Integrand size = 33, antiderivative size = 375 \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {f^2 (d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {f^2 (d+e x) \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) (d+e x)^2\right )}{8 a \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) f^2 \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) f^2 \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}} e} \]
-1/4*f^2*(e*x+d)*(b+2*c*(e*x+d)^2)/(-4*a*c+b^2)/e/(a+b*(e*x+d)^2+c*(e*x+d) ^4)^2+1/8*f^2*(e*x+d)*(b*(8*a*c+b^2)+c*(20*a*c+b^2)*(e*x+d)^2)/a/(-4*a*c+b ^2)^2/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/16*f^2*arctan((e*x+d)*2^(1/2)*c^(1/2 )/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^2+20*a*c+b*(-52*a*c+b^2)/(-4*a* c+b^2)^(1/2))/a/(-4*a*c+b^2)^2/e*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/16 *f^2*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)* (b^2+20*a*c-b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^2/e*2^(1/2) /(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 2.78 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.03 \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f^2 \left (-\frac {4 \left (b (d+e x)+2 c (d+e x)^3\right )}{\left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 (d+e x) \left (b^3+8 a b c+b^2 c (d+e x)^2+20 a c^2 (d+e x)^2\right )}{a \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right )}{16 e} \]
(f^2*((-4*(b*(d + e*x) + 2*c*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x) ^2 + c*(d + e*x)^4)^2) + (2*(d + e*x)*(b^3 + 8*a*b*c + b^2*c*(d + e*x)^2 + 20*a*c^2*(d + e*x)^2))/(a*(b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x )^4)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c* Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*S qrt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c] )*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(16*e)
Time = 0.54 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1462, 1439, 1492, 25, 1480, 218}
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.
| \(\displaystyle \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1462 |
| \(\displaystyle \frac {f^2 \int \frac {(d+e x)^2}{\left (c (d+e x)^4+b (d+e x)^2+a\right )^3}d(d+e x)}{e}\) |
| \(\Big \downarrow \) 1439 |
| \(\displaystyle \frac {f^2 \left (\frac {\int \frac {b-10 c (d+e x)^2}{\left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)}{4 \left (b^2-4 a c\right )}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}\right )}{e}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {(d+e x) \left (c \left (20 a c+b^2\right ) (d+e x)^2+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\int -\frac {c \left (b^2+20 a c\right ) (d+e x)^2+b \left (b^2-16 a c\right )}{c (d+e x)^4+b (d+e x)^2+a}d(d+e x)}{2 a \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}\right )}{e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\int \frac {c \left (b^2+20 a c\right ) (d+e x)^2+b \left (b^2-16 a c\right )}{c (d+e x)^4+b (d+e x)^2+a}d(d+e x)}{2 a \left (b^2-4 a c\right )}+\frac {(d+e x) \left (c \left (20 a c+b^2\right ) (d+e x)^2+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}\right )}{e}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\frac {1}{2} c \left (-\frac {52 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d(d+e x)+\frac {1}{2} c \left (\frac {52 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d(d+e x)}{2 a \left (b^2-4 a c\right )}+\frac {(d+e x) \left (c \left (20 a c+b^2\right ) (d+e x)^2+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}\right )}{e}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {f^2 \left (\frac {\frac {\frac {\sqrt {c} \left (-\frac {52 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (\frac {52 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {(d+e x) \left (c \left (20 a c+b^2\right ) (d+e x)^2+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x) \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}\right )}{e}\) |
(f^2*(-1/4*((d + e*x)*(b + 2*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + b*(d + e* x)^2 + c*(d + e*x)^4)^2) + (((d + e*x)*(b*(b^2 + 8*a*c) + c*(b^2 + 20*a*c) *(d + e*x)^2))/(2*a*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + ( (Sqrt[c]*(b^2 + 20*a*c + b^3/Sqrt[b^2 - 4*a*c] - (52*a*b*c)/Sqrt[b^2 - 4*a *c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqr t[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b^2 + 20*a*c - b^3/Sqrt[b^2 - 4*a*c] + (52*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x) )/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2* a*(b^2 - 4*a*c)))/(4*(b^2 - 4*a*c))))/e
3.7.56.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(d*x)^(m - 1)*(b + 2*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[d^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^(m - 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x ] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.37
| method | result | size |
| default | \(f^{2} \left (\frac {\frac {c^{2} e^{6} \left (20 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {7 c^{2} d \,e^{5} \left (20 a c +b^{2}\right ) x^{6}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (420 a \,c^{2} d^{2}+21 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) c \,e^{4} x^{5}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {5 c d \,e^{3} \left (140 a \,c^{2} d^{2}+7 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {e^{2} \left (700 a \,c^{3} d^{4}+35 b^{2} c^{2} d^{4}+280 b \,c^{2} d^{2} a +20 b^{3} c \,d^{2}+36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d e \left (420 a \,c^{3} d^{4}+21 b^{2} c^{2} d^{4}+280 b \,c^{2} d^{2} a +20 b^{3} c \,d^{2}+108 a^{2} c^{2}+15 a \,b^{2} c +3 b^{4}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (140 a \,c^{3} d^{6}+7 b^{2} c^{2} d^{6}+140 a b \,c^{2} d^{4}+10 b^{3} c \,d^{4}+108 a^{2} c^{2} d^{2}+15 b^{2} a c \,d^{2}+3 b^{4} d^{2}+16 c b \,a^{2}-a \,b^{3}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d \left (20 a \,c^{3} d^{6}+b^{2} c^{2} d^{6}+28 a b \,c^{2} d^{4}+2 b^{3} c \,d^{4}+36 a^{2} c^{2} d^{2}+5 b^{2} a c \,d^{2}+b^{4} d^{2}+16 c b \,a^{2}-a \,b^{3}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\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 )^{2}}+\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 (c \,e^{2} \left (20 a c +b^{2}\right ) \textit {\_R}^{2}+2 d c e \left (20 a c +b^{2}\right ) \textit {\_R} +20 a \,c^{2} d^{2}+b^{2} c \,d^{2}-16 a b c +b^{3}\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}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a e}\right )\) | \(889\) |
| risch | \(\frac {\frac {c^{2} e^{6} f^{2} \left (20 a c +b^{2}\right ) x^{7}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {7 c^{2} d \,e^{5} f^{2} \left (20 a c +b^{2}\right ) x^{6}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {\left (420 a \,c^{2} d^{2}+21 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) c \,e^{4} f^{2} x^{5}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {5 c d \,e^{3} f^{2} \left (140 a \,c^{2} d^{2}+7 b^{2} c \,d^{2}+28 a b c +2 b^{3}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {e^{2} f^{2} \left (700 a \,c^{3} d^{4}+35 b^{2} c^{2} d^{4}+280 b \,c^{2} d^{2} a +20 b^{3} c \,d^{2}+36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d e \,f^{2} \left (420 a \,c^{3} d^{4}+21 b^{2} c^{2} d^{4}+280 b \,c^{2} d^{2} a +20 b^{3} c \,d^{2}+108 a^{2} c^{2}+15 a \,b^{2} c +3 b^{4}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {f^{2} \left (140 a \,c^{3} d^{6}+7 b^{2} c^{2} d^{6}+140 a b \,c^{2} d^{4}+10 b^{3} c \,d^{4}+108 a^{2} c^{2} d^{2}+15 b^{2} a c \,d^{2}+3 b^{4} d^{2}+16 c b \,a^{2}-a \,b^{3}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {d \,f^{2} \left (20 a \,c^{3} d^{6}+b^{2} c^{2} d^{6}+28 a b \,c^{2} d^{4}+2 b^{3} c \,d^{4}+36 a^{2} c^{2} d^{2}+5 b^{2} a c \,d^{2}+b^{4} d^{2}+16 c b \,a^{2}-a \,b^{3}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\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 )^{2}}+\frac {f^{2} \left (\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 (\frac {c \,e^{2} \left (20 a c +b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 d c e \left (20 a c +b^{2}\right ) \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {-20 a \,c^{2} d^{2}-b^{2} c \,d^{2}+16 a b c -b^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\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}\right )}{16 a e}\) | \(960\) |
f^2*((1/8*c^2*e^6*(20*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^7+7/8*c^2*d* e^5*(20*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^6+1/8*(420*a*c^2*d^2+21*b^ 2*c*d^2+28*a*b*c+2*b^3)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^5+5/8*c*d*e^3 *(140*a*c^2*d^2+7*b^2*c*d^2+28*a*b*c+2*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x ^4+1/8*e^2*(700*a*c^3*d^4+35*b^2*c^2*d^4+280*a*b*c^2*d^2+20*b^3*c*d^2+36*a ^2*c^2+5*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^3+1/8*d*e*(420*a*c^3* d^4+21*b^2*c^2*d^4+280*a*b*c^2*d^2+20*b^3*c*d^2+108*a^2*c^2+15*a*b^2*c+3*b ^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^2+1/8*(140*a*c^3*d^6+7*b^2*c^2*d^6+140* a*b*c^2*d^4+10*b^3*c*d^4+108*a^2*c^2*d^2+15*a*b^2*c*d^2+3*b^4*d^2+16*a^2*b *c-a*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/8*d/e*(20*a*c^3*d^6+b^2*c^2*d^6 +28*a*b*c^2*d^4+2*b^3*c*d^4+36*a^2*c^2*d^2+5*a*b^2*c*d^2+b^4*d^2+16*a^2*b* c-a*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)/a)/(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)^2+1/16/(16*a^2*c^2-8* a*b^2*c+b^4)/a/e*sum((c*e^2*(20*a*c+b^2)*_R^2+2*d*c*e*(20*a*c+b^2)*_R+20*a *c^2*d^2+b^2*c*d^2-16*a*b*c+b^3)/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 7838 vs. \(2 (331) = 662\).
Time = 0.53 (sec) , antiderivative size = 7838, normalized size of antiderivative = 20.90 \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\int { \frac {{\left (e f x + d f\right )}^{2}}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{3}} \,d x } \]
1/8*f^2*integrate(((b^2*c + 20*a*c^2)*e^2*x^2 + 2*(b^2*c + 20*a*c^2)*d*e*x + b^3 - 16*a*b*c + (b^2*c + 20*a*c^2)*d^2)/(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), x)/(a*b ^4 - 8*a^2*b^2*c + 16*a^3*c^2) + 1/8*((b^2*c^2 + 20*a*c^3)*e^7*f^2*x^7 + 7 *(b^2*c^2 + 20*a*c^3)*d*e^6*f^2*x^6 + (2*b^3*c + 28*a*b*c^2 + 21*(b^2*c^2 + 20*a*c^3)*d^2)*e^5*f^2*x^5 + 5*(7*(b^2*c^2 + 20*a*c^3)*d^3 + 2*(b^3*c + 14*a*b*c^2)*d)*e^4*f^2*x^4 + (35*(b^2*c^2 + 20*a*c^3)*d^4 + b^4 + 5*a*b^2* c + 36*a^2*c^2 + 20*(b^3*c + 14*a*b*c^2)*d^2)*e^3*f^2*x^3 + (21*(b^2*c^2 + 20*a*c^3)*d^5 + 20*(b^3*c + 14*a*b*c^2)*d^3 + 3*(b^4 + 5*a*b^2*c + 36*a^2 *c^2)*d)*e^2*f^2*x^2 + (7*(b^2*c^2 + 20*a*c^3)*d^6 + 10*(b^3*c + 14*a*b*c^ 2)*d^4 - a*b^3 + 16*a^2*b*c + 3*(b^4 + 5*a*b^2*c + 36*a^2*c^2)*d^2)*e*f^2* x + ((b^2*c^2 + 20*a*c^3)*d^7 + 2*(b^3*c + 14*a*b*c^2)*d^5 + (b^4 + 5*a*b^ 2*c + 36*a^2*c^2)*d^3 - (a*b^3 - 16*a^2*b*c)*d)*f^2)/((a*b^4*c^2 - 8*a^2*b ^2*c^3 + 16*a^3*c^4)*e^9*x^8 + 8*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)* d*e^8*x^7 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3 + 14*(a*b^4*c^2 - 8* a^2*b^2*c^3 + 16*a^3*c^4)*d^2)*e^7*x^6 + 4*(14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^3 + 3*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d)*e^6*x^5 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3 + 70*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a ^3*c^4)*d^4 + 30*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d^2)*e^5*x^4 + 4 *(14*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d^5 + 10*(a*b^5*c - 8*a^2...
Leaf count of result is larger than twice the leaf count of optimal. 2679 vs. \(2 (331) = 662\).
Time = 0.33 (sec) , antiderivative size = 2679, normalized size of antiderivative = 7.14 \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \]
-1/16*((b^2*c*e^2*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c* e^4)) + d/e)^2 + 20*a*c^2*e^2*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a *c)*e^2)/(c*e^4)) + d/e)^2 - 2*b^2*c*d*e*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqr t(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e) - 40*a*c^2*d*e*f^2*(sqrt(1/2)*sqrt(-(b *e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e) + b^2*c*d^2*f^2 + 20*a*c^2*d ^2*f^2 + b^3*f^2 - 16*a*b*c*f^2)*log(x + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e)/(2*c*e^4*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e)^3 - 6*c*d*e^3*(sqrt(1/2)*sqrt(-(b*e^2 + sqr t(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e)^2 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e)) - (b^2*c*e^2*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e)^2 + 20*a*c^2*e^2*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e ^2)/(c*e^4)) - d/e)^2 + 2*b^2*c*d*e*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e) + 40*a*c^2*d*e*f^2*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e) + b^2*c*d^2*f^2 + 20*a*c^2*d^2*f^ 2 + b^3*f^2 - 16*a*b*c*f^2)*log(x - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4* a*c)*e^2)/(c*e^4)) + d/e)/(2*c*e^4*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4* a*c)*e^2)/(c*e^4)) - d/e)^3 + 6*c*d*e^3*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e)^2 + 2*c*d^3*e + b*d*e + (6*c*d^2*e^2 + b*e^ 2)*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e)) + ...
Time = 12.30 (sec) , antiderivative size = 16025, normalized size of antiderivative = 42.73 \[ \int \frac {(d f+e f x)^2}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \]
atan((((((67108864*a^9*b*c^9*d*e^13 - 4096*a^2*b^15*c^2*d*e^13 + 114688*a^ 3*b^13*c^3*d*e^13 - 1376256*a^4*b^11*c^4*d*e^13 + 9175040*a^5*b^9*c^5*d*e^ 13 - 36700160*a^6*b^7*c^6*d*e^13 + 88080384*a^7*b^5*c^7*d*e^13 - 117440512 *a^8*b^3*c^8*d*e^13)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10*c + 240*a ^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b^2*c^5)) + (x *(262144*a^7*b*c^7*e^14 - 256*a^2*b^11*c^2*e^14 + 5120*a^3*b^9*c^3*e^14 - 40960*a^4*b^7*c^4*e^14 + 163840*a^5*b^5*c^5*e^14 - 327680*a^6*b^3*c^6*e^14 ))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256*a^5*b^ 2*c^3)))*(-(b^17*f^4 + b^2*f^4*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c ^8*f^4 + 1140*a^2*b^13*c^2*f^4 - 10160*a^3*b^11*c^3*f^4 + 34880*a^4*b^9*c^ 4*f^4 + 43776*a^5*b^7*c^5*f^4 - 680960*a^6*b^5*c^6*f^4 + 1863680*a^7*b^3*c ^7*f^4 - 55*a*b^15*c*f^4 - 25*a*c*f^4*(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3 *b^20*e^2 + 1048576*a^13*c^10*e^2 - 40*a^4*b^18*c*e^2 + 720*a^5*b^16*c^2*e ^2 - 7680*a^6*b^14*c^3*e^2 + 53760*a^7*b^12*c^4*e^2 - 258048*a^8*b^10*c^5* e^2 + 860160*a^9*b^8*c^6*e^2 - 1966080*a^10*b^6*c^7*e^2 + 2949120*a^11*b^4 *c^8*e^2 - 2621440*a^12*b^2*c^9*e^2)))^(1/2) - (122880*a^3*b^9*c^4*e^12*f^ 2 - 9216*a^2*b^11*c^3*e^12*f^2 - 819200*a^4*b^7*c^5*e^12*f^2 + 2949120*a^5 *b^5*c^6*e^12*f^2 - 5505024*a^6*b^3*c^7*e^12*f^2 + 256*a*b^13*c^2*e^12*f^2 + 4194304*a^7*b*c^8*e^12*f^2)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3*b^10 *c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7*b...