Integrand size = 33, antiderivative size = 353 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f^4 (d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {f^4 (d+e x) \left (7 b^2-4 a c+12 b c (d+e x)^2\right )}{8 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 \sqrt {c} \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) f^4 \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {3 \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) f^4 \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \] Output:
1/4*f^4*(e*x+d)*(2*a+b*(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^4*(e*x+d)*(7*b^2-4*a*c+12*b*c*(e*x+d)^2)/(-4*a*c+b^2)^2/e/(a+b* (e*x+d)^2+c*(e*x+d)^4)+3/8*c^(1/2)*(3*b^2+4*a*c-2*b*(-4*a*c+b^2)^(1/2))*f^ 4*arctan(2^(1/2)*c^(1/2)*(e*x+d)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/(-4 *a*c+b^2)^(5/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/e-3/8*c^(1/2)*(3*b^2+4*a*c+2* b*(-4*a*c+b^2)^(1/2))*f^4*arctan(2^(1/2)*c^(1/2)*(e*x+d)/(b+(-4*a*c+b^2)^( 1/2))^(1/2))*2^(1/2)/(-4*a*c+b^2)^(5/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/e
Time = 4.37 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.94 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f^4 \left (-\frac {2 \left (-2 a (d+e x)-b (d+e x)^3\right )}{\left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(d+e x) \left (-7 b^2+4 a c-12 b c (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (3 b^2+4 a c-2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (3 b^2+4 a c+2 b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 e} \] Input:
Integrate[(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]
Output:
(f^4*((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x )^2 + c*(d + e*x)^4)^2) + ((d + e*x)*(-7*b^2 + 4*a*c - 12*b*c*(d + e*x)^2) )/((b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[2]*Sqrt[ c]*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e* x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c + 2*b*Sqrt[b^2 - 4*a*c])*Arc Tan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a* c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(8*e)
Time = 0.49 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1462, 1440, 1492, 27, 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.
| \(\displaystyle \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1462 |
| \(\displaystyle \frac {f^4 \int \frac {(d+e x)^4}{\left (c (d+e x)^4+b (d+e x)^2+a\right )^3}d(d+e x)}{e}\) |
| \(\Big \downarrow \) 1440 |
| \(\displaystyle \frac {f^4 \left (\frac {(d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\int \frac {2 a-5 b (d+e x)^2}{\left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)}{4 \left (b^2-4 a c\right )}\right )}{e}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {f^4 \left (\frac {(d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\frac {(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\int \frac {3 a \left (b^2-4 c (d+e x)^2 b+4 a c\right )}{c (d+e x)^4+b (d+e x)^2+a}d(d+e x)}{2 a \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\right )}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^4 \left (\frac {(d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\frac {(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \int \frac {b^2-4 c (d+e x)^2 b+4 a c}{c (d+e x)^4+b (d+e x)^2+a}d(d+e x)}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\right )}{e}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {f^4 \left (\frac {(d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\frac {(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \left (-c \left (2 b-\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d(d+e x)-c \left (\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}+2 b\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d(d+e x)\right )}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\right )}{e}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {f^4 \left (\frac {(d+e x) \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\frac {(d+e x) \left (-4 a c+7 b^2+12 b c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \left (-\frac {\sqrt {2} \sqrt {c} \left (2 b-\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (\frac {4 a c+3 b^2}{\sqrt {b^2-4 a c}}+2 b\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}\right )}{e}\) |
Input:
Int[(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]
Output:
(f^4*(((d + e*x)*(2*a + b*(d + e*x)^2))/(4*(b^2 - 4*a*c)*(a + b*(d + e*x)^ 2 + c*(d + e*x)^4)^2) - (((d + e*x)*(7*b^2 - 4*a*c + 12*b*c*(d + e*x)^2))/ (2*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*(-((Sqrt[2]*Sqr t[c]*(2*b - (3*b^2 + 4*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt [2]*Sqrt[c]*(2*b + (3*b^2 + 4*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt [c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]])) /(2*(b^2 - 4*a*c)))/(4*(b^2 - 4*a*c))))/e
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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^3)*(d*x)^(m - 3)*(2*a + b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2* (p + 1)*(b^2 - 4*a*c))), x] + Simp[d^4/(2*(p + 1)*(b^2 - 4*a*c)) Int[(d*x )^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*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] && Gt Q[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 = 0.25 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.01
| method | result | size |
| default | \(f^{4} \left (\frac {-\frac {3 c^{2} e^{6} b \,x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {21 c^{2} d \,e^{5} b \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (-252 b c \,d^{2}+4 a c -19 b^{2}\right ) c \,e^{4} x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {5 c d \,e^{3} \left (-84 b c \,d^{2}+4 a c -19 b^{2}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {e^{2} \left (420 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+16 a b c +5 b^{3}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d e \left (252 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+48 a b c +15 b^{3}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (84 b \,c^{2} d^{6}-20 a \,c^{2} d^{4}+95 b^{2} c \,d^{4}+48 a b c \,d^{2}+15 b^{3} d^{2}+12 a^{2} c +3 b^{2} a \right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d \left (12 b \,c^{2} d^{6}-4 a \,c^{2} d^{4}+19 b^{2} c \,d^{4}+16 a b c \,d^{2}+5 b^{3} d^{2}+12 a^{2} c +3 b^{2} a \right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 b d e \right ) \textit {\_Z} +c \,d^{4}+b \,d^{2}+a \right )}{\sum }\frac {\left (-4 \textit {\_R}^{2} b c \,e^{2}-8 \textit {\_R} b c d e -4 b c \,d^{2}+4 a c +b^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,e^{3} \textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\right )\) | \(708\) |
| risch | \(\frac {-\frac {3 c^{2} e^{6} b \,f^{4} x^{7}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {21 c^{2} d \,e^{5} b \,f^{4} x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (-252 b c \,d^{2}+4 a c -19 b^{2}\right ) c \,e^{4} f^{4} x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {5 c d \,e^{3} f^{4} \left (-84 b c \,d^{2}+4 a c -19 b^{2}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {e^{2} f^{4} \left (420 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+16 a b c +5 b^{3}\right ) x^{3}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d e \,f^{4} \left (252 b \,c^{2} d^{4}-40 a \,c^{2} d^{2}+190 b^{2} c \,d^{2}+48 a b c +15 b^{3}\right ) x^{2}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {f^{4} \left (84 b \,c^{2} d^{6}-20 a \,c^{2} d^{4}+95 b^{2} c \,d^{4}+48 a b c \,d^{2}+15 b^{3} d^{2}+12 a^{2} c +3 b^{2} a \right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d \,f^{4} \left (12 b \,c^{2} d^{6}-4 a \,c^{2} d^{4}+19 b^{2} c \,d^{4}+16 a b c \,d^{2}+5 b^{3} d^{2}+12 a^{2} c +3 b^{2} a \right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\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 +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}+\frac {3 f^{4} \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 c \,d^{3} e +2 b d e \right ) \textit {\_Z} +c \,d^{4}+b \,d^{2}+a \right )}{\sum }\frac {\left (-\frac {4 b c \,e^{2} \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {8 b c d e \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {-4 b c \,d^{2}+4 a c +b^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,e^{3} \textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{16 e}\) | \(775\) |
Input:
int((e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)
Output:
f^4*((-3/2*c^2*e^6*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7-21/2*c^2*d*e^5*b/(16*a ^2*c^2-8*a*b^2*c+b^4)*x^6+1/8*(-252*b*c*d^2+4*a*c-19*b^2)*c*e^4/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x^5+5/8*c*d*e^3*(-84*b*c*d^2+4*a*c-19*b^2)/(16*a^2*c^2-8* a*b^2*c+b^4)*x^4-1/8*e^2*(420*b*c^2*d^4-40*a*c^2*d^2+190*b^2*c*d^2+16*a*b* c+5*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/8*d*e*(252*b*c^2*d^4-40*a*c^2*d^ 2+190*b^2*c*d^2+48*a*b*c+15*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/8*(84*b* c^2*d^6-20*a*c^2*d^4+95*b^2*c*d^4+48*a*b*c*d^2+15*b^3*d^2+12*a^2*c+3*a*b^2 )/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/8*d/e*(12*b*c^2*d^6-4*a*c^2*d^4+19*b^2*c* d^4+16*a*b*c*d^2+5*b^3*d^2+12*a^2*c+3*a*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4))/( 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+3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((-4*_R^2*b*c*e^2-8*_R *b*c*d*e-4*b*c*d^2+4*a*c+b^2)/(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+c*d^4+b*d^2+a)))
Leaf count of result is larger than twice the leaf count of optimal. 6770 vs. \(2 (305) = 610\).
Time = 0.37 (sec) , antiderivative size = 6770, normalized size of antiderivative = 19.18 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas ")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 2237 vs. \(2 (328) = 656\).
Time = 140.36 (sec) , antiderivative size = 2237, normalized size of antiderivative = 6.34 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*f*x+d*f)**4/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)
Output:
(-12*a**2*c*d*f**4 - 3*a*b**2*d*f**4 - 16*a*b*c*d**3*f**4 + 4*a*c**2*d**5* f**4 - 5*b**3*d**3*f**4 - 19*b**2*c*d**5*f**4 - 12*b*c**2*d**7*f**4 - 84*b *c**2*d*e**6*f**4*x**6 - 12*b*c**2*e**7*f**4*x**7 + x**5*(4*a*c**2*e**5*f* *4 - 19*b**2*c*e**5*f**4 - 252*b*c**2*d**2*e**5*f**4) + x**4*(20*a*c**2*d* e**4*f**4 - 95*b**2*c*d*e**4*f**4 - 420*b*c**2*d**3*e**4*f**4) + x**3*(-16 *a*b*c*e**3*f**4 + 40*a*c**2*d**2*e**3*f**4 - 5*b**3*e**3*f**4 - 190*b**2* c*d**2*e**3*f**4 - 420*b*c**2*d**4*e**3*f**4) + x**2*(-48*a*b*c*d*e**2*f** 4 + 40*a*c**2*d**3*e**2*f**4 - 15*b**3*d*e**2*f**4 - 190*b**2*c*d**3*e**2* f**4 - 252*b*c**2*d**5*e**2*f**4) + x*(-12*a**2*c*e*f**4 - 3*a*b**2*e*f**4 - 48*a*b*c*d**2*e*f**4 + 20*a*c**2*d**4*e*f**4 - 15*b**3*d**2*e*f**4 - 95 *b**2*c*d**4*e*f**4 - 84*b*c**2*d**6*e*f**4))/(128*a**4*c**2*e - 64*a**3*b **2*c*e + 256*a**3*b*c**2*d**2*e + 256*a**3*c**3*d**4*e + 8*a**2*b**4*e - 128*a**2*b**3*c*d**2*e + 256*a**2*b*c**3*d**6*e + 128*a**2*c**4*d**8*e + 1 6*a*b**5*d**2*e - 48*a*b**4*c*d**4*e - 128*a*b**3*c**2*d**6*e - 64*a*b**2* c**3*d**8*e + 8*b**6*d**4*e + 16*b**5*c*d**6*e + 8*b**4*c**2*d**8*e + x**8 *(128*a**2*c**4*e**9 - 64*a*b**2*c**3*e**9 + 8*b**4*c**2*e**9) + x**7*(102 4*a**2*c**4*d*e**8 - 512*a*b**2*c**3*d*e**8 + 64*b**4*c**2*d*e**8) + x**6* (256*a**2*b*c**3*e**7 + 3584*a**2*c**4*d**2*e**7 - 128*a*b**3*c**2*e**7 - 1792*a*b**2*c**3*d**2*e**7 + 16*b**5*c*e**7 + 224*b**4*c**2*d**2*e**7) + x **5*(1536*a**2*b*c**3*d*e**6 + 7168*a**2*c**4*d**3*e**6 - 768*a*b**3*c*...
\[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\int { \frac {{\left (e f x + d f\right )}^{4}}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{3}} \,d x } \] Input:
integrate((e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima ")
Output:
-3/8*f^4*integrate((4*b*c*e^2*x^2 + 8*b*c*d*e*x + 4*b*c*d^2 - 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), x)/(b^4 - 8*a*b^2*c + 16*a^2*c^2) - 1/8*(12*b*c^2* e^7*f^4*x^7 + 84*b*c^2*d*e^6*f^4*x^6 + (252*b*c^2*d^2 + 19*b^2*c - 4*a*c^2 )*e^5*f^4*x^5 + 5*(84*b*c^2*d^3 + (19*b^2*c - 4*a*c^2)*d)*e^4*f^4*x^4 + (4 20*b*c^2*d^4 + 5*b^3 + 16*a*b*c + 10*(19*b^2*c - 4*a*c^2)*d^2)*e^3*f^4*x^3 + (252*b*c^2*d^5 + 10*(19*b^2*c - 4*a*c^2)*d^3 + 3*(5*b^3 + 16*a*b*c)*d)* e^2*f^4*x^2 + (84*b*c^2*d^6 + 5*(19*b^2*c - 4*a*c^2)*d^4 + 3*a*b^2 + 12*a^ 2*c + 3*(5*b^3 + 16*a*b*c)*d^2)*e*f^4*x + (12*b*c^2*d^7 + (19*b^2*c - 4*a* c^2)*d^5 + (5*b^3 + 16*a*b*c)*d^3 + 3*(a*b^2 + 4*a^2*c)*d)*f^4)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^ 4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b ^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2* c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a* b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5 *c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c ^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a ^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + ...
Leaf count of result is larger than twice the leaf count of optimal. 1958 vs. \(2 (305) = 610\).
Time = 0.18 (sec) , antiderivative size = 1958, normalized size of antiderivative = 5.55 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")
Output:
3/16*((4*b*c*e^2*f^4*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e ^4)) + d/e)^2 - 8*b*c*d*e*f^4*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)* e^2)/(c*e^4)) + d/e) + 4*b*c*d^2*f^4 - b^2*f^4 - 4*a*c*f^4)*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*(s qrt(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)) - (4*b*c*e^2*f^4*(sqrt(1/2)*sqrt(-(b*e^2 + sqrt (b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e)^2 + 8*b*c*d*e*f^4*(sqrt(1/2)*sqrt(-(b*e ^2 + sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) - d/e) + 4*b*c*d^2*f^4 - b^2*f^4 - 4* a*c*f^4)*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)) + (4*b*c*e^2*f^4*(sqrt(1 /2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e)^2 - 8*b*c*d*e*f^ 4*(sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)/(c*e^4)) + d/e) + 4*b*c *d^2*f^4 - b^2*f^4 - 4*a*c*f^4)*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 - s...
Time = 15.58 (sec) , antiderivative size = 13840, normalized size of antiderivative = 39.21 \[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:
int((d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x)
Output:
atan(((-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b*c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4*f^8 - 1 024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(512*(a*b^ 20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16*c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10*c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a^9*b^4*c^8* e^2 - 2621440*a^10*b^2*c^9*e^2)))^(1/2)*((((1024*b^15*c^2*d*e^13 - 28672*a *b^13*c^3*d*e^13 - 16777216*a^7*b*c^9*d*e^13 + 344064*a^2*b^11*c^4*d*e^13 - 2293760*a^3*b^9*c^5*d*e^13 + 9175040*a^4*b^7*c^6*d*e^13 - 22020096*a^5*b ^5*c^7*d*e^13 + 29360128*a^6*b^3*c^8*d*e^13)/(128*(b^12 + 4096*a^6*c^6 + 2 40*a^2*b^8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10*c)) + (x*(128*b^11*c^2*e^14 - 2560*a*b^9*c^3*e^14 - 131072*a^5*b *c^7*e^14 + 20480*a^2*b^7*c^4*e^14 - 81920*a^3*b^5*c^5*e^14 + 163840*a^4*b ^3*c^6*e^14))/(16*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(-(9*(b^15*f^8 + f^8*(-(4*a*c - b^2)^15)^(1/2) - 81920*a^7*b *c^7*f^8 - 560*a^2*b^11*c^2*f^8 + 4160*a^3*b^9*c^3*f^8 - 11520*a^4*b^7*c^4 *f^8 - 1024*a^5*b^5*c^5*f^8 + 61440*a^6*b^3*c^6*f^8 + 20*a*b^13*c*f^8))/(5 12*(a*b^20*e^2 + 1048576*a^11*c^10*e^2 - 40*a^2*b^18*c*e^2 + 720*a^3*b^16* c^2*e^2 - 7680*a^4*b^14*c^3*e^2 + 53760*a^5*b^12*c^4*e^2 - 258048*a^6*b^10 *c^5*e^2 + 860160*a^7*b^8*c^6*e^2 - 1966080*a^8*b^6*c^7*e^2 + 2949120*a...
\[ \int \frac {(d f+e f x)^4}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {too large to display} \] Input:
int((e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)
Output:
(f**4*( - 12*int(x**5/(a**3*b**2 + 12*a**3*b*c*d**2 + 36*a**3*c**2*d**4 + 3*a**2*b**3*d**2 + 6*a**2*b**3*d*e*x + 3*a**2*b**3*e**2*x**2 + 39*a**2*b** 2*c*d**4 + 84*a**2*b**2*c*d**3*e*x + 54*a**2*b**2*c*d**2*e**2*x**2 + 12*a* *2*b**2*c*d*e**3*x**3 + 3*a**2*b**2*c*e**4*x**4 + 144*a**2*b*c**2*d**6 + 3 60*a**2*b*c**2*d**5*e*x + 324*a**2*b*c**2*d**4*e**2*x**2 + 144*a**2*b*c**2 *d**3*e**3*x**3 + 36*a**2*b*c**2*d**2*e**4*x**4 + 108*a**2*c**3*d**8 + 432 *a**2*c**3*d**7*e*x + 648*a**2*c**3*d**6*e**2*x**2 + 432*a**2*c**3*d**5*e* *3*x**3 + 108*a**2*c**3*d**4*e**4*x**4 + 3*a*b**4*d**4 + 12*a*b**4*d**3*e* x + 18*a*b**4*d**2*e**2*x**2 + 12*a*b**4*d*e**3*x**3 + 3*a*b**4*e**4*x**4 + 42*a*b**3*c*d**6 + 180*a*b**3*c*d**5*e*x + 306*a*b**3*c*d**4*e**2*x**2 + 264*a*b**3*c*d**3*e**3*x**3 + 126*a*b**3*c*d**2*e**4*x**4 + 36*a*b**3*c*d *e**5*x**5 + 6*a*b**3*c*e**6*x**6 + 183*a*b**2*c**2*d**8 + 888*a*b**2*c**2 *d**7*e*x + 1812*a*b**2*c**2*d**6*e**2*x**2 + 2040*a*b**2*c**2*d**5*e**3*x **3 + 1398*a*b**2*c**2*d**4*e**4*x**4 + 600*a*b**2*c**2*d**3*e**5*x**5 + 1 56*a*b**2*c**2*d**2*e**6*x**6 + 24*a*b**2*c**2*d*e**7*x**7 + 3*a*b**2*c**2 *e**8*x**8 + 252*a*b*c**3*d**10 + 1584*a*b*c**3*d**9*e*x + 4248*a*b*c**3*d **8*e**2*x**2 + 6336*a*b*c**3*d**7*e**3*x**3 + 5760*a*b*c**3*d**6*e**4*x** 4 + 3312*a*b*c**3*d**5*e**5*x**5 + 1224*a*b*c**3*d**4*e**6*x**6 + 288*a*b* c**3*d**3*e**7*x**7 + 36*a*b*c**3*d**2*e**8*x**8 + 108*a*c**4*d**12 + 864* a*c**4*d**11*e*x + 3024*a*c**4*d**10*e**2*x**2 + 6048*a*c**4*d**9*e**3*...