Integrand size = 27, antiderivative size = 466 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=-\frac {c \left (3 a^2 e^4 (3 C d-B e)+c^2 d^3 \left (21 C d^2-5 e (3 B d-2 A e)\right )+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )\right ) x}{e^8}+\frac {c \left (3 a^2 C e^4+c^2 d^2 \left (15 C d^2-2 e (5 B d-3 A e)\right )+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )\right ) x^2}{2 e^7}-\frac {c^2 \left (3 a e^2 (3 C d-B e)+c d \left (10 C d^2-3 e (2 B d-A e)\right )\right ) x^3}{3 e^6}+\frac {c^2 \left (3 a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right ) x^4}{4 e^5}-\frac {c^3 (3 C d-B e) x^5}{5 e^4}+\frac {c^3 C x^6}{6 e^3}-\frac {\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right )}{2 e^9 (d+e x)^2}+\frac {\left (c d^2+a e^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right )}{e^9 (d+e x)}+\frac {\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 d^2 \left (28 C d^2-3 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right ) \log (d+e x)}{e^9} \]
-c*(3*a^2*e^4*(-B*e+3*C*d)+c^2*d^3*(21*C*d^2-5*e*(-2*A*e+3*B*d))+3*a*c*d*e ^2*(10*C*d^2-3*e*(-A*e+2*B*d)))*x/e^8+1/2*c*(3*a^2*C*e^4+c^2*d^2*(15*C*d^2 -2*e*(-3*A*e+5*B*d))+3*a*c*e^2*(6*C*d^2-e*(-A*e+3*B*d)))*x^2/e^7-1/3*c^2*( 3*a*e^2*(-B*e+3*C*d)+c*d*(10*C*d^2-3*e*(-A*e+2*B*d)))*x^3/e^6+1/4*c^2*(3*a *C*e^2+c*(6*C*d^2-e*(-A*e+3*B*d)))*x^4/e^5-1/5*c^3*(-B*e+3*C*d)*x^5/e^4+1/ 6*c^3*C*x^6/e^3-1/2*(a*e^2+c*d^2)^3*(A*e^2-B*d*e+C*d^2)/e^9/(e*x+d)^2+(a*e ^2+c*d^2)^2*(a*e^2*(-B*e+2*C*d)+c*d*(8*C*d^2-e*(-6*A*e+7*B*d)))/e^9/(e*x+d )+(a*e^2+c*d^2)*(a^2*C*e^4+c^2*d^2*(28*C*d^2-3*e*(-5*A*e+7*B*d))+a*c*e^2*( 17*C*d^2-3*e*(-A*e+3*B*d)))*ln(e*x+d)/e^9
Time = 0.13 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {-60 c e \left (-3 a^2 e^4 (-3 C d+B e)+3 a c d e^2 \left (10 C d^2+3 e (-2 B d+A e)\right )+c^2 \left (21 C d^5+5 d^3 e (-3 B d+2 A e)\right )\right ) x+30 c e^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2+e (-3 B d+A e)\right )+c^2 \left (15 C d^4+2 d^2 e (-5 B d+3 A e)\right )\right ) x^2-20 c^2 e^3 \left (10 c C d^3+3 c d e (-2 B d+A e)-3 a e^2 (-3 C d+B e)\right ) x^3+15 c^2 e^4 \left (6 c C d^2+3 a C e^2+c e (-3 B d+A e)\right ) x^4+12 c^3 e^5 (-3 C d+B e) x^5+10 c^3 C e^6 x^6-\frac {30 \left (c d^2+a e^2\right )^3 \left (C d^2+e (-B d+A e)\right )}{(d+e x)^2}+\frac {60 \left (c d^2+a e^2\right )^2 \left (8 c C d^3+c d e (-7 B d+6 A e)+a e^2 (2 C d-B e)\right )}{d+e x}+60 \left (c d^2+a e^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2+3 e (-3 B d+A e)\right )+c^2 \left (28 C d^4+3 d^2 e (-7 B d+5 A e)\right )\right ) \log (d+e x)}{60 e^9} \]
(-60*c*e*(-3*a^2*e^4*(-3*C*d + B*e) + 3*a*c*d*e^2*(10*C*d^2 + 3*e*(-2*B*d + A*e)) + c^2*(21*C*d^5 + 5*d^3*e*(-3*B*d + 2*A*e)))*x + 30*c*e^2*(3*a^2*C *e^4 + 3*a*c*e^2*(6*C*d^2 + e*(-3*B*d + A*e)) + c^2*(15*C*d^4 + 2*d^2*e*(- 5*B*d + 3*A*e)))*x^2 - 20*c^2*e^3*(10*c*C*d^3 + 3*c*d*e*(-2*B*d + A*e) - 3 *a*e^2*(-3*C*d + B*e))*x^3 + 15*c^2*e^4*(6*c*C*d^2 + 3*a*C*e^2 + c*e*(-3*B *d + A*e))*x^4 + 12*c^3*e^5*(-3*C*d + B*e)*x^5 + 10*c^3*C*e^6*x^6 - (30*(c *d^2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 + (60*(c*d^2 + a*e ^2)^2*(8*c*C*d^3 + c*d*e*(-7*B*d + 6*A*e) + a*e^2*(2*C*d - B*e)))/(d + e*x ) + 60*(c*d^2 + a*e^2)*(a^2*C*e^4 + a*c*e^2*(17*C*d^2 + 3*e*(-3*B*d + A*e) ) + c^2*(28*C*d^4 + 3*d^2*e*(-7*B*d + 5*A*e)))*Log[d + e*x])/(60*e^9)
Time = 1.08 (sec) , antiderivative size = 463, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {\left (a e^2+c d^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{e^8 (d+e x)}+\frac {c x \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{e^7}+\frac {c \left (-3 a^2 e^4 (3 C d-B e)-3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )-c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )\right )}{e^8}+\frac {c^2 x^2 \left (-3 a e^2 (3 C d-B e)+3 c d e (2 B d-A e)-10 c C d^3\right )}{e^6}+\frac {c^2 x^3 \left (3 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{e^5}+\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{e^8 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 \left (-a e^2 (2 C d-B e)+c d e (7 B d-6 A e)-8 c C d^3\right )}{e^8 (d+e x)^2}+\frac {c^3 x^4 (B e-3 C d)}{e^4}+\frac {c^3 C x^5}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{e^9}+\frac {c x^2 \left (3 a^2 C e^4+3 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )\right )}{2 e^7}-\frac {c x \left (3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left (10 C d^2-3 e (2 B d-A e)\right )+c^2 \left (21 C d^5-5 d^3 e (3 B d-2 A e)\right )\right )}{e^8}-\frac {c^2 x^3 \left (3 a e^2 (3 C d-B e)-3 c d e (2 B d-A e)+10 c C d^3\right )}{3 e^6}+\frac {c^2 x^4 \left (3 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{4 e^5}-\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{2 e^9 (d+e x)^2}+\frac {\left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^9 (d+e x)}-\frac {c^3 x^5 (3 C d-B e)}{5 e^4}+\frac {c^3 C x^6}{6 e^3}\) |
-((c*(3*a^2*e^4*(3*C*d - B*e) + c^2*(21*C*d^5 - 5*d^3*e*(3*B*d - 2*A*e)) + 3*a*c*d*e^2*(10*C*d^2 - 3*e*(2*B*d - A*e)))*x)/e^8) + (c*(3*a^2*C*e^4 + c ^2*(15*C*d^4 - 2*d^2*e*(5*B*d - 3*A*e)) + 3*a*c*e^2*(6*C*d^2 - e*(3*B*d - A*e)))*x^2)/(2*e^7) - (c^2*(10*c*C*d^3 - 3*c*d*e*(2*B*d - A*e) + 3*a*e^2*( 3*C*d - B*e))*x^3)/(3*e^6) + (c^2*(6*c*C*d^2 + 3*a*C*e^2 - c*e*(3*B*d - A* e))*x^4)/(4*e^5) - (c^3*(3*C*d - B*e)*x^5)/(5*e^4) + (c^3*C*x^6)/(6*e^3) - ((c*d^2 + a*e^2)^3*(C*d^2 - B*d*e + A*e^2))/(2*e^9*(d + e*x)^2) + ((c*d^2 + a*e^2)^2*(8*c*C*d^3 - c*d*e*(7*B*d - 6*A*e) + a*e^2*(2*C*d - B*e)))/(e^ 9*(d + e*x)) + ((c*d^2 + a*e^2)*(a^2*C*e^4 + c^2*(28*C*d^4 - 3*d^2*e*(7*B* d - 5*A*e)) + a*c*e^2*(17*C*d^2 - 3*e*(3*B*d - A*e)))*Log[d + e*x])/e^9
3.1.38.3.1 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 = 0.48 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.50
| method | result | size |
| norman | \(\frac {\frac {\left (6 A \,a^{2} c d \,e^{6}+36 A a \,c^{2} d^{3} e^{4}+30 A \,c^{3} d^{5} e^{2}-B \,a^{3} e^{7}-18 B \,a^{2} c \,d^{2} e^{5}-60 B a \,c^{2} d^{4} e^{3}-42 B \,c^{3} d^{6} e +2 C \,a^{3} d \,e^{6}+36 C \,a^{2} c \,d^{3} e^{4}+90 C a \,c^{2} d^{5} e^{2}+56 C \,c^{3} d^{7}\right ) x}{e^{8}}-\frac {A \,a^{3} e^{8}-9 A \,a^{2} c \,d^{2} e^{6}-54 A a \,c^{2} d^{4} e^{4}-45 A \,c^{3} d^{6} e^{2}+B \,a^{3} d \,e^{7}+27 B \,a^{2} c \,d^{3} e^{5}+90 B a \,c^{2} d^{5} e^{3}+63 B \,c^{3} d^{7} e -3 C \,a^{3} d^{2} e^{6}-54 C \,a^{2} c \,d^{4} e^{4}-135 C a \,c^{2} d^{6} e^{2}-84 C \,c^{3} d^{8}}{2 e^{9}}+\frac {C \,c^{3} x^{8}}{6 e}+\frac {c \left (18 A a c \,e^{4}+15 A \,c^{2} d^{2} e^{2}-30 B a c d \,e^{3}-21 B \,c^{2} d^{3} e +18 a^{2} C \,e^{4}+45 C a c \,d^{2} e^{2}+28 C \,c^{2} d^{4}\right ) x^{4}}{12 e^{5}}-\frac {c \left (18 A a c d \,e^{4}+15 A \,c^{2} d^{3} e^{2}-9 B \,e^{5} a^{2}-30 B a c \,d^{2} e^{3}-21 B \,c^{2} d^{4} e +18 C \,a^{2} d \,e^{4}+45 C a c \,d^{3} e^{2}+28 C \,c^{2} d^{5}\right ) x^{3}}{3 e^{6}}+\frac {c^{2} \left (15 A c \,e^{2}-21 B c d e +45 a C \,e^{2}+28 C c \,d^{2}\right ) x^{6}}{60 e^{3}}-\frac {c^{2} \left (15 A c d \,e^{2}-30 B \,e^{3} a -21 B c \,d^{2} e +45 C a d \,e^{2}+28 C c \,d^{3}\right ) x^{5}}{30 e^{4}}+\frac {c^{3} \left (3 B e -4 C d \right ) x^{7}}{15 e^{2}}}{\left (e x +d \right )^{2}}+\frac {\left (3 A \,a^{2} c \,e^{6}+18 A a \,c^{2} d^{2} e^{4}+15 A \,c^{3} d^{4} e^{2}-9 B \,a^{2} c d \,e^{5}-30 B a \,c^{2} d^{3} e^{3}-21 B \,c^{3} d^{5} e +a^{3} C \,e^{6}+18 C \,a^{2} c \,d^{2} e^{4}+45 C a \,c^{2} d^{4} e^{2}+28 C \,c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(700\) |
| default | \(-\frac {c \left (30 C a c \,d^{3} e^{2} x -18 B x a c \,d^{2} e^{3}+9 A x a c d \,e^{4}+\frac {9}{2} B \,x^{2} a c d \,e^{4}-\frac {3}{2} C \,a^{2} e^{5} x^{2}-\frac {1}{6} c^{2} C \,x^{6} e^{5}-3 B x \,a^{2} e^{5}-\frac {1}{4} A \,x^{4} c^{2} e^{5}-\frac {1}{5} B \,x^{5} c^{2} e^{5}+21 C \,c^{2} d^{5} x -9 C a c \,d^{2} e^{3} x^{2}+3 C a c d \,e^{4} x^{3}+\frac {3}{5} C \,c^{2} d \,e^{4} x^{5}-B \,x^{3} a c \,e^{5}-\frac {3}{2} A \,x^{2} a c \,e^{5}-\frac {3}{2} C \,c^{2} d^{2} e^{3} x^{4}+9 C \,a^{2} d \,e^{4} x -\frac {3}{4} C a c \,e^{5} x^{4}+A \,x^{3} c^{2} d \,e^{4}-2 B \,x^{3} c^{2} d^{2} e^{3}-3 A \,x^{2} c^{2} d^{2} e^{3}+5 B \,x^{2} c^{2} d^{3} e^{2}+10 A x \,c^{2} d^{3} e^{2}-15 B x \,c^{2} d^{4} e +\frac {3}{4} B \,x^{4} c^{2} d \,e^{4}+\frac {10}{3} C \,c^{2} d^{3} e^{2} x^{3}-\frac {15}{2} C \,c^{2} d^{4} e \,x^{2}\right )}{e^{8}}-\frac {-6 A \,a^{2} c d \,e^{6}-12 A a \,c^{2} d^{3} e^{4}-6 A \,c^{3} d^{5} e^{2}+B \,a^{3} e^{7}+9 B \,a^{2} c \,d^{2} e^{5}+15 B a \,c^{2} d^{4} e^{3}+7 B \,c^{3} d^{6} e -2 C \,a^{3} d \,e^{6}-12 C \,a^{2} c \,d^{3} e^{4}-18 C a \,c^{2} d^{5} e^{2}-8 C \,c^{3} d^{7}}{e^{9} \left (e x +d \right )}+\frac {\left (3 A \,a^{2} c \,e^{6}+18 A a \,c^{2} d^{2} e^{4}+15 A \,c^{3} d^{4} e^{2}-9 B \,a^{2} c d \,e^{5}-30 B a \,c^{2} d^{3} e^{3}-21 B \,c^{3} d^{5} e +a^{3} C \,e^{6}+18 C \,a^{2} c \,d^{2} e^{4}+45 C a \,c^{2} d^{4} e^{2}+28 C \,c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}-\frac {A \,a^{3} e^{8}+3 A \,a^{2} c \,d^{2} e^{6}+3 A a \,c^{2} d^{4} e^{4}+A \,c^{3} d^{6} e^{2}-B \,a^{3} d \,e^{7}-3 B \,a^{2} c \,d^{3} e^{5}-3 B a \,c^{2} d^{5} e^{3}-B \,c^{3} d^{7} e +C \,a^{3} d^{2} e^{6}+3 C \,a^{2} c \,d^{4} e^{4}+3 C a \,c^{2} d^{6} e^{2}+C \,c^{3} d^{8}}{2 e^{9} \left (e x +d \right )^{2}}\) | \(753\) |
| risch | \(\frac {3 c C \,a^{2} x^{2}}{2 e^{3}}+\frac {3 c B x \,a^{2}}{e^{3}}-\frac {10 c^{3} A x \,d^{3}}{e^{6}}+\frac {15 c^{3} B x \,d^{4}}{e^{7}}-\frac {3 c^{3} B \,x^{4} d}{4 e^{4}}-\frac {10 c^{3} C \,d^{3} x^{3}}{3 e^{6}}+\frac {15 c^{3} C \,d^{4} x^{2}}{2 e^{7}}-\frac {3 c^{2} C a d \,x^{3}}{e^{4}}-\frac {9 c C \,a^{2} d x}{e^{4}}+\frac {18 \ln \left (e x +d \right ) A a \,c^{2} d^{2}}{e^{5}}-\frac {9 \ln \left (e x +d \right ) B \,a^{2} c d}{e^{4}}-\frac {30 \ln \left (e x +d \right ) B a \,c^{2} d^{3}}{e^{6}}+\frac {18 \ln \left (e x +d \right ) C \,a^{2} c \,d^{2}}{e^{5}}+\frac {45 \ln \left (e x +d \right ) C a \,c^{2} d^{4}}{e^{7}}+\frac {\ln \left (e x +d \right ) a^{3} C}{e^{3}}+\frac {c^{3} A \,x^{4}}{4 e^{3}}+\frac {c^{3} B \,x^{5}}{5 e^{3}}+\frac {3 \ln \left (e x +d \right ) A \,a^{2} c}{e^{3}}+\frac {15 \ln \left (e x +d \right ) A \,c^{3} d^{4}}{e^{7}}-\frac {21 \ln \left (e x +d \right ) B \,c^{3} d^{5}}{e^{8}}+\frac {28 \ln \left (e x +d \right ) C \,c^{3} d^{6}}{e^{9}}-\frac {c^{3} A \,x^{3} d}{e^{4}}+\frac {2 c^{3} B \,x^{3} d^{2}}{e^{5}}+\frac {3 c^{3} A \,x^{2} d^{2}}{e^{5}}-\frac {5 c^{3} B \,x^{2} d^{3}}{e^{6}}+\frac {9 c^{2} C a \,d^{2} x^{2}}{e^{5}}+\frac {\left (6 A \,a^{2} c d \,e^{6}+12 A a \,c^{2} d^{3} e^{4}+6 A \,c^{3} d^{5} e^{2}-B \,a^{3} e^{7}-9 B \,a^{2} c \,d^{2} e^{5}-15 B a \,c^{2} d^{4} e^{3}-7 B \,c^{3} d^{6} e +2 C \,a^{3} d \,e^{6}+12 C \,a^{2} c \,d^{3} e^{4}+18 C a \,c^{2} d^{5} e^{2}+8 C \,c^{3} d^{7}\right ) x -\frac {A \,a^{3} e^{8}-9 A \,a^{2} c \,d^{2} e^{6}-21 A a \,c^{2} d^{4} e^{4}-11 A \,c^{3} d^{6} e^{2}+B \,a^{3} d \,e^{7}+15 B \,a^{2} c \,d^{3} e^{5}+27 B a \,c^{2} d^{5} e^{3}+13 B \,c^{3} d^{7} e -3 C \,a^{3} d^{2} e^{6}-21 C \,a^{2} c \,d^{4} e^{4}-33 C a \,c^{2} d^{6} e^{2}-15 C \,c^{3} d^{8}}{2 e}}{e^{8} \left (e x +d \right )^{2}}-\frac {21 c^{3} C \,d^{5} x}{e^{8}}-\frac {3 c^{3} C d \,x^{5}}{5 e^{4}}+\frac {c^{2} B \,x^{3} a}{e^{3}}+\frac {3 c^{2} A \,x^{2} a}{2 e^{3}}+\frac {3 c^{3} C \,d^{2} x^{4}}{2 e^{5}}+\frac {3 c^{2} C a \,x^{4}}{4 e^{3}}-\frac {30 c^{2} C a \,d^{3} x}{e^{6}}+\frac {18 c^{2} B x a \,d^{2}}{e^{5}}-\frac {9 c^{2} A x a d}{e^{4}}-\frac {9 c^{2} B \,x^{2} a d}{2 e^{4}}+\frac {c^{3} C \,x^{6}}{6 e^{3}}\) | \(826\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1252\) |
((6*A*a^2*c*d*e^6+36*A*a*c^2*d^3*e^4+30*A*c^3*d^5*e^2-B*a^3*e^7-18*B*a^2*c *d^2*e^5-60*B*a*c^2*d^4*e^3-42*B*c^3*d^6*e+2*C*a^3*d*e^6+36*C*a^2*c*d^3*e^ 4+90*C*a*c^2*d^5*e^2+56*C*c^3*d^7)/e^8*x-1/2*(A*a^3*e^8-9*A*a^2*c*d^2*e^6- 54*A*a*c^2*d^4*e^4-45*A*c^3*d^6*e^2+B*a^3*d*e^7+27*B*a^2*c*d^3*e^5+90*B*a* c^2*d^5*e^3+63*B*c^3*d^7*e-3*C*a^3*d^2*e^6-54*C*a^2*c*d^4*e^4-135*C*a*c^2* d^6*e^2-84*C*c^3*d^8)/e^9+1/6*C*c^3/e*x^8+1/12*c*(18*A*a*c*e^4+15*A*c^2*d^ 2*e^2-30*B*a*c*d*e^3-21*B*c^2*d^3*e+18*C*a^2*e^4+45*C*a*c*d^2*e^2+28*C*c^2 *d^4)/e^5*x^4-1/3*c*(18*A*a*c*d*e^4+15*A*c^2*d^3*e^2-9*B*a^2*e^5-30*B*a*c* d^2*e^3-21*B*c^2*d^4*e+18*C*a^2*d*e^4+45*C*a*c*d^3*e^2+28*C*c^2*d^5)/e^6*x ^3+1/60*c^2*(15*A*c*e^2-21*B*c*d*e+45*C*a*e^2+28*C*c*d^2)/e^3*x^6-1/30*c^2 *(15*A*c*d*e^2-30*B*a*e^3-21*B*c*d^2*e+45*C*a*d*e^2+28*C*c*d^3)/e^4*x^5+1/ 15*c^3*(3*B*e-4*C*d)/e^2*x^7)/(e*x+d)^2+1/e^9*(3*A*a^2*c*e^6+18*A*a*c^2*d^ 2*e^4+15*A*c^3*d^4*e^2-9*B*a^2*c*d*e^5-30*B*a*c^2*d^3*e^3-21*B*c^3*d^5*e+C *a^3*e^6+18*C*a^2*c*d^2*e^4+45*C*a*c^2*d^4*e^2+28*C*c^3*d^6)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (454) = 908\).
Time = 0.30 (sec) , antiderivative size = 1025, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {10 \, C c^{3} e^{8} x^{8} + 450 \, C c^{3} d^{8} - 390 \, B c^{3} d^{7} e - 810 \, B a c^{2} d^{5} e^{3} - 450 \, B a^{2} c d^{3} e^{5} - 30 \, B a^{3} d e^{7} - 30 \, A a^{3} e^{8} + 330 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 630 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + 90 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} - 4 \, {\left (4 \, C c^{3} d e^{7} - 3 \, B c^{3} e^{8}\right )} x^{7} + {\left (28 \, C c^{3} d^{2} e^{6} - 21 \, B c^{3} d e^{7} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} e^{8}\right )} x^{6} - 2 \, {\left (28 \, C c^{3} d^{3} e^{5} - 21 \, B c^{3} d^{2} e^{6} - 30 \, B a c^{2} e^{8} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{7}\right )} x^{5} + 5 \, {\left (28 \, C c^{3} d^{4} e^{4} - 21 \, B c^{3} d^{3} e^{5} - 30 \, B a c^{2} d e^{7} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{6} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} e^{8}\right )} x^{4} - 20 \, {\left (28 \, C c^{3} d^{5} e^{3} - 21 \, B c^{3} d^{4} e^{4} - 30 \, B a c^{2} d^{2} e^{6} - 9 \, B a^{2} c e^{8} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{5} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{7}\right )} x^{3} - 30 \, {\left (69 \, C c^{3} d^{6} e^{2} - 50 \, B c^{3} d^{5} e^{3} - 63 \, B a c^{2} d^{3} e^{5} - 12 \, B a^{2} c d e^{7} + 34 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{4} + 33 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{6}\right )} x^{2} - 60 \, {\left (13 \, C c^{3} d^{7} e - 8 \, B c^{3} d^{6} e^{2} - 3 \, B a c^{2} d^{4} e^{4} + 6 \, B a^{2} c d^{2} e^{6} + B a^{3} e^{8} + 4 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} - 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} - 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x + 60 \, {\left (28 \, C c^{3} d^{8} - 21 \, B c^{3} d^{7} e - 30 \, B a c^{2} d^{5} e^{3} - 9 \, B a^{2} c d^{3} e^{5} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} + {\left (28 \, C c^{3} d^{6} e^{2} - 21 \, B c^{3} d^{5} e^{3} - 30 \, B a c^{2} d^{3} e^{5} - 9 \, B a^{2} c d e^{7} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{4} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{6} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{8}\right )} x^{2} + 2 \, {\left (28 \, C c^{3} d^{7} e - 21 \, B c^{3} d^{6} e^{2} - 30 \, B a c^{2} d^{4} e^{4} - 9 \, B a^{2} c d^{2} e^{6} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} \]
1/60*(10*C*c^3*e^8*x^8 + 450*C*c^3*d^8 - 390*B*c^3*d^7*e - 810*B*a*c^2*d^5 *e^3 - 450*B*a^2*c*d^3*e^5 - 30*B*a^3*d*e^7 - 30*A*a^3*e^8 + 330*(3*C*a*c^ 2 + A*c^3)*d^6*e^2 + 630*(C*a^2*c + A*a*c^2)*d^4*e^4 + 90*(C*a^3 + 3*A*a^2 *c)*d^2*e^6 - 4*(4*C*c^3*d*e^7 - 3*B*c^3*e^8)*x^7 + (28*C*c^3*d^2*e^6 - 21 *B*c^3*d*e^7 + 15*(3*C*a*c^2 + A*c^3)*e^8)*x^6 - 2*(28*C*c^3*d^3*e^5 - 21* B*c^3*d^2*e^6 - 30*B*a*c^2*e^8 + 15*(3*C*a*c^2 + A*c^3)*d*e^7)*x^5 + 5*(28 *C*c^3*d^4*e^4 - 21*B*c^3*d^3*e^5 - 30*B*a*c^2*d*e^7 + 15*(3*C*a*c^2 + A*c ^3)*d^2*e^6 + 18*(C*a^2*c + A*a*c^2)*e^8)*x^4 - 20*(28*C*c^3*d^5*e^3 - 21* B*c^3*d^4*e^4 - 30*B*a*c^2*d^2*e^6 - 9*B*a^2*c*e^8 + 15*(3*C*a*c^2 + A*c^3 )*d^3*e^5 + 18*(C*a^2*c + A*a*c^2)*d*e^7)*x^3 - 30*(69*C*c^3*d^6*e^2 - 50* B*c^3*d^5*e^3 - 63*B*a*c^2*d^3*e^5 - 12*B*a^2*c*d*e^7 + 34*(3*C*a*c^2 + A* c^3)*d^4*e^4 + 33*(C*a^2*c + A*a*c^2)*d^2*e^6)*x^2 - 60*(13*C*c^3*d^7*e - 8*B*c^3*d^6*e^2 - 3*B*a*c^2*d^4*e^4 + 6*B*a^2*c*d^2*e^6 + B*a^3*e^8 + 4*(3 *C*a*c^2 + A*c^3)*d^5*e^3 - 3*(C*a^2*c + A*a*c^2)*d^3*e^5 - 2*(C*a^3 + 3*A *a^2*c)*d*e^7)*x + 60*(28*C*c^3*d^8 - 21*B*c^3*d^7*e - 30*B*a*c^2*d^5*e^3 - 9*B*a^2*c*d^3*e^5 + 15*(3*C*a*c^2 + A*c^3)*d^6*e^2 + 18*(C*a^2*c + A*a*c ^2)*d^4*e^4 + (C*a^3 + 3*A*a^2*c)*d^2*e^6 + (28*C*c^3*d^6*e^2 - 21*B*c^3*d ^5*e^3 - 30*B*a*c^2*d^3*e^5 - 9*B*a^2*c*d*e^7 + 15*(3*C*a*c^2 + A*c^3)*d^4 *e^4 + 18*(C*a^2*c + A*a*c^2)*d^2*e^6 + (C*a^3 + 3*A*a^2*c)*e^8)*x^2 + 2*( 28*C*c^3*d^7*e - 21*B*c^3*d^6*e^2 - 30*B*a*c^2*d^4*e^4 - 9*B*a^2*c*d^2*...
Time = 9.23 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {C c^{3} x^{6}}{6 e^{3}} + x^{5} \left (\frac {B c^{3}}{5 e^{3}} - \frac {3 C c^{3} d}{5 e^{4}}\right ) + x^{4} \left (\frac {A c^{3}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}} + \frac {3 C a c^{2}}{4 e^{3}} + \frac {3 C c^{3} d^{2}}{2 e^{5}}\right ) + x^{3} \left (- \frac {A c^{3} d}{e^{4}} + \frac {B a c^{2}}{e^{3}} + \frac {2 B c^{3} d^{2}}{e^{5}} - \frac {3 C a c^{2} d}{e^{4}} - \frac {10 C c^{3} d^{3}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {3 A a c^{2}}{2 e^{3}} + \frac {3 A c^{3} d^{2}}{e^{5}} - \frac {9 B a c^{2} d}{2 e^{4}} - \frac {5 B c^{3} d^{3}}{e^{6}} + \frac {3 C a^{2} c}{2 e^{3}} + \frac {9 C a c^{2} d^{2}}{e^{5}} + \frac {15 C c^{3} d^{4}}{2 e^{7}}\right ) + x \left (- \frac {9 A a c^{2} d}{e^{4}} - \frac {10 A c^{3} d^{3}}{e^{6}} + \frac {3 B a^{2} c}{e^{3}} + \frac {18 B a c^{2} d^{2}}{e^{5}} + \frac {15 B c^{3} d^{4}}{e^{7}} - \frac {9 C a^{2} c d}{e^{4}} - \frac {30 C a c^{2} d^{3}}{e^{6}} - \frac {21 C c^{3} d^{5}}{e^{8}}\right ) + \frac {- A a^{3} e^{8} + 9 A a^{2} c d^{2} e^{6} + 21 A a c^{2} d^{4} e^{4} + 11 A c^{3} d^{6} e^{2} - B a^{3} d e^{7} - 15 B a^{2} c d^{3} e^{5} - 27 B a c^{2} d^{5} e^{3} - 13 B c^{3} d^{7} e + 3 C a^{3} d^{2} e^{6} + 21 C a^{2} c d^{4} e^{4} + 33 C a c^{2} d^{6} e^{2} + 15 C c^{3} d^{8} + x \left (12 A a^{2} c d e^{7} + 24 A a c^{2} d^{3} e^{5} + 12 A c^{3} d^{5} e^{3} - 2 B a^{3} e^{8} - 18 B a^{2} c d^{2} e^{6} - 30 B a c^{2} d^{4} e^{4} - 14 B c^{3} d^{6} e^{2} + 4 C a^{3} d e^{7} + 24 C a^{2} c d^{3} e^{5} + 36 C a c^{2} d^{5} e^{3} + 16 C c^{3} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac {\left (a e^{2} + c d^{2}\right ) \left (3 A a c e^{4} + 15 A c^{2} d^{2} e^{2} - 9 B a c d e^{3} - 21 B c^{2} d^{3} e + C a^{2} e^{4} + 17 C a c d^{2} e^{2} + 28 C c^{2} d^{4}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
C*c**3*x**6/(6*e**3) + x**5*(B*c**3/(5*e**3) - 3*C*c**3*d/(5*e**4)) + x**4 *(A*c**3/(4*e**3) - 3*B*c**3*d/(4*e**4) + 3*C*a*c**2/(4*e**3) + 3*C*c**3*d **2/(2*e**5)) + x**3*(-A*c**3*d/e**4 + B*a*c**2/e**3 + 2*B*c**3*d**2/e**5 - 3*C*a*c**2*d/e**4 - 10*C*c**3*d**3/(3*e**6)) + x**2*(3*A*a*c**2/(2*e**3) + 3*A*c**3*d**2/e**5 - 9*B*a*c**2*d/(2*e**4) - 5*B*c**3*d**3/e**6 + 3*C*a **2*c/(2*e**3) + 9*C*a*c**2*d**2/e**5 + 15*C*c**3*d**4/(2*e**7)) + x*(-9*A *a*c**2*d/e**4 - 10*A*c**3*d**3/e**6 + 3*B*a**2*c/e**3 + 18*B*a*c**2*d**2/ e**5 + 15*B*c**3*d**4/e**7 - 9*C*a**2*c*d/e**4 - 30*C*a*c**2*d**3/e**6 - 2 1*C*c**3*d**5/e**8) + (-A*a**3*e**8 + 9*A*a**2*c*d**2*e**6 + 21*A*a*c**2*d **4*e**4 + 11*A*c**3*d**6*e**2 - B*a**3*d*e**7 - 15*B*a**2*c*d**3*e**5 - 2 7*B*a*c**2*d**5*e**3 - 13*B*c**3*d**7*e + 3*C*a**3*d**2*e**6 + 21*C*a**2*c *d**4*e**4 + 33*C*a*c**2*d**6*e**2 + 15*C*c**3*d**8 + x*(12*A*a**2*c*d*e** 7 + 24*A*a*c**2*d**3*e**5 + 12*A*c**3*d**5*e**3 - 2*B*a**3*e**8 - 18*B*a** 2*c*d**2*e**6 - 30*B*a*c**2*d**4*e**4 - 14*B*c**3*d**6*e**2 + 4*C*a**3*d*e **7 + 24*C*a**2*c*d**3*e**5 + 36*C*a*c**2*d**5*e**3 + 16*C*c**3*d**7*e))/( 2*d**2*e**9 + 4*d*e**10*x + 2*e**11*x**2) + (a*e**2 + c*d**2)*(3*A*a*c*e** 4 + 15*A*c**2*d**2*e**2 - 9*B*a*c*d*e**3 - 21*B*c**2*d**3*e + C*a**2*e**4 + 17*C*a*c*d**2*e**2 + 28*C*c**2*d**4)*log(d + e*x)/e**9
Time = 0.21 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e - 27 \, B a c^{2} d^{5} e^{3} - 15 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} - A a^{3} e^{8} + 11 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 21 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + 3 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6} + 2 \, {\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} - 15 \, B a c^{2} d^{4} e^{4} - 9 \, B a^{2} c d^{2} e^{6} - B a^{3} e^{8} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 12 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x}{2 \, {\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac {10 \, C c^{3} e^{5} x^{6} - 12 \, {\left (3 \, C c^{3} d e^{4} - B c^{3} e^{5}\right )} x^{5} + 15 \, {\left (6 \, C c^{3} d^{2} e^{3} - 3 \, B c^{3} d e^{4} + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{5}\right )} x^{4} - 20 \, {\left (10 \, C c^{3} d^{3} e^{2} - 6 \, B c^{3} d^{2} e^{3} - 3 \, B a c^{2} e^{5} + 3 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{4}\right )} x^{3} + 30 \, {\left (15 \, C c^{3} d^{4} e - 10 \, B c^{3} d^{3} e^{2} - 9 \, B a c^{2} d e^{4} + 6 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{3} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{5}\right )} x^{2} - 60 \, {\left (21 \, C c^{3} d^{5} - 15 \, B c^{3} d^{4} e - 18 \, B a c^{2} d^{2} e^{3} - 3 \, B a^{2} c e^{5} + 10 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{2} + 9 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{4}\right )} x}{60 \, e^{8}} + \frac {{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e - 30 \, B a c^{2} d^{3} e^{3} - 9 \, B a^{2} c d e^{5} + 15 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{2} + 18 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \]
1/2*(15*C*c^3*d^8 - 13*B*c^3*d^7*e - 27*B*a*c^2*d^5*e^3 - 15*B*a^2*c*d^3*e ^5 - B*a^3*d*e^7 - A*a^3*e^8 + 11*(3*C*a*c^2 + A*c^3)*d^6*e^2 + 21*(C*a^2* c + A*a*c^2)*d^4*e^4 + 3*(C*a^3 + 3*A*a^2*c)*d^2*e^6 + 2*(8*C*c^3*d^7*e - 7*B*c^3*d^6*e^2 - 15*B*a*c^2*d^4*e^4 - 9*B*a^2*c*d^2*e^6 - B*a^3*e^8 + 6*( 3*C*a*c^2 + A*c^3)*d^5*e^3 + 12*(C*a^2*c + A*a*c^2)*d^3*e^5 + 2*(C*a^3 + 3 *A*a^2*c)*d*e^7)*x)/(e^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/60*(10*C*c^3*e^5 *x^6 - 12*(3*C*c^3*d*e^4 - B*c^3*e^5)*x^5 + 15*(6*C*c^3*d^2*e^3 - 3*B*c^3* d*e^4 + (3*C*a*c^2 + A*c^3)*e^5)*x^4 - 20*(10*C*c^3*d^3*e^2 - 6*B*c^3*d^2* e^3 - 3*B*a*c^2*e^5 + 3*(3*C*a*c^2 + A*c^3)*d*e^4)*x^3 + 30*(15*C*c^3*d^4* e - 10*B*c^3*d^3*e^2 - 9*B*a*c^2*d*e^4 + 6*(3*C*a*c^2 + A*c^3)*d^2*e^3 + 3 *(C*a^2*c + A*a*c^2)*e^5)*x^2 - 60*(21*C*c^3*d^5 - 15*B*c^3*d^4*e - 18*B*a *c^2*d^2*e^3 - 3*B*a^2*c*e^5 + 10*(3*C*a*c^2 + A*c^3)*d^3*e^2 + 9*(C*a^2*c + A*a*c^2)*d*e^4)*x)/e^8 + (28*C*c^3*d^6 - 21*B*c^3*d^5*e - 30*B*a*c^2*d^ 3*e^3 - 9*B*a^2*c*d*e^5 + 15*(3*C*a*c^2 + A*c^3)*d^4*e^2 + 18*(C*a^2*c + A *a*c^2)*d^2*e^4 + (C*a^3 + 3*A*a^2*c)*e^6)*log(e*x + d)/e^9
Time = 0.29 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\frac {{\left (28 \, C c^{3} d^{6} - 21 \, B c^{3} d^{5} e + 45 \, C a c^{2} d^{4} e^{2} + 15 \, A c^{3} d^{4} e^{2} - 30 \, B a c^{2} d^{3} e^{3} + 18 \, C a^{2} c d^{2} e^{4} + 18 \, A a c^{2} d^{2} e^{4} - 9 \, B a^{2} c d e^{5} + C a^{3} e^{6} + 3 \, A a^{2} c e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} + \frac {15 \, C c^{3} d^{8} - 13 \, B c^{3} d^{7} e + 33 \, C a c^{2} d^{6} e^{2} + 11 \, A c^{3} d^{6} e^{2} - 27 \, B a c^{2} d^{5} e^{3} + 21 \, C a^{2} c d^{4} e^{4} + 21 \, A a c^{2} d^{4} e^{4} - 15 \, B a^{2} c d^{3} e^{5} + 3 \, C a^{3} d^{2} e^{6} + 9 \, A a^{2} c d^{2} e^{6} - B a^{3} d e^{7} - A a^{3} e^{8} + 2 \, {\left (8 \, C c^{3} d^{7} e - 7 \, B c^{3} d^{6} e^{2} + 18 \, C a c^{2} d^{5} e^{3} + 6 \, A c^{3} d^{5} e^{3} - 15 \, B a c^{2} d^{4} e^{4} + 12 \, C a^{2} c d^{3} e^{5} + 12 \, A a c^{2} d^{3} e^{5} - 9 \, B a^{2} c d^{2} e^{6} + 2 \, C a^{3} d e^{7} + 6 \, A a^{2} c d e^{7} - B a^{3} e^{8}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{9}} + \frac {10 \, C c^{3} e^{15} x^{6} - 36 \, C c^{3} d e^{14} x^{5} + 12 \, B c^{3} e^{15} x^{5} + 90 \, C c^{3} d^{2} e^{13} x^{4} - 45 \, B c^{3} d e^{14} x^{4} + 45 \, C a c^{2} e^{15} x^{4} + 15 \, A c^{3} e^{15} x^{4} - 200 \, C c^{3} d^{3} e^{12} x^{3} + 120 \, B c^{3} d^{2} e^{13} x^{3} - 180 \, C a c^{2} d e^{14} x^{3} - 60 \, A c^{3} d e^{14} x^{3} + 60 \, B a c^{2} e^{15} x^{3} + 450 \, C c^{3} d^{4} e^{11} x^{2} - 300 \, B c^{3} d^{3} e^{12} x^{2} + 540 \, C a c^{2} d^{2} e^{13} x^{2} + 180 \, A c^{3} d^{2} e^{13} x^{2} - 270 \, B a c^{2} d e^{14} x^{2} + 90 \, C a^{2} c e^{15} x^{2} + 90 \, A a c^{2} e^{15} x^{2} - 1260 \, C c^{3} d^{5} e^{10} x + 900 \, B c^{3} d^{4} e^{11} x - 1800 \, C a c^{2} d^{3} e^{12} x - 600 \, A c^{3} d^{3} e^{12} x + 1080 \, B a c^{2} d^{2} e^{13} x - 540 \, C a^{2} c d e^{14} x - 540 \, A a c^{2} d e^{14} x + 180 \, B a^{2} c e^{15} x}{60 \, e^{18}} \]
(28*C*c^3*d^6 - 21*B*c^3*d^5*e + 45*C*a*c^2*d^4*e^2 + 15*A*c^3*d^4*e^2 - 3 0*B*a*c^2*d^3*e^3 + 18*C*a^2*c*d^2*e^4 + 18*A*a*c^2*d^2*e^4 - 9*B*a^2*c*d* e^5 + C*a^3*e^6 + 3*A*a^2*c*e^6)*log(abs(e*x + d))/e^9 + 1/2*(15*C*c^3*d^8 - 13*B*c^3*d^7*e + 33*C*a*c^2*d^6*e^2 + 11*A*c^3*d^6*e^2 - 27*B*a*c^2*d^5 *e^3 + 21*C*a^2*c*d^4*e^4 + 21*A*a*c^2*d^4*e^4 - 15*B*a^2*c*d^3*e^5 + 3*C* a^3*d^2*e^6 + 9*A*a^2*c*d^2*e^6 - B*a^3*d*e^7 - A*a^3*e^8 + 2*(8*C*c^3*d^7 *e - 7*B*c^3*d^6*e^2 + 18*C*a*c^2*d^5*e^3 + 6*A*c^3*d^5*e^3 - 15*B*a*c^2*d ^4*e^4 + 12*C*a^2*c*d^3*e^5 + 12*A*a*c^2*d^3*e^5 - 9*B*a^2*c*d^2*e^6 + 2*C *a^3*d*e^7 + 6*A*a^2*c*d*e^7 - B*a^3*e^8)*x)/((e*x + d)^2*e^9) + 1/60*(10* C*c^3*e^15*x^6 - 36*C*c^3*d*e^14*x^5 + 12*B*c^3*e^15*x^5 + 90*C*c^3*d^2*e^ 13*x^4 - 45*B*c^3*d*e^14*x^4 + 45*C*a*c^2*e^15*x^4 + 15*A*c^3*e^15*x^4 - 2 00*C*c^3*d^3*e^12*x^3 + 120*B*c^3*d^2*e^13*x^3 - 180*C*a*c^2*d*e^14*x^3 - 60*A*c^3*d*e^14*x^3 + 60*B*a*c^2*e^15*x^3 + 450*C*c^3*d^4*e^11*x^2 - 300*B *c^3*d^3*e^12*x^2 + 540*C*a*c^2*d^2*e^13*x^2 + 180*A*c^3*d^2*e^13*x^2 - 27 0*B*a*c^2*d*e^14*x^2 + 90*C*a^2*c*e^15*x^2 + 90*A*a*c^2*e^15*x^2 - 1260*C* c^3*d^5*e^10*x + 900*B*c^3*d^4*e^11*x - 1800*C*a*c^2*d^3*e^12*x - 600*A*c^ 3*d^3*e^12*x + 1080*B*a*c^2*d^2*e^13*x - 540*C*a^2*c*d*e^14*x - 540*A*a*c^ 2*d*e^14*x + 180*B*a^2*c*e^15*x)/e^18
Time = 12.38 (sec) , antiderivative size = 1290, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]
x^3*((d*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/e - (d^2*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e^2 + (B* a*c^2)/e^3 - (C*c^3*d^3)/(3*e^6)) + x*((3*d*((3*d*((3*d*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/e - (3*d^2*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e^2 + (3*B*a*c^2)/e^3 - (C*c^3*d^ 3)/e^6))/e - (3*d^2*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e - (A*c^3 + 3* C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/e^2 + (d^3*((B*c^3)/e^3 - (3*C*c^3*d)/e ^4))/e^3 - (3*a*c*(A*c + C*a))/e^3))/e + (d^3*((3*d*((B*c^3)/e^3 - (3*C*c^ 3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/e^3 - (3*d^2* ((3*d*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/e - (3*d^2*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e^2 + (3* B*a*c^2)/e^3 - (C*c^3*d^3)/e^6))/e^2 + (3*B*a^2*c)/e^3) + x^5*((B*c^3)/(5* e^3) - (3*C*c^3*d)/(5*e^4)) - x^4*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/( 4*e) - (A*c^3 + 3*C*a*c^2)/(4*e^3) + (3*C*c^3*d^2)/(4*e^5)) - x^2*((3*d*(( 3*d*((3*d*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + ( 3*C*c^3*d^2)/e^5))/e - (3*d^2*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/e^2 + (3*B* a*c^2)/e^3 - (C*c^3*d^3)/e^6))/(2*e) - (3*d^2*((3*d*((B*c^3)/e^3 - (3*C*c^ 3*d)/e^4))/e - (A*c^3 + 3*C*a*c^2)/e^3 + (3*C*c^3*d^2)/e^5))/(2*e^2) + (d^ 3*((B*c^3)/e^3 - (3*C*c^3*d)/e^4))/(2*e^3) - (3*a*c*(A*c + C*a))/(2*e^3)) + ((15*C*c^3*d^8 - A*a^3*e^8 - B*a^3*d*e^7 - 13*B*c^3*d^7*e + 11*A*c^3*...