Integrand size = 33, antiderivative size = 174 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2} e f}+\frac {\log (d+e x)}{a^2 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f} \]
1/2*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f/(a+b*(e*x+d)^2+c*(e*x+d)^ 4)+1/2*b*(-6*a*c+b^2)*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^2/(- 4*a*c+b^2)^(3/2)/e/f+ln(e*x+d)/a^2/e/f-1/4*ln(a+b*(e*x+d)^2+c*(e*x+d)^4)/a ^2/e/f
Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\frac {\frac {2 a \left (b^2-2 a c+b c (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^3-6 a b c+b^2 \sqrt {b^2-4 a c}-4 a c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (b^3-6 a b c-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 a^2 e f} \]
((2*a*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a *c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2 ])/(b^2 - 4*a*c)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*S qrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a *c)^(3/2))/(4*a^2*e*f)
Time = 0.40 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1462, 1434, 1165, 25, 1200, 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.
| \(\displaystyle \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1462 |
| \(\displaystyle \frac {\int \frac {1}{(d+e x) \left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)}{e f}\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {\int \frac {1}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)^2}{2 e f}\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {\frac {-2 a c+b^2+b c (d+e x)^2}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\int -\frac {b^2+c (d+e x)^2 b-4 a c}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)^2}{a \left (b^2-4 a c\right )}}{2 e f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b^2+c (d+e x)^2 b-4 a c}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 e f}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {\frac {\int \left (\frac {b^2-4 a c}{a (d+e x)^2}+\frac {-c \left (b^2-4 a c\right ) (d+e x)^2-b \left (b^2-5 a c\right )}{a \left (c (d+e x)^4+b (d+e x)^2+a\right )}\right )d(d+e x)^2}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 e f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\left (b^2-4 a c\right ) \log \left ((d+e x)^2\right )}{a}-\frac {\left (b^2-4 a c\right ) \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 e f}\) |
((b^2 - 2*a*c + b*c*(d + e*x)^2)/(a*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*( d + e*x)^4)) + ((b*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)*Log[(d + e*x)^2])/a - ((b^ 2 - 4*a*c)*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(2*a))/(a*(b^2 - 4*a*c) ))/(2*e*f)
3.7.50.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Si mp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p , x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.74 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {-\frac {\frac {\frac {a e b c \,x^{2}}{8 a c -2 b^{2}}+\frac {b c d a x}{4 a c -b^{2}}-\frac {a \left (-b c \,d^{2}+2 a c -b^{2}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (e^{3} c \left (4 a c -b^{2}\right ) \textit {\_R}^{3}+3 c d \,e^{2} \left (4 a c -b^{2}\right ) \textit {\_R}^{2}+e \left (12 a \,c^{2} d^{2}-3 b^{2} c \,d^{2}+5 a b c -b^{3}\right ) \textit {\_R} +4 a \,c^{2} d^{3}-b^{2} c \,d^{3}+5 a b c d -b^{3} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 \left (4 a c -b^{2}\right ) e}}{a^{2}}+\frac {\ln \left (e x +d \right )}{a^{2} e}}{f}\) | \(403\) |
| risch | \(\frac {-\frac {c \,x^{2} b e}{2 a \left (4 a c -b^{2}\right )}-\frac {x b c d}{\left (4 a c -b^{2}\right ) a}+\frac {-b c \,d^{2}+2 a c -b^{2}}{2 e a \left (4 a c -b^{2}\right )}}{f \left (c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a \right )}+\frac {\ln \left (e x +d \right )}{a^{2} e f}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} e^{2} f^{2}-48 a^{4} b^{2} c^{2} e^{2} f^{2}+12 a^{3} b^{4} c \,e^{2} f^{2}-a^{2} b^{6} e^{2} f^{2}\right ) \textit {\_Z}^{2}+\left (64 c^{3} a^{3} e f -48 a^{2} b^{2} c^{2} e f +12 a \,b^{4} c e f -b^{6} e f \right ) \textit {\_Z} +16 a \,c^{3}-3 b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{5} c^{3} e^{4} f^{2}-128 a^{4} b^{2} c^{2} e^{4} f^{2}+34 a^{3} b^{4} c \,e^{4} f^{2}-3 a^{2} b^{6} e^{4} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} e^{3} f -36 a^{2} b^{2} c^{2} e^{3} f +4 a \,b^{4} c \,e^{3} f \right ) \textit {\_R} +2 b^{2} c^{2} e^{2}\right ) x^{2}+\left (\left (320 a^{5} c^{3} d \,e^{3} f^{2}-256 a^{4} b^{2} c^{2} d \,e^{3} f^{2}+68 a^{3} b^{4} c d \,e^{3} f^{2}-6 a^{2} b^{6} d \,e^{3} f^{2}\right ) \textit {\_R}^{2}+\left (160 a^{3} c^{3} d \,e^{2} f -72 a^{2} b^{2} c^{2} d \,e^{2} f +8 a \,b^{4} c d \,e^{2} f \right ) \textit {\_R} +4 b^{2} c^{2} d e \right ) x +\left (160 a^{5} c^{3} d^{2} e^{2} f^{2}-128 a^{4} b^{2} c^{2} d^{2} e^{2} f^{2}+34 a^{3} b^{4} c \,d^{2} e^{2} f^{2}-3 a^{2} b^{6} d^{2} e^{2} f^{2}-16 a^{5} b \,c^{2} e^{2} f^{2}+8 a^{4} b^{3} c \,e^{2} f^{2}-a^{3} b^{5} e^{2} f^{2}\right ) \textit {\_R}^{2}+\left (80 a^{3} c^{3} d^{2} e f -36 a^{2} b^{2} c^{2} d^{2} e f +4 a \,b^{4} c \,d^{2} e f +36 a^{3} b \,c^{2} e f -17 a^{2} b^{3} c e f +2 a \,b^{5} e f \right ) \textit {\_R} +2 d^{2} c^{2} b^{2}-8 a b \,c^{2}+2 b^{3} c \right )\right )}{2}\) | \(769\) |
1/f*(-1/a^2*((1/2*a/(4*a*c-b^2)*e*b*c*x^2+b*c*d*a/(4*a*c-b^2)*x-1/2/e*a*(- b*c*d^2+2*a*c-b^2)/(4*a*c-b^2))/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4 *c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b*d^2+a)+1/2/(4*a*c-b^2)/e*sum((e^3*c *(4*a*c-b^2)*_R^3+3*c*d*e^2*(4*a*c-b^2)*_R^2+e*(12*a*c^2*d^2-3*b^2*c*d^2+5 *a*b*c-b^3)*_R+4*a*c^2*d^3-b^2*c*d^3+5*a*b*c*d-b^3*d)/(2*_R^3*c*e^3+6*_R^2 *c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(c*e^4*_Z^4+4* c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+b*d^2+a )))+ln(e*x+d)/a^2/e)
Leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (164) = 328\).
Time = 0.54 (sec) , antiderivative size = 2486, normalized size of antiderivative = 14.29 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*e^2* x^2 + 4*(a*b^3*c - 4*a^2*b*c^2)*d*e*x + 2*(a*b^3*c - 4*a^2*b*c^2)*d^2 + (( b^3*c - 6*a*b*c^2)*e^4*x^4 + 4*(b^3*c - 6*a*b*c^2)*d*e^3*x^3 + (b^3*c - 6* a*b*c^2)*d^4 + (b^4 - 6*a*b^2*c + 6*(b^3*c - 6*a*b*c^2)*d^2)*e^2*x^2 + a*b ^3 - 6*a^2*b*c + (b^4 - 6*a*b^2*c)*d^2 + 2*(2*(b^3*c - 6*a*b*c^2)*d^3 + (b ^4 - 6*a*b^2*c)*d)*e*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*e^4*x^4 + 8*c^2*d*e^3 *x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2 + 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c + (2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt( b^2 - 4*a*c))/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a)) - ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3) *e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2* b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b ^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2 + 2*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2* c^3)*d^3 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d)*e*x)*log(c*e^4*x^4 + 4*c*d* e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a) + 4*((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e^4*x^4 + 4*(b^4*c - 8*a*b^2*c^ 2 + 16*a^2*c^3)*d*e^3*x^3 + a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8* a*b^2*c^2 + 16*a^2*c^3)*d^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 6*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2)*e^2*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c...
Timed out. \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\int { \frac {1}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{2} {\left (e f x + d f\right )}} \,d x } \]
1/2*(b*c*e^2*x^2 + 2*b*c*d*e*x + b*c*d^2 + b^2 - 2*a*c)/((a*b^2*c - 4*a^2* c^2)*e^5*f*x^4 + 4*(a*b^2*c - 4*a^2*c^2)*d*e^4*f*x^3 + (a*b^3 - 4*a^2*b*c + 6*(a*b^2*c - 4*a^2*c^2)*d^2)*e^3*f*x^2 + 2*(2*(a*b^2*c - 4*a^2*c^2)*d^3 + (a*b^3 - 4*a^2*b*c)*d)*e^2*f*x + ((a*b^2*c - 4*a^2*c^2)*d^4 + a^2*b^2 - 4*a^3*c + (a*b^3 - 4*a^2*b*c)*d^2)*e*f) - integrate(((b^2*c - 4*a*c^2)*e^3 *x^3 + 3*(b^2*c - 4*a*c^2)*d*e^2*x^2 + (b^2*c - 4*a*c^2)*d^3 + (b^3 - 5*a* b*c + 3*(b^2*c - 4*a*c^2)*d^2)*e*x + (b^3 - 5*a*b*c)*d)/((b^2*c - 4*a*c^2) *e^4*x^4 + 4*(b^2*c - 4*a*c^2)*d*e^3*x^3 + (b^2*c - 4*a*c^2)*d^4 + (b^3 - 4*a*b*c + 6*(b^2*c - 4*a*c^2)*d^2)*e^2*x^2 + a*b^2 - 4*a^2*c + (b^3 - 4*a* b*c)*d^2 + 2*(2*(b^2*c - 4*a*c^2)*d^3 + (b^3 - 4*a*b*c)*d)*e*x), x)/(a^2*f ) + log(e*x + d)/(a^2*e*f)
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (164) = 328\).
Time = 0.40 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.81 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=-\frac {{\left (a^{2} b^{3} c e^{3} f - 6 \, a^{3} b c^{2} e^{3} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} + 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right ) - {\left (a^{2} b^{3} c e^{3} f - 6 \, a^{3} b c^{2} e^{3} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left ({\left | -b e^{2} x^{2} + \sqrt {b^{2} - 4 \, a c} e^{2} x^{2} - 2 \, b d e x + 2 \, \sqrt {b^{2} - 4 \, a c} d e x - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{4 \, {\left (a^{4} b^{4} c e^{4} f^{2} - 8 \, a^{5} b^{2} c^{2} e^{4} f^{2} + 16 \, a^{6} c^{3} e^{4} f^{2}\right )}} - \frac {\log \left ({\left | c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a \right |}\right )}{4 \, a^{2} e f} + \frac {\log \left ({\left | e x + d \right |}\right )}{a^{2} e f} + \frac {a b c e^{2} x^{2} + 2 \, a b c d e x + a b c d^{2} + a b^{2} - 2 \, a^{2} c}{2 \, {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4} + b e^{2} x^{2} + 2 \, b d e x + b d^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2} e f} \]
-1/4*((a^2*b^3*c*e^3*f - 6*a^3*b*c^2*e^3*f)*sqrt(b^2 - 4*a*c)*log(abs(b*e^ 2*x^2 + sqrt(b^2 - 4*a*c)*e^2*x^2 + 2*b*d*e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x + b*d^2 + sqrt(b^2 - 4*a*c)*d^2 + 2*a)) - (a^2*b^3*c*e^3*f - 6*a^3*b*c^2*e ^3*f)*sqrt(b^2 - 4*a*c)*log(abs(-b*e^2*x^2 + sqrt(b^2 - 4*a*c)*e^2*x^2 - 2 *b*d*e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x - b*d^2 + sqrt(b^2 - 4*a*c)*d^2 - 2*a )))/(a^4*b^4*c*e^4*f^2 - 8*a^5*b^2*c^2*e^4*f^2 + 16*a^6*c^3*e^4*f^2) - 1/4 *log(abs(c*e^4*x^4 + 4*c*d*e^3*x^3 + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*x + c*d^4 + b*e^2*x^2 + 2*b*d*e*x + b*d^2 + a))/(a^2*e*f) + log(abs(e*x + d))/(a^2* e*f) + 1/2*(a*b*c*e^2*x^2 + 2*a*b*c*d*e*x + a*b*c*d^2 + a*b^2 - 2*a^2*c)/( (c*e^4*x^4 + 4*c*d*e^3*x^3 + 6*c*d^2*e^2*x^2 + 4*c*d^3*e*x + c*d^4 + b*e^2 *x^2 + 2*b*d*e*x + b*d^2 + a)*(b^2 - 4*a*c)*a^2*e*f)
Time = 14.69 (sec) , antiderivative size = 13434, normalized size of antiderivative = 77.21 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx=\text {Too large to display} \]
((b^2 - 2*a*c + b*c*d^2)/(2*e*(a*b^2 - 4*a^2*c)) + (b*c*e*x^2)/(2*(a*b^2 - 4*a^2*c)) + (b*c*d*x)/(a*b^2 - 4*a^2*c))/(a*f + x^2*(b*e^2*f + 6*c*d^2*e^ 2*f) + x*(4*c*d^3*e*f + 2*b*d*e*f) + b*d^2*f + c*d^4*f + c*e^4*f*x^4 + 4*c *d*e^3*f*x^3) - (log((((a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a* c - b^2)^3))^(1/2) - 1)*(((a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4 *a*c - b^2)^3))^(1/2) - 1)*((2*b*c^2*e^16*(2*b^3 - 10*a*c^2*d^2 + b^2*c*d^ 2 - 10*a*b*c))/(a*f*(4*a*c - b^2)) + (b*c^2*e^16*(a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) - 1)*(a*b + 3*b^2*d^2 + 3*b^2 *e^2*x^2 - 10*a*c*d^2 + 6*b^2*d*e*x - 10*a*c*e^2*x^2 - 20*a*c*d*e*x))/(a^2 *f) - (2*b*c^3*e^18*x^2*(10*a*c - b^2))/(a*f*(4*a*c - b^2)) - (4*b*c^3*d*e ^17*x*(10*a*c - b^2))/(a*f*(4*a*c - b^2))))/(4*a^2*e*f) - (b*c^3*e^15*(4*b ^3 - 20*a*c^2*d^2 + 6*b^2*c*d^2 - 17*a*b*c))/(a^2*f^2*(4*a*c - b^2)^2) + ( 2*b*c^4*e^17*x^2*(10*a*c - 3*b^2))/(a^2*f^2*(4*a*c - b^2)^2) + (4*b*c^4*d* e^16*x*(10*a*c - 3*b^2))/(a^2*f^2*(4*a*c - b^2)^2)))/(4*a^2*e*f) + (b^3*c^ 5*e^16*x^2)/(a^3*f^3*(4*a*c - b^2)^3) + (b^2*c^4*e^14*(b^2 - 4*a*c + b*c*d ^2))/(a^3*f^3*(4*a*c - b^2)^3) + (2*b^3*c^5*d*e^15*x)/(a^3*f^3*(4*a*c - b^ 2)^3))*((b^3*c^5*e^16*x^2)/(a^3*f^3*(4*a*c - b^2)^3) - ((a^2*e*f*(-(b^2*(6 *a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + 1)*(((a^2*e*f*(-(b^2 *(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + 1)*((b*c^2*e^16*( a^2*e*f*(-(b^2*(6*a*c - b^2)^2)/(a^4*e^2*f^2*(4*a*c - b^2)^3))^(1/2) + ...