Integrand size = 27, antiderivative size = 292 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=\frac {\left (a^2 C e^4+c^2 d^2 \left (5 C d^2-e (4 B d-3 A e)\right )+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )\right ) x}{e^6}-\frac {c \left (2 a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right ) x^2}{2 e^5}+\frac {c \left (2 a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right ) x^3}{3 e^4}-\frac {c^2 (2 C d-B e) x^4}{4 e^3}+\frac {c^2 C x^5}{5 e^2}-\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right )}{e^7 (d+e x)}-\frac {\left (c d^2+a e^2\right ) \left (a e^2 (2 C d-B e)+c d \left (6 C d^2-e (5 B d-4 A e)\right )\right ) \log (d+e x)}{e^7} \]
(a^2*C*e^4+c^2*d^2*(5*C*d^2-e*(-3*A*e+4*B*d))+2*a*c*e^2*(3*C*d^2-e*(-A*e+2 *B*d)))*x/e^6-1/2*c*(2*a*e^2*(-B*e+2*C*d)+c*d*(4*C*d^2-e*(-2*A*e+3*B*d)))* x^2/e^5+1/3*c*(2*a*C*e^2+c*(3*C*d^2-e*(-A*e+2*B*d)))*x^3/e^4-1/4*c^2*(-B*e +2*C*d)*x^4/e^3+1/5*c^2*C*x^5/e^2-(a*e^2+c*d^2)^2*(A*e^2-B*d*e+C*d^2)/e^7/ (e*x+d)-(a*e^2+c*d^2)*(a*e^2*(-B*e+2*C*d)+c*d*(6*C*d^2-e*(-4*A*e+5*B*d)))* ln(e*x+d)/e^7
Time = 0.18 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=\frac {60 e \left (a^2 C e^4+2 a c e^2 \left (3 C d^2+e (-2 B d+A e)\right )+c^2 \left (5 C d^4+d^2 e (-4 B d+3 A e)\right )\right ) x-30 c e^2 \left (4 c C d^3+c d e (-3 B d+2 A e)-2 a e^2 (-2 C d+B e)\right ) x^2+20 c e^3 \left (3 c C d^2+2 a C e^2+c e (-2 B d+A e)\right ) x^3+15 c^2 e^4 (-2 C d+B e) x^4+12 c^2 C e^5 x^5-\frac {60 \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right )}{d+e x}-60 \left (c d^2+a e^2\right ) \left (6 c C d^3+c d e (-5 B d+4 A e)+a e^2 (2 C d-B e)\right ) \log (d+e x)}{60 e^7} \]
(60*e*(a^2*C*e^4 + 2*a*c*e^2*(3*C*d^2 + e*(-2*B*d + A*e)) + c^2*(5*C*d^4 + d^2*e*(-4*B*d + 3*A*e)))*x - 30*c*e^2*(4*c*C*d^3 + c*d*e*(-3*B*d + 2*A*e) - 2*a*e^2*(-2*C*d + B*e))*x^2 + 20*c*e^3*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(-2 *B*d + A*e))*x^3 + 15*c^2*e^4*(-2*C*d + B*e)*x^4 + 12*c^2*C*e^5*x^5 - (60* (c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) - 60*(c*d^2 + a*e^ 2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d + e*x] )/(60*e^7)
Time = 0.68 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )}{e^6}+\frac {c x \left (-2 a e^2 (2 C d-B e)+c d e (3 B d-2 A e)-4 c C d^3\right )}{e^5}+\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^6 (d+e x)^2}+\frac {c x^2 \left (2 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{e^4}+\frac {\left (a e^2+c d^2\right ) \left (-a e^2 (2 C d-B e)+c d e (5 B d-4 A e)-6 c C d^3\right )}{e^6 (d+e x)}+\frac {c^2 x^3 (B e-2 C d)}{e^3}+\frac {c^2 C x^4}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )\right )}{e^6}-\frac {c x^2 \left (2 a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{2 e^5}-\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^7 (d+e x)}+\frac {c x^3 \left (2 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{3 e^4}-\frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)-c d e (5 B d-4 A e)+6 c C d^3\right )}{e^7}-\frac {c^2 x^4 (2 C d-B e)}{4 e^3}+\frac {c^2 C x^5}{5 e^2}\) |
((a^2*C*e^4 + c^2*(5*C*d^4 - d^2*e*(4*B*d - 3*A*e)) + 2*a*c*e^2*(3*C*d^2 - e*(2*B*d - A*e)))*x)/e^6 - (c*(4*c*C*d^3 - c*d*e*(3*B*d - 2*A*e) + 2*a*e^ 2*(2*C*d - B*e))*x^2)/(2*e^5) + (c*(3*c*C*d^2 + 2*a*C*e^2 - c*e*(2*B*d - A *e))*x^3)/(3*e^4) - (c^2*(2*C*d - B*e)*x^4)/(4*e^3) + (c^2*C*x^5)/(5*e^2) - ((c*d^2 + a*e^2)^2*(C*d^2 - B*d*e + A*e^2))/(e^7*(d + e*x)) - ((c*d^2 + a*e^2)*(6*c*C*d^3 - c*d*e*(5*B*d - 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d + e *x])/e^7
3.1.30.3.1 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]
Time = 0.49 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.37
| method | result | size |
| norman | \(\frac {\frac {\left (A \,a^{2} e^{6}+4 A a c \,d^{2} e^{4}+4 A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-6 B a c \,d^{3} e^{3}-5 B \,c^{2} d^{5} e +2 C \,a^{2} d^{2} e^{4}+8 C a c \,d^{4} e^{2}+6 C \,c^{2} d^{6}\right ) x}{e^{6} d}+\frac {\left (4 A a c \,e^{4}+4 A \,c^{2} d^{2} e^{2}-6 B a c d \,e^{3}-5 B \,c^{2} d^{3} e +2 a^{2} C \,e^{4}+8 C a c \,d^{2} e^{2}+6 C \,c^{2} d^{4}\right ) x^{2}}{2 e^{5}}+\frac {c \left (4 A c \,e^{2}-5 B c d e +8 a C \,e^{2}+6 C c \,d^{2}\right ) x^{4}}{12 e^{3}}-\frac {c \left (4 A c d \,e^{2}-6 B \,e^{3} a -5 B c \,d^{2} e +8 C a d \,e^{2}+6 C c \,d^{3}\right ) x^{3}}{6 e^{4}}+\frac {c^{2} C \,x^{6}}{5 e}+\frac {c^{2} \left (5 B e -6 C d \right ) x^{5}}{20 e^{2}}}{e x +d}-\frac {\left (4 A a c d \,e^{4}+4 A \,c^{2} d^{3} e^{2}-B \,e^{5} a^{2}-6 B a c \,d^{2} e^{3}-5 B \,c^{2} d^{4} e +2 C \,a^{2} d \,e^{4}+8 C a c \,d^{3} e^{2}+6 C \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(401\) |
| default | \(\frac {\frac {1}{5} c^{2} C \,x^{5} e^{4}+\frac {1}{4} B \,c^{2} e^{4} x^{4}-\frac {1}{2} C \,c^{2} d \,e^{3} x^{4}+\frac {1}{3} A \,c^{2} e^{4} x^{3}-\frac {2}{3} B \,c^{2} d \,e^{3} x^{3}+\frac {2}{3} C a c \,e^{4} x^{3}+C \,c^{2} d^{2} e^{2} x^{3}-A \,c^{2} d \,e^{3} x^{2}+B a c \,e^{4} x^{2}+\frac {3}{2} B \,c^{2} d^{2} e^{2} x^{2}-2 C a c d \,e^{3} x^{2}-2 C \,c^{2} d^{3} e \,x^{2}+2 A a c \,e^{4} x +3 A \,c^{2} d^{2} e^{2} x -4 B a c d \,e^{3} x -4 B \,c^{2} d^{3} e x +a^{2} C \,e^{4} x +6 C a c \,d^{2} e^{2} x +5 C \,c^{2} d^{4} x}{e^{6}}-\frac {A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}}{e^{7} \left (e x +d \right )}+\frac {\left (-4 A a c d \,e^{4}-4 A \,c^{2} d^{3} e^{2}+B \,e^{5} a^{2}+6 B a c \,d^{2} e^{3}+5 B \,c^{2} d^{4} e -2 C \,a^{2} d \,e^{4}-8 C a c \,d^{3} e^{2}-6 C \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(426\) |
| risch | \(-\frac {C \,c^{2} d \,x^{4}}{2 e^{3}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}+\frac {B \,c^{2} d^{5}}{e^{6} \left (e x +d \right )}-\frac {C \,a^{2} d^{2}}{e^{3} \left (e x +d \right )}-\frac {C \,c^{2} d^{6}}{e^{7} \left (e x +d \right )}+\frac {6 C a c \,d^{2} x}{e^{4}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}+\frac {a^{2} C x}{e^{2}}+\frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {\ln \left (e x +d \right ) a^{2} B}{e^{2}}-\frac {A \,a^{2}}{e \left (e x +d \right )}-\frac {2 C a c d \,x^{2}}{e^{3}}-\frac {4 B a c d x}{e^{3}}-\frac {2 A a c \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 B a c \,d^{3}}{e^{4} \left (e x +d \right )}-\frac {2 C a c \,d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 \ln \left (e x +d \right ) A \,c^{2} d^{3}}{e^{5}}+\frac {5 \ln \left (e x +d \right ) B \,c^{2} d^{4}}{e^{6}}-\frac {2 \ln \left (e x +d \right ) C \,a^{2} d}{e^{3}}-\frac {6 \ln \left (e x +d \right ) C \,c^{2} d^{5}}{e^{7}}+\frac {c^{2} C \,x^{5}}{5 e^{2}}+\frac {2 C a c \,x^{3}}{3 e^{2}}+\frac {C \,c^{2} d^{2} x^{3}}{e^{4}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {B a c \,x^{2}}{e^{2}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {2 C \,c^{2} d^{3} x^{2}}{e^{5}}+\frac {2 A a c x}{e^{2}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}}+\frac {5 C \,c^{2} d^{4} x}{e^{6}}-\frac {A \,c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 \ln \left (e x +d \right ) A a c d}{e^{3}}+\frac {6 \ln \left (e x +d \right ) B a c \,d^{2}}{e^{4}}-\frac {8 \ln \left (e x +d \right ) C a c \,d^{3}}{e^{5}}+\frac {B \,a^{2} d}{e^{2} \left (e x +d \right )}\) | \(527\) |
| parallelrisch | \(-\frac {150 B \,x^{2} c^{2} d^{3} e^{3}+18 C \,x^{5} c^{2} d \,e^{5}+25 B \,x^{4} c^{2} d \,e^{5}+120 C \ln \left (e x +d \right ) a^{2} d^{2} e^{4}+240 A \ln \left (e x +d \right ) c^{2} d^{4} e^{2}-60 B \ln \left (e x +d \right ) a^{2} d \,e^{5}-300 B \ln \left (e x +d \right ) c^{2} d^{5} e -40 C \,x^{4} a c \,e^{6}-30 C \,x^{4} c^{2} d^{2} e^{4}+240 A a c \,d^{2} e^{4}-360 B a c \,d^{3} e^{3}+480 C a c \,d^{4} e^{2}+60 A \,a^{2} e^{6}+360 C \,c^{2} d^{6}-15 B \,x^{5} c^{2} e^{6}-20 A \,x^{4} c^{2} e^{6}-12 C \,x^{6} c^{2} e^{6}+360 C \ln \left (e x +d \right ) c^{2} d^{6}-60 C \,x^{2} a^{2} e^{6}-300 B \,c^{2} d^{5} e +120 C \,a^{2} d^{2} e^{4}-60 B \,a^{2} d \,e^{5}+240 A \,c^{2} d^{4} e^{2}+80 C \,x^{3} a c d \,e^{5}+180 B \,x^{2} a c d \,e^{5}-50 B \,x^{3} c^{2} d^{2} e^{4}+60 C \,x^{3} c^{2} d^{3} e^{3}+480 C \ln \left (e x +d \right ) a c \,d^{4} e^{2}-60 B \ln \left (e x +d \right ) x \,a^{2} e^{6}+240 A \ln \left (e x +d \right ) x a c d \,e^{5}-360 B \ln \left (e x +d \right ) x a c \,d^{2} e^{4}+480 C \ln \left (e x +d \right ) x a c \,d^{3} e^{3}+240 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{3}-120 A \,x^{2} a c \,e^{6}-120 A \,x^{2} c^{2} d^{2} e^{4}-360 B \ln \left (e x +d \right ) a c \,d^{3} e^{3}-60 B \,x^{3} a c \,e^{6}-180 C \,x^{2} c^{2} d^{4} e^{2}+40 A \,x^{3} c^{2} d \,e^{5}-300 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e^{2}+120 C \ln \left (e x +d \right ) x \,a^{2} d \,e^{5}+360 C \ln \left (e x +d \right ) x \,c^{2} d^{5} e -240 C \,x^{2} a c \,d^{2} e^{4}+240 A \ln \left (e x +d \right ) a c \,d^{2} e^{4}}{60 e^{7} \left (e x +d \right )}\) | \(631\) |
((A*a^2*e^6+4*A*a*c*d^2*e^4+4*A*c^2*d^4*e^2-B*a^2*d*e^5-6*B*a*c*d^3*e^3-5* B*c^2*d^5*e+2*C*a^2*d^2*e^4+8*C*a*c*d^4*e^2+6*C*c^2*d^6)/e^6/d*x+1/2*(4*A* a*c*e^4+4*A*c^2*d^2*e^2-6*B*a*c*d*e^3-5*B*c^2*d^3*e+2*C*a^2*e^4+8*C*a*c*d^ 2*e^2+6*C*c^2*d^4)/e^5*x^2+1/12*c*(4*A*c*e^2-5*B*c*d*e+8*C*a*e^2+6*C*c*d^2 )/e^3*x^4-1/6*c*(4*A*c*d*e^2-6*B*a*e^3-5*B*c*d^2*e+8*C*a*d*e^2+6*C*c*d^3)/ e^4*x^3+1/5*c^2*C*x^6/e+1/20*c^2*(5*B*e-6*C*d)/e^2*x^5)/(e*x+d)-(4*A*a*c*d *e^4+4*A*c^2*d^3*e^2-B*a^2*e^5-6*B*a*c*d^2*e^3-5*B*c^2*d^4*e+2*C*a^2*d*e^4 +8*C*a*c*d^3*e^2+6*C*c^2*d^5)/e^7*ln(e*x+d)
Time = 0.28 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=\frac {12 \, C c^{2} e^{6} x^{6} - 60 \, C c^{2} d^{6} + 60 \, B c^{2} d^{5} e + 120 \, B a c d^{3} e^{3} + 60 \, B a^{2} d e^{5} - 60 \, A a^{2} e^{6} - 60 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} - 60 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} - 3 \, {\left (6 \, C c^{2} d e^{5} - 5 \, B c^{2} e^{6}\right )} x^{5} + 5 \, {\left (6 \, C c^{2} d^{2} e^{4} - 5 \, B c^{2} d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (6 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} - 6 \, B a c e^{6} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} + 30 \, {\left (6 \, C c^{2} d^{4} e^{2} - 5 \, B c^{2} d^{3} e^{3} - 6 \, B a c d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} e^{6}\right )} x^{2} + 60 \, {\left (5 \, C c^{2} d^{5} e - 4 \, B c^{2} d^{4} e^{2} - 4 \, B a c d^{2} e^{4} + 3 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x - 60 \, {\left (6 \, C c^{2} d^{6} - 5 \, B c^{2} d^{5} e - 6 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} + {\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} - B a^{2} e^{6} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{8} x + d e^{7}\right )}} \]
1/60*(12*C*c^2*e^6*x^6 - 60*C*c^2*d^6 + 60*B*c^2*d^5*e + 120*B*a*c*d^3*e^3 + 60*B*a^2*d*e^5 - 60*A*a^2*e^6 - 60*(2*C*a*c + A*c^2)*d^4*e^2 - 60*(C*a^ 2 + 2*A*a*c)*d^2*e^4 - 3*(6*C*c^2*d*e^5 - 5*B*c^2*e^6)*x^5 + 5*(6*C*c^2*d^ 2*e^4 - 5*B*c^2*d*e^5 + 4*(2*C*a*c + A*c^2)*e^6)*x^4 - 10*(6*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 - 6*B*a*c*e^6 + 4*(2*C*a*c + A*c^2)*d*e^5)*x^3 + 30*(6* C*c^2*d^4*e^2 - 5*B*c^2*d^3*e^3 - 6*B*a*c*d*e^5 + 4*(2*C*a*c + A*c^2)*d^2* e^4 + 2*(C*a^2 + 2*A*a*c)*e^6)*x^2 + 60*(5*C*c^2*d^5*e - 4*B*c^2*d^4*e^2 - 4*B*a*c*d^2*e^4 + 3*(2*C*a*c + A*c^2)*d^3*e^3 + (C*a^2 + 2*A*a*c)*d*e^5)* x - 60*(6*C*c^2*d^6 - 5*B*c^2*d^5*e - 6*B*a*c*d^3*e^3 - B*a^2*d*e^5 + 4*(2 *C*a*c + A*c^2)*d^4*e^2 + 2*(C*a^2 + 2*A*a*c)*d^2*e^4 + (6*C*c^2*d^5*e - 5 *B*c^2*d^4*e^2 - 6*B*a*c*d^2*e^4 - B*a^2*e^6 + 4*(2*C*a*c + A*c^2)*d^3*e^3 + 2*(C*a^2 + 2*A*a*c)*d*e^5)*x)*log(e*x + d))/(e^8*x + d*e^7)
Time = 1.09 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=\frac {C c^{2} x^{5}}{5 e^{2}} + x^{4} \left (\frac {B c^{2}}{4 e^{2}} - \frac {C c^{2} d}{2 e^{3}}\right ) + x^{3} \left (\frac {A c^{2}}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}} + \frac {2 C a c}{3 e^{2}} + \frac {C c^{2} d^{2}}{e^{4}}\right ) + x^{2} \left (- \frac {A c^{2} d}{e^{3}} + \frac {B a c}{e^{2}} + \frac {3 B c^{2} d^{2}}{2 e^{4}} - \frac {2 C a c d}{e^{3}} - \frac {2 C c^{2} d^{3}}{e^{5}}\right ) + x \left (\frac {2 A a c}{e^{2}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {4 B a c d}{e^{3}} - \frac {4 B c^{2} d^{3}}{e^{5}} + \frac {C a^{2}}{e^{2}} + \frac {6 C a c d^{2}}{e^{4}} + \frac {5 C c^{2} d^{4}}{e^{6}}\right ) + \frac {- A a^{2} e^{6} - 2 A a c d^{2} e^{4} - A c^{2} d^{4} e^{2} + B a^{2} d e^{5} + 2 B a c d^{3} e^{3} + B c^{2} d^{5} e - C a^{2} d^{2} e^{4} - 2 C a c d^{4} e^{2} - C c^{2} d^{6}}{d e^{7} + e^{8} x} - \frac {\left (a e^{2} + c d^{2}\right ) \left (4 A c d e^{2} - B a e^{3} - 5 B c d^{2} e + 2 C a d e^{2} + 6 C c d^{3}\right ) \log {\left (d + e x \right )}}{e^{7}} \]
C*c**2*x**5/(5*e**2) + x**4*(B*c**2/(4*e**2) - C*c**2*d/(2*e**3)) + x**3*( A*c**2/(3*e**2) - 2*B*c**2*d/(3*e**3) + 2*C*a*c/(3*e**2) + C*c**2*d**2/e** 4) + x**2*(-A*c**2*d/e**3 + B*a*c/e**2 + 3*B*c**2*d**2/(2*e**4) - 2*C*a*c* d/e**3 - 2*C*c**2*d**3/e**5) + x*(2*A*a*c/e**2 + 3*A*c**2*d**2/e**4 - 4*B* a*c*d/e**3 - 4*B*c**2*d**3/e**5 + C*a**2/e**2 + 6*C*a*c*d**2/e**4 + 5*C*c* *2*d**4/e**6) + (-A*a**2*e**6 - 2*A*a*c*d**2*e**4 - A*c**2*d**4*e**2 + B*a **2*d*e**5 + 2*B*a*c*d**3*e**3 + B*c**2*d**5*e - C*a**2*d**2*e**4 - 2*C*a* c*d**4*e**2 - C*c**2*d**6)/(d*e**7 + e**8*x) - (a*e**2 + c*d**2)*(4*A*c*d* e**2 - B*a*e**3 - 5*B*c*d**2*e + 2*C*a*d*e**2 + 6*C*c*d**3)*log(d + e*x)/e **7
Time = 0.20 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=-\frac {C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac {12 \, C c^{2} e^{4} x^{5} - 15 \, {\left (2 \, C c^{2} d e^{3} - B c^{2} e^{4}\right )} x^{4} + 20 \, {\left (3 \, C c^{2} d^{2} e^{2} - 2 \, B c^{2} d e^{3} + {\left (2 \, C a c + A c^{2}\right )} e^{4}\right )} x^{3} - 30 \, {\left (4 \, C c^{2} d^{3} e - 3 \, B c^{2} d^{2} e^{2} - 2 \, B a c e^{4} + 2 \, {\left (2 \, C a c + A c^{2}\right )} d e^{3}\right )} x^{2} + 60 \, {\left (5 \, C c^{2} d^{4} - 4 \, B c^{2} d^{3} e - 4 \, B a c d e^{3} + 3 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} e^{4}\right )} x}{60 \, e^{6}} - \frac {{\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e - 6 \, B a c d^{2} e^{3} - B a^{2} e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]
-(C*c^2*d^6 - B*c^2*d^5*e - 2*B*a*c*d^3*e^3 - B*a^2*d*e^5 + A*a^2*e^6 + (2 *C*a*c + A*c^2)*d^4*e^2 + (C*a^2 + 2*A*a*c)*d^2*e^4)/(e^8*x + d*e^7) + 1/6 0*(12*C*c^2*e^4*x^5 - 15*(2*C*c^2*d*e^3 - B*c^2*e^4)*x^4 + 20*(3*C*c^2*d^2 *e^2 - 2*B*c^2*d*e^3 + (2*C*a*c + A*c^2)*e^4)*x^3 - 30*(4*C*c^2*d^3*e - 3* B*c^2*d^2*e^2 - 2*B*a*c*e^4 + 2*(2*C*a*c + A*c^2)*d*e^3)*x^2 + 60*(5*C*c^2 *d^4 - 4*B*c^2*d^3*e - 4*B*a*c*d*e^3 + 3*(2*C*a*c + A*c^2)*d^2*e^2 + (C*a^ 2 + 2*A*a*c)*e^4)*x)/e^6 - (6*C*c^2*d^5 - 5*B*c^2*d^4*e - 6*B*a*c*d^2*e^3 - B*a^2*e^5 + 4*(2*C*a*c + A*c^2)*d^3*e^2 + 2*(C*a^2 + 2*A*a*c)*d*e^4)*log (e*x + d)/e^7
Time = 0.28 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=\frac {{\left (12 \, C c^{2} - \frac {15 \, {\left (6 \, C c^{2} d e - B c^{2} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {20 \, {\left (15 \, C c^{2} d^{2} e^{2} - 5 \, B c^{2} d e^{3} + 2 \, C a c e^{4} + A c^{2} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {60 \, {\left (10 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} + 4 \, C a c d e^{5} + 2 \, A c^{2} d e^{5} - B a c e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {60 \, {\left (15 \, C c^{2} d^{4} e^{4} - 10 \, B c^{2} d^{3} e^{5} + 12 \, C a c d^{2} e^{6} + 6 \, A c^{2} d^{2} e^{6} - 6 \, B a c d e^{7} + C a^{2} e^{8} + 2 \, A a c e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}}\right )} {\left (e x + d\right )}^{5}}{60 \, e^{7}} + \frac {{\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e + 8 \, C a c d^{3} e^{2} + 4 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} + 2 \, C a^{2} d e^{4} + 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{7}} - \frac {\frac {C c^{2} d^{6} e^{5}}{e x + d} - \frac {B c^{2} d^{5} e^{6}}{e x + d} + \frac {2 \, C a c d^{4} e^{7}}{e x + d} + \frac {A c^{2} d^{4} e^{7}}{e x + d} - \frac {2 \, B a c d^{3} e^{8}}{e x + d} + \frac {C a^{2} d^{2} e^{9}}{e x + d} + \frac {2 \, A a c d^{2} e^{9}}{e x + d} - \frac {B a^{2} d e^{10}}{e x + d} + \frac {A a^{2} e^{11}}{e x + d}}{e^{12}} \]
1/60*(12*C*c^2 - 15*(6*C*c^2*d*e - B*c^2*e^2)/((e*x + d)*e) + 20*(15*C*c^2 *d^2*e^2 - 5*B*c^2*d*e^3 + 2*C*a*c*e^4 + A*c^2*e^4)/((e*x + d)^2*e^2) - 60 *(10*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 + 4*C*a*c*d*e^5 + 2*A*c^2*d*e^5 - B*a *c*e^6)/((e*x + d)^3*e^3) + 60*(15*C*c^2*d^4*e^4 - 10*B*c^2*d^3*e^5 + 12*C *a*c*d^2*e^6 + 6*A*c^2*d^2*e^6 - 6*B*a*c*d*e^7 + C*a^2*e^8 + 2*A*a*c*e^8)/ ((e*x + d)^4*e^4))*(e*x + d)^5/e^7 + (6*C*c^2*d^5 - 5*B*c^2*d^4*e + 8*C*a* c*d^3*e^2 + 4*A*c^2*d^3*e^2 - 6*B*a*c*d^2*e^3 + 2*C*a^2*d*e^4 + 4*A*a*c*d* e^4 - B*a^2*e^5)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^7 - (C*c^2*d^6*e ^5/(e*x + d) - B*c^2*d^5*e^6/(e*x + d) + 2*C*a*c*d^4*e^7/(e*x + d) + A*c^2 *d^4*e^7/(e*x + d) - 2*B*a*c*d^3*e^8/(e*x + d) + C*a^2*d^2*e^9/(e*x + d) + 2*A*a*c*d^2*e^9/(e*x + d) - B*a^2*d*e^10/(e*x + d) + A*a^2*e^11/(e*x + d) )/e^12
Time = 0.12 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a+c x^2\right )^2 \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx=x^4\,\left (\frac {B\,c^2}{4\,e^2}-\frac {C\,c^2\,d}{2\,e^3}\right )+x\,\left (\frac {C\,a^2+2\,A\,c\,a}{e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,B\,a\,c}{e^2}\right )}{e}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{3\,e}-\frac {A\,c^2+2\,C\,a\,c}{3\,e^2}+\frac {C\,c^2\,d^2}{3\,e^4}\right )+x^2\,\left (\frac {d\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{2\,e^2}+\frac {B\,a\,c}{e^2}\right )-\frac {C\,a^2\,d^2\,e^4-B\,a^2\,d\,e^5+A\,a^2\,e^6+2\,C\,a\,c\,d^4\,e^2-2\,B\,a\,c\,d^3\,e^3+2\,A\,a\,c\,d^2\,e^4+C\,c^2\,d^6-B\,c^2\,d^5\,e+A\,c^2\,d^4\,e^2}{e\,\left (x\,e^7+d\,e^6\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,C\,a^2\,d\,e^4-B\,a^2\,e^5+8\,C\,a\,c\,d^3\,e^2-6\,B\,a\,c\,d^2\,e^3+4\,A\,a\,c\,d\,e^4+6\,C\,c^2\,d^5-5\,B\,c^2\,d^4\,e+4\,A\,c^2\,d^3\,e^2\right )}{e^7}+\frac {C\,c^2\,x^5}{5\,e^2} \]
x^4*((B*c^2)/(4*e^2) - (C*c^2*d)/(2*e^3)) + x*((C*a^2 + 2*A*a*c)/e^2 + (d^ 2*((2*d*((B*c^2)/e^2 - (2*C*c^2*d)/e^3))/e - (A*c^2 + 2*C*a*c)/e^2 + (C*c^ 2*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((B*c^2)/e^2 - (2*C*c^2*d)/e^3))/e - ( A*c^2 + 2*C*a*c)/e^2 + (C*c^2*d^2)/e^4))/e - (d^2*((B*c^2)/e^2 - (2*C*c^2* d)/e^3))/e^2 + (2*B*a*c)/e^2))/e) - x^3*((2*d*((B*c^2)/e^2 - (2*C*c^2*d)/e ^3))/(3*e) - (A*c^2 + 2*C*a*c)/(3*e^2) + (C*c^2*d^2)/(3*e^4)) + x^2*((d*(( 2*d*((B*c^2)/e^2 - (2*C*c^2*d)/e^3))/e - (A*c^2 + 2*C*a*c)/e^2 + (C*c^2*d^ 2)/e^4))/e - (d^2*((B*c^2)/e^2 - (2*C*c^2*d)/e^3))/(2*e^2) + (B*a*c)/e^2) - (A*a^2*e^6 + C*c^2*d^6 - B*a^2*d*e^5 - B*c^2*d^5*e + A*c^2*d^4*e^2 + C*a ^2*d^2*e^4 + 2*A*a*c*d^2*e^4 - 2*B*a*c*d^3*e^3 + 2*C*a*c*d^4*e^2)/(e*(d*e^ 6 + e^7*x)) - (log(d + e*x)*(6*C*c^2*d^5 - B*a^2*e^5 + 2*C*a^2*d*e^4 - 5*B *c^2*d^4*e + 4*A*c^2*d^3*e^2 + 4*A*a*c*d*e^4 - 6*B*a*c*d^2*e^3 + 8*C*a*c*d ^3*e^2))/e^7 + (C*c^2*x^5)/(5*e^2)