Integrand size = 27, antiderivative size = 156 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=-\frac {(a B-(A c-a C) x) (d+e x)^2}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) (a (A c+3 a C) e-c (3 A c d+a C d+2 a B e) x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (a (A c+3 a C) e^2+c d (3 A c d+a C d+2 a B e)\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \]
-1/4*(a*B-(A*c-C*a)*x)*(e*x+d)^2/a/c/(c*x^2+a)^2-1/8*(e*x+d)*(a*(A*c+3*C*a )*e-c*(3*A*c*d+2*B*a*e+C*a*d)*x)/a^2/c^2/(c*x^2+a)+1/8*(a*(A*c+3*C*a)*e^2+ c*d*(3*A*c*d+2*B*a*e+C*a*d))*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/c^(5/2)
Time = 0.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=\frac {3 A c^2 d^2 x+a c \left (C d^2+e (2 B d+A e)\right ) x-a^2 e (8 C d+4 B e+5 C e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {A c^2 d^2 x+a^2 e (2 C d+B e+C e x)-a c \left (C d^2 x+A e (2 d+e x)+B d (d+2 e x)\right )}{4 a c^2 \left (a+c x^2\right )^2}+\frac {\left (A c \left (3 c d^2+a e^2\right )+a \left (3 a C e^2+c d (C d+2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}} \]
(3*A*c^2*d^2*x + a*c*(C*d^2 + e*(2*B*d + A*e))*x - a^2*e*(8*C*d + 4*B*e + 5*C*e*x))/(8*a^2*c^2*(a + c*x^2)) + (A*c^2*d^2*x + a^2*e*(2*C*d + B*e + C* e*x) - a*c*(C*d^2*x + A*e*(2*d + e*x) + B*d*(d + 2*e*x)))/(4*a*c^2*(a + c* x^2)^2) + ((A*c*(3*c*d^2 + a*e^2) + a*(3*a*C*e^2 + c*d*(C*d + 2*B*e)))*Arc Tan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))
Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2176, 25, 675, 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.
| \(\displaystyle \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle -\frac {\int -\frac {(d+e x) (3 A c d+a C d+2 a B e+(A c+3 a C) e x)}{\left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x) (3 A c d+a C d+2 a B e+(A c+3 a C) e x)}{\left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 675 |
| \(\displaystyle \frac {\frac {\left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right ) \int \frac {1}{c x^2+a}dx}{2 a c}-\frac {x \left (a e^2 (3 a C+A c)-c d (2 a B e+a C d+3 A c d)\right )}{2 a c \left (a+c x^2\right )}-\frac {e (a B e+2 a C d+2 A c d)}{c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right )}{2 a^{3/2} c^{3/2}}-\frac {x \left (a e^2 (3 a C+A c)-c d (2 a B e+a C d+3 A c d)\right )}{2 a c \left (a+c x^2\right )}-\frac {e (a B e+2 a C d+2 A c d)}{c \left (a+c x^2\right )}}{4 a c}-\frac {(d+e x)^2 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}\) |
-1/4*((a*B - (A*c - a*C)*x)*(d + e*x)^2)/(a*c*(a + c*x^2)^2) + (-((e*(2*A* c*d + 2*a*C*d + a*B*e))/(c*(a + c*x^2))) - ((a*(A*c + 3*a*C)*e^2 - c*d*(3* A*c*d + a*C*d + 2*a*B*e))*x)/(2*a*c*(a + c*x^2)) + ((a*(A*c + 3*a*C)*e^2 + c*d*(3*A*c*d + a*C*d + 2*a*B*e))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)* c^(3/2)))/(4*a*c)
3.1.58.3.1 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_) + (e_.)*(x_))*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[a*(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + (- Simp[(c*d*f - a*e*g)*x*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Simp[(a* e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)) Int[(a + c*x^2)^(p + 1), x], x]) / ; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && !(IntegerQ[p] && NiceSqrtQ [(-a)*c])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.59 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(\frac {\frac {\left (A a c \,e^{2}+3 A \,c^{2} d^{2}+2 B a c d e -5 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{3}}{8 c \,a^{2}}-\frac {e \left (B e +2 C d \right ) x^{2}}{2 c}-\frac {\left (A a c \,e^{2}-5 A \,c^{2} d^{2}+2 B a c d e +3 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{8 a \,c^{2}}-\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}+2 a d e C}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\left (A a c \,e^{2}+3 A \,c^{2} d^{2}+2 B a c d e +3 a^{2} C \,e^{2}+C a c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 a^{2} c^{2} \sqrt {a c}}\) | \(222\) |
| risch | \(\frac {\frac {\left (A a c \,e^{2}+3 A \,c^{2} d^{2}+2 B a c d e -5 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x^{3}}{8 c \,a^{2}}-\frac {e \left (B e +2 C d \right ) x^{2}}{2 c}-\frac {\left (A a c \,e^{2}-5 A \,c^{2} d^{2}+2 B a c d e +3 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{8 a \,c^{2}}-\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}+2 a d e C}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) A \,e^{2}}{16 \sqrt {-a c}\, c a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) A \,d^{2}}{16 \sqrt {-a c}\, a^{2}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) B d e}{8 \sqrt {-a c}\, c a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) C \,e^{2}}{16 \sqrt {-a c}\, c^{2}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) C \,d^{2}}{16 \sqrt {-a c}\, c a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) A \,e^{2}}{16 \sqrt {-a c}\, c a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) A \,d^{2}}{16 \sqrt {-a c}\, a^{2}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) B d e}{8 \sqrt {-a c}\, c a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) C \,e^{2}}{16 \sqrt {-a c}\, c^{2}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) C \,d^{2}}{16 \sqrt {-a c}\, c a}\) | \(441\) |
(1/8*(A*a*c*e^2+3*A*c^2*d^2+2*B*a*c*d*e-5*C*a^2*e^2+C*a*c*d^2)/c/a^2*x^3-1 /2*e*(B*e+2*C*d)*x^2/c-1/8*(A*a*c*e^2-5*A*c^2*d^2+2*B*a*c*d*e+3*C*a^2*e^2+ C*a*c*d^2)/a/c^2*x-1/4*(2*A*c*d*e+B*a*e^2+B*c*d^2+2*C*a*d*e)/c^2)/(c*x^2+a )^2+1/8*(A*a*c*e^2+3*A*c^2*d^2+2*B*a*c*d*e+3*C*a^2*e^2+C*a*c*d^2)/a^2/c^2/ (a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (141) = 282\).
Time = 0.50 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.17 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {4 \, B a^{3} c^{2} d^{2} + 4 \, B a^{4} c e^{2} - 2 \, {\left (2 \, B a^{2} c^{3} d e + {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{2} - {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} e^{2}\right )} x^{3} + 8 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d e + 8 \, {\left (2 \, C a^{3} c^{2} d e + B a^{3} c^{2} e^{2}\right )} x^{2} + {\left (2 \, B a^{3} c d e + {\left (2 \, B a c^{3} d e + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{2} + {\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} e^{2}\right )} x^{4} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} e^{2} + 2 \, {\left (2 \, B a^{2} c^{2} d e + {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{2} + {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (2 \, B a^{3} c^{2} d e + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{2} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} e^{2}\right )} x}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{2} + 2 \, B a^{4} c e^{2} - {\left (2 \, B a^{2} c^{3} d e + {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{2} - {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} e^{2}\right )} x^{3} + 4 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d e + 4 \, {\left (2 \, C a^{3} c^{2} d e + B a^{3} c^{2} e^{2}\right )} x^{2} - {\left (2 \, B a^{3} c d e + {\left (2 \, B a c^{3} d e + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{2} + {\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} e^{2}\right )} x^{4} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} e^{2} + 2 \, {\left (2 \, B a^{2} c^{2} d e + {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{2} + {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (2 \, B a^{3} c^{2} d e + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{2} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} e^{2}\right )} x}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \]
[-1/16*(4*B*a^3*c^2*d^2 + 4*B*a^4*c*e^2 - 2*(2*B*a^2*c^3*d*e + (C*a^2*c^3 + 3*A*a*c^4)*d^2 - (5*C*a^3*c^2 - A*a^2*c^3)*e^2)*x^3 + 8*(C*a^4*c + A*a^3 *c^2)*d*e + 8*(2*C*a^3*c^2*d*e + B*a^3*c^2*e^2)*x^2 + (2*B*a^3*c*d*e + (2* B*a*c^3*d*e + (C*a*c^3 + 3*A*c^4)*d^2 + (3*C*a^2*c^2 + A*a*c^3)*e^2)*x^4 + (C*a^3*c + 3*A*a^2*c^2)*d^2 + (3*C*a^4 + A*a^3*c)*e^2 + 2*(2*B*a^2*c^2*d* e + (C*a^2*c^2 + 3*A*a*c^3)*d^2 + (3*C*a^3*c + A*a^2*c^2)*e^2)*x^2)*sqrt(- a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(2*B*a^3*c^2*d*e + (C*a^3*c^2 - 5*A*a^2*c^3)*d^2 + (3*C*a^4*c + A*a^3*c^2)*e^2)*x)/(a^3*c^5*x ^4 + 2*a^4*c^4*x^2 + a^5*c^3), -1/8*(2*B*a^3*c^2*d^2 + 2*B*a^4*c*e^2 - (2* B*a^2*c^3*d*e + (C*a^2*c^3 + 3*A*a*c^4)*d^2 - (5*C*a^3*c^2 - A*a^2*c^3)*e^ 2)*x^3 + 4*(C*a^4*c + A*a^3*c^2)*d*e + 4*(2*C*a^3*c^2*d*e + B*a^3*c^2*e^2) *x^2 - (2*B*a^3*c*d*e + (2*B*a*c^3*d*e + (C*a*c^3 + 3*A*c^4)*d^2 + (3*C*a^ 2*c^2 + A*a*c^3)*e^2)*x^4 + (C*a^3*c + 3*A*a^2*c^2)*d^2 + (3*C*a^4 + A*a^3 *c)*e^2 + 2*(2*B*a^2*c^2*d*e + (C*a^2*c^2 + 3*A*a*c^3)*d^2 + (3*C*a^3*c + A*a^2*c^2)*e^2)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (2*B*a^3*c^2*d*e + (C*a^3*c^2 - 5*A*a^2*c^3)*d^2 + (3*C*a^4*c + A*a^3*c^2)*e^2)*x)/(a^3*c^5*x ^4 + 2*a^4*c^4*x^2 + a^5*c^3)]
Timed out. \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.62 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=-\frac {2 \, B a^{2} c d^{2} + 2 \, B a^{3} e^{2} - {\left (2 \, B a c^{2} d e + {\left (C a c^{2} + 3 \, A c^{3}\right )} d^{2} - {\left (5 \, C a^{2} c - A a c^{2}\right )} e^{2}\right )} x^{3} + 4 \, {\left (C a^{3} + A a^{2} c\right )} d e + 4 \, {\left (2 \, C a^{2} c d e + B a^{2} c e^{2}\right )} x^{2} + {\left (2 \, B a^{2} c d e + {\left (C a^{2} c - 5 \, A a c^{2}\right )} d^{2} + {\left (3 \, C a^{3} + A a^{2} c\right )} e^{2}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {{\left (2 \, B a c d e + {\left (C a c + 3 \, A c^{2}\right )} d^{2} + {\left (3 \, C a^{2} + A a c\right )} e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \]
-1/8*(2*B*a^2*c*d^2 + 2*B*a^3*e^2 - (2*B*a*c^2*d*e + (C*a*c^2 + 3*A*c^3)*d ^2 - (5*C*a^2*c - A*a*c^2)*e^2)*x^3 + 4*(C*a^3 + A*a^2*c)*d*e + 4*(2*C*a^2 *c*d*e + B*a^2*c*e^2)*x^2 + (2*B*a^2*c*d*e + (C*a^2*c - 5*A*a*c^2)*d^2 + ( 3*C*a^3 + A*a^2*c)*e^2)*x)/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2) + 1/8*( 2*B*a*c*d*e + (C*a*c + 3*A*c^2)*d^2 + (3*C*a^2 + A*a*c)*e^2)*arctan(c*x/sq rt(a*c))/(sqrt(a*c)*a^2*c^2)
Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=\frac {{\left (C a c d^{2} + 3 \, A c^{2} d^{2} + 2 \, B a c d e + 3 \, C a^{2} e^{2} + A a c e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {C a c^{2} d^{2} x^{3} + 3 \, A c^{3} d^{2} x^{3} + 2 \, B a c^{2} d e x^{3} - 5 \, C a^{2} c e^{2} x^{3} + A a c^{2} e^{2} x^{3} - 8 \, C a^{2} c d e x^{2} - 4 \, B a^{2} c e^{2} x^{2} - C a^{2} c d^{2} x + 5 \, A a c^{2} d^{2} x - 2 \, B a^{2} c d e x - 3 \, C a^{3} e^{2} x - A a^{2} c e^{2} x - 2 \, B a^{2} c d^{2} - 4 \, C a^{3} d e - 4 \, A a^{2} c d e - 2 \, B a^{3} e^{2}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]
1/8*(C*a*c*d^2 + 3*A*c^2*d^2 + 2*B*a*c*d*e + 3*C*a^2*e^2 + A*a*c*e^2)*arct an(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2) + 1/8*(C*a*c^2*d^2*x^3 + 3*A*c^3*d^2 *x^3 + 2*B*a*c^2*d*e*x^3 - 5*C*a^2*c*e^2*x^3 + A*a*c^2*e^2*x^3 - 8*C*a^2*c *d*e*x^2 - 4*B*a^2*c*e^2*x^2 - C*a^2*c*d^2*x + 5*A*a*c^2*d^2*x - 2*B*a^2*c *d*e*x - 3*C*a^3*e^2*x - A*a^2*c*e^2*x - 2*B*a^2*c*d^2 - 4*C*a^3*d*e - 4*A *a^2*c*d*e - 2*B*a^3*e^2)/((c*x^2 + a)^2*a^2*c^2)
Time = 12.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^2 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+3\,A\,c^2\,d^2\right )}{8\,a^{5/2}\,c^{5/2}}-\frac {\frac {B\,a\,e^2+B\,c\,d^2+2\,A\,c\,d\,e+2\,C\,a\,d\,e}{4\,c^2}+\frac {x^2\,\left (B\,e^2+2\,C\,d\,e\right )}{2\,c}+\frac {x\,\left (3\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2-5\,A\,c^2\,d^2\right )}{8\,a\,c^2}-\frac {x^3\,\left (-5\,C\,a^2\,e^2+C\,a\,c\,d^2+2\,B\,a\,c\,d\,e+A\,a\,c\,e^2+3\,A\,c^2\,d^2\right )}{8\,a^2\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \]
(atan((c^(1/2)*x)/a^(1/2))*(3*A*c^2*d^2 + 3*C*a^2*e^2 + A*a*c*e^2 + C*a*c* d^2 + 2*B*a*c*d*e))/(8*a^(5/2)*c^(5/2)) - ((B*a*e^2 + B*c*d^2 + 2*A*c*d*e + 2*C*a*d*e)/(4*c^2) + (x^2*(B*e^2 + 2*C*d*e))/(2*c) + (x*(3*C*a^2*e^2 - 5 *A*c^2*d^2 + A*a*c*e^2 + C*a*c*d^2 + 2*B*a*c*d*e))/(8*a*c^2) - (x^3*(3*A*c ^2*d^2 - 5*C*a^2*e^2 + A*a*c*e^2 + C*a*c*d^2 + 2*B*a*c*d*e))/(8*a^2*c))/(a ^2 + c^2*x^4 + 2*a*c*x^2)