a^3*x^2) + ((3*b^5*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + 6*a^2*b*c*(5*c*d - a*f) - a*b^3*(20*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^2 - 4*a*c)*(3 *b^2*d - 2*a*b*e - a*(2*c*d - a*f))*Log[x^2])/a^4 - ((b^2 - 4*a*c)*(3*b^2* d - 2*a*b*e - a*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(2*a^4))/(b^2 - 4*a *c))/2
3.1.67.3.1 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_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
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 = 0.23 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(-\frac {d}{4 a^{2} x^{4}}-\frac {a e -2 b d}{2 a^{3} x^{2}}+\frac {\left (f \,a^{2}-2 a b e -2 a c d +3 b^{2} d \right ) \ln \left (x \right )}{a^{4}}-\frac {\frac {\frac {a c \left (a^{2} b f +2 a^{2} c e -a \,b^{2} e -3 a b c d +b^{3} d \right ) x^{2}}{4 a c -b^{2}}-\frac {a \left (2 a^{3} c f -a^{2} b^{2} f -3 a^{2} b c e -2 a^{2} c^{2} d +a \,b^{3} e +4 a \,b^{2} c d -d \,b^{4}\right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 a^{3} c^{2} f -a^{2} b^{2} c f -8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (5 a^{3} b c f +6 a^{3} c^{2} e -a^{2} b^{3} f -10 a^{2} b^{2} c e -19 a^{2} b \,c^{2} d +2 a \,b^{4} e +17 a \,b^{3} c d -3 b^{5} d -\frac {\left (4 a^{3} c^{2} f -a^{2} b^{2} c f -8 a^{2} b \,c^{2} e -8 a^{2} c^{3} d +2 a \,b^{3} c e +14 a \,b^{2} c^{2} d -3 b^{4} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{4}}\) | \(466\) |
| risch | \(\text {Expression too large to display}\) | \(1227\) |
-1/4*d/a^2/x^4-1/2*(a*e-2*b*d)/a^3/x^2+(a^2*f-2*a*b*e-2*a*c*d+3*b^2*d)/a^4 *ln(x)-1/2/a^4*((a*c*(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^2-a*(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))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(4*a^3*c^2*f-a^2*b^ 2*c*f-8*a^2*b*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2*c^2*d-3*b^4*c*d)/c*ln (c*x^4+b*x^2+a)+2*(5*a^3*b*c*f+6*a^3*c^2*e-a^2*b^3*f-10*a^2*b^2*c*e-19*a^2 *b*c^2*d+2*a*b^4*e+17*a*b^3*c*d-3*b^5*d-1/2*(4*a^3*c^2*f-a^2*b^2*c*f-8*a^2 *b*c^2*e-8*a^2*c^3*d+2*a*b^3*c*e+14*a*b^2*c^2*d-3*b^4*c*d)*b/c)/(4*a*c-b^2 )^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1272 vs. \(2 (315) = 630\).
Time = 4.47 (sec) , antiderivative size = 2567, normalized size of antiderivative = 7.80 \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*((3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*d - 2*(a^2*b^4*c - 7* a^3*b^2*c^2 + 12*a^4*c^3)*e + (a^3*b^3*c - 4*a^4*b*c^2)*f)*x^6 + ((6*a*b^6 - 49*a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(2*a^2*b^5 - 15*a^3* b^3*c + 28*a^4*b*c^2)*e + 2*(a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*f)*x^4 + ( 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d - 2*(a^3*b^4 - 8*a^4*b^2*c + 16 *a^5*c^2)*e)*x^2 + (((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*d - 2*(a*b^4* c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (a^2*b^3*c - 6*a^3*b*c^2)*f)*x^8 + ((3* b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*d - 2*(a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^ 2)*e + (a^2*b^4 - 6*a^3*b^2*c)*f)*x^6 + ((3*a*b^5 - 20*a^2*b^3*c + 30*a^3* b*c^2)*d - 2*(a^2*b^4 - 6*a^3*b^2*c + 6*a^4*c^2)*e + (a^3*b^3 - 6*a^4*b*c) *f)*x^4)*sqrt(b^2 - 4*a*c)*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)) - (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*d - (((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4 )*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (a^2*b^4*c - 8*a^3*b^ 2*c^2 + 16*a^4*c^3)*f)*x^8 + ((3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^ 3*b*c^3)*d - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e + (a^2*b^5 - 8*a^3 *b^3*c + 16*a^4*b*c^2)*f)*x^6 + ((3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*d - 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e + (a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f)*x^4)*log(c*x^4 + b*x^2 + a) + 4*(((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*...
Timed out. \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.58 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.58 \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=-\frac {{\left (3 \, b^{5} d - 20 \, a b^{3} c d + 30 \, a^{2} b c^{2} d - 2 \, a b^{4} e + 12 \, a^{2} b^{2} c e - 12 \, a^{3} c^{2} e + a^{2} b^{3} f - 6 \, a^{3} b c f\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c d x^{4} - 14 \, a b^{2} c^{2} d x^{4} + 8 \, a^{2} c^{3} d x^{4} - 2 \, a b^{3} c e x^{4} + 8 \, a^{2} b c^{2} e x^{4} + a^{2} b^{2} c f x^{4} - 4 \, a^{3} c^{2} f x^{4} + 3 \, b^{5} d x^{2} - 12 \, a b^{3} c d x^{2} + 2 \, a^{2} b c^{2} d x^{2} - 2 \, a b^{4} e x^{2} + 6 \, a^{2} b^{2} c e x^{2} + 4 \, a^{3} c^{2} e x^{2} + a^{2} b^{3} f x^{2} - 2 \, a^{3} b c f x^{2} + 5 \, a b^{4} d - 22 \, a^{2} b^{2} c d + 12 \, a^{3} c^{2} d - 4 \, a^{2} b^{3} e + 14 \, a^{3} b c e + 3 \, a^{3} b^{2} f - 8 \, a^{4} c f}{4 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} {\left (c x^{4} + b x^{2} + a\right )}} - \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e + a^{2} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac {{\left (3 \, b^{2} d - 2 \, a c d - 2 \, a b e + a^{2} f\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {9 \, b^{2} d x^{4} - 6 \, a c d x^{4} - 6 \, a b e x^{4} + 3 \, a^{2} f x^{4} - 4 \, a b d x^{2} + 2 \, a^{2} e x^{2} + a^{2} d}{4 \, a^{4} x^{4}} \]
-1/2*(3*b^5*d - 20*a*b^3*c*d + 30*a^2*b*c^2*d - 2*a*b^4*e + 12*a^2*b^2*c*e - 12*a^3*c^2*e + a^2*b^3*f - 6*a^3*b*c*f)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4*a*c)) + 1/4*(3*b^4*c*d*x^4 - 14*a*b^2*c^2*d*x^4 + 8*a^2*c^3*d*x^4 - 2*a*b^3*c*e*x^4 + 8*a^2*b*c^2*e*x^4 + a^2*b^2*c*f*x^4 - 4*a^3*c^2*f*x^4 + 3*b^5*d*x^2 - 12*a*b^3*c*d*x^2 + 2* a^2*b*c^2*d*x^2 - 2*a*b^4*e*x^2 + 6*a^2*b^2*c*e*x^2 + 4*a^3*c^2*e*x^2 + a^ 2*b^3*f*x^2 - 2*a^3*b*c*f*x^2 + 5*a*b^4*d - 22*a^2*b^2*c*d + 12*a^3*c^2*d - 4*a^2*b^3*e + 14*a^3*b*c*e + 3*a^3*b^2*f - 8*a^4*c*f)/((a^4*b^2 - 4*a^5* c)*(c*x^4 + b*x^2 + a)) - 1/4*(3*b^2*d - 2*a*c*d - 2*a*b*e + a^2*f)*log(c* x^4 + b*x^2 + a)/a^4 + 1/2*(3*b^2*d - 2*a*c*d - 2*a*b*e + a^2*f)*log(x^2)/ a^4 - 1/4*(9*b^2*d*x^4 - 6*a*c*d*x^4 - 6*a*b*e*x^4 + 3*a^2*f*x^4 - 4*a*b*d *x^2 + 2*a^2*e*x^2 + a^2*d)/(a^4*x^4)
Time = 24.98 (sec) , antiderivative size = 15905, normalized size of antiderivative = 48.34 \[ \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
(log(x)*(3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*d))/a^4 - (log(((((((4*b*c^2*(3 *b^5*d + a^2*b^3*f - 6*a^3*c^2*e - 2*a*b^4*e - 17*a*b^3*c*d - 5*a^3*b*c*f + 19*a^2*b*c^2*d + 10*a^2*b^2*c*e))/(a^3*(4*a*c - b^2)) - (b*c^2*(a*b + 3* b^2*x^2 - 10*a*c*x^2)*(a^4*(-(3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4 *e - 20*a*b^3*c*d - 6*a^3*b*c*f + 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8* (4*a*c - b^2)^3))^(1/2) + 3*b^2*d + a^2*f - 2*a*b*e - 2*a*c*d))/a^4 + (2*c ^3*x^2*(3*b^5*d + a^2*b^3*f + 60*a^3*c^2*e - 2*a*b^4*e + 4*a*b^3*c*d - 10* a^3*b*c*f - 70*a^2*b*c^2*d - 4*a^2*b^2*c*e))/(a^3*(4*a*c - b^2)))*(a^4*(-( 3*b^5*d + a^2*b^3*f - 12*a^3*c^2*e - 2*a*b^4*e - 20*a*b^3*c*d - 6*a^3*b*c* f + 30*a^2*b*c^2*d + 12*a^2*b^2*c*e)^2/(a^8*(4*a*c - b^2)^3))^(1/2) + 3*b^ 2*d + a^2*f - 2*a*b*e - 2*a*c*d))/(4*a^4) + (c^3*(36*b^8*d^2 + 16*a^2*b^6* e^2 + 4*a^4*b^4*f^2 - 36*a^5*c^3*e^2 - 116*a^3*b^4*c*e^2 - 17*a^5*b^2*c*f^ 2 - 48*a*b^7*d*e + 778*a^2*b^4*c^2*d^2 - 473*a^3*b^2*c^3*d^2 + 216*a^4*b^2 *c^2*e^2 - 309*a*b^6*c*d^2 + 24*a^2*b^6*d*f - 16*a^3*b^5*e*f + 380*a^2*b^5 *c*d*e + 324*a^4*b*c^3*d*e - 154*a^3*b^4*c*d*f + 92*a^4*b^3*c*e*f - 108*a^ 5*b*c^2*e*f - 832*a^3*b^3*c^2*d*e + 230*a^4*b^2*c^2*d*f))/(a^6*(4*a*c - b^ 2)^2) + (c^4*x^2*(54*b^7*d^2 + 24*a^2*b^5*e^2 + 6*a^4*b^3*f^2 - 440*a^3*b* c^3*d^2 - 164*a^3*b^3*c*e^2 + 276*a^4*b*c^2*e^2 - 72*a*b^6*d*e + 1011*a^2* b^3*c^2*d^2 - 441*a*b^5*c*d^2 - 20*a^5*b*c*f^2 + 36*a^2*b^5*d*f + 240*a^4* c^3*d*e - 24*a^3*b^4*e*f - 120*a^5*c^2*e*f + 540*a^2*b^4*c*d*e - 207*a^...