Integrand size = 27, antiderivative size = 490 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=-\frac {\left (c d^2+a e^2\right )^2 \left (a e^2 (2 C d-B e)+c d \left (8 C d^2-e (7 B d-6 A e)\right )\right ) x}{e^8}+\frac {\left (c d^2+a e^2\right ) \left (a^2 C e^4+c^2 d^2 \left (28 C d^2-3 e (7 B d-5 A e)\right )+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )\right ) (d+e x)^2}{2 e^9}-\frac {c \left (3 a^2 e^4 (4 C d-B e)+c^2 d^3 \left (56 C d^2-5 e (7 B d-4 A e)\right )+6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )\right ) (d+e x)^3}{3 e^9}+\frac {c \left (3 a^2 C e^4+5 c^2 d^2 \left (14 C d^2-e (7 B d-3 A e)\right )+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )\right ) (d+e x)^4}{4 e^9}-\frac {c^2 \left (3 a e^2 (6 C d-B e)+c d \left (56 C d^2-3 e (7 B d-2 A e)\right )\right ) (d+e x)^5}{5 e^9}+\frac {c^2 \left (3 a C e^2+c \left (28 C d^2-e (7 B d-A e)\right )\right ) (d+e x)^6}{6 e^9}-\frac {c^3 (8 C d-B e) (d+e x)^7}{7 e^9}+\frac {c^3 C (d+e x)^8}{8 e^9}+\frac {\left (c d^2+a e^2\right )^3 \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^9} \]
-(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)))*x/e^8 +1/2*(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)))*(e*x+d)^2/e^9-1/3*c*(3*a^2*e^4*(-B*e+4*C*d) +c^2*d^3*(56*C*d^2-5*e*(-4*A*e+7*B*d))+6*a*c*d*e^2*(10*C*d^2-e*(-2*A*e+5*B *d)))*(e*x+d)^3/e^9+1/4*c*(3*a^2*C*e^4+5*c^2*d^2*(14*C*d^2-e*(-3*A*e+7*B*d ))+3*a*c*e^2*(15*C*d^2-e*(-A*e+5*B*d)))*(e*x+d)^4/e^9-1/5*c^2*(3*a*e^2*(-B *e+6*C*d)+c*d*(56*C*d^2-3*e*(-2*A*e+7*B*d)))*(e*x+d)^5/e^9+1/6*c^2*(3*a*C* e^2+c*(28*C*d^2-e*(-A*e+7*B*d)))*(e*x+d)^6/e^9-1/7*c^3*(-B*e+8*C*d)*(e*x+d )^7/e^9+1/8*c^3*C*(e*x+d)^8/e^9+(a*e^2+c*d^2)^3*(A*e^2-B*d*e+C*d^2)*ln(e*x +d)/e^9
Time = 0.30 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=\frac {x \left (420 a^3 e^6 (-2 C d+2 B e+C e x)+210 a^2 c e^4 \left (C \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 e \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )+42 a c^2 e^2 \left (C \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )+c^3 \left (C \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )+2 e \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )\right )}{840 e^8}+\frac {\left (c d^2+a e^2\right )^3 \left (C d^2+e (-B d+A e)\right ) \log (d+e x)}{e^9} \]
(x*(420*a^3*e^6*(-2*C*d + 2*B*e + C*e*x) + 210*a^2*c*e^4*(C*(-12*d^3 + 6*d ^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3 *d*e*x + 2*e^2*x^2))) + 42*a*c^2*e^2*(C*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2 *x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + e*(5*A*e*(-12*d^3 + 6 *d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2* x^2 - 15*d*e^3*x^3 + 12*e^4*x^4))) + c^3*(C*(-840*d^7 + 420*d^6*e*x - 280* d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d* e^6*x^6 + 105*e^7*x^7) + 2*e*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e ^6*x^6)))))/(840*e^8) + ((c*d^2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e))*Log[ d + e*x])/e^9
Time = 1.18 (sec) , antiderivative size = 487, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {c (d+e x)^3 \left (3 a^2 C e^4+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )+5 c^2 \left (14 C d^4-d^2 e (7 B d-3 A e)\right )\right )}{e^8}+\frac {(d+e x) \left (a e^2+c d^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{e^8}+\frac {c (d+e x)^2 \left (-3 a^2 e^4 (4 C d-B e)-6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )-c^2 \left (56 C d^5-5 d^3 e (7 B d-4 A e)\right )\right )}{e^8}+\frac {c^2 (d+e x)^4 \left (-3 a e^2 (6 C d-B e)+3 c d e (7 B d-2 A e)-56 c C d^3\right )}{e^8}+\frac {c^2 (d+e x)^5 \left (3 a C e^2-c e (7 B d-A e)+28 c C d^2\right )}{e^8}+\frac {\left (a e^2+c d^2\right )^3 \left (A e^2-B d e+C d^2\right )}{e^8 (d+e x)}+\frac {\left (a e^2+c d^2\right )^2 \left (-a e^2 (2 C d-B e)+c d e (7 B d-6 A e)-8 c C d^3\right )}{e^8}+\frac {c^3 (d+e x)^6 (B e-8 C d)}{e^8}+\frac {c^3 C (d+e x)^7}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c (d+e x)^4 \left (3 a^2 C e^4+3 a c e^2 \left (15 C d^2-e (5 B d-A e)\right )+5 c^2 \left (14 C d^4-d^2 e (7 B d-3 A e)\right )\right )}{4 e^9}+\frac {(d+e x)^2 \left (a e^2+c d^2\right ) \left (a^2 C e^4+a c e^2 \left (17 C d^2-3 e (3 B d-A e)\right )+c^2 \left (28 C d^4-3 d^2 e (7 B d-5 A e)\right )\right )}{2 e^9}-\frac {c (d+e x)^3 \left (3 a^2 e^4 (4 C d-B e)+6 a c d e^2 \left (10 C d^2-e (5 B d-2 A e)\right )+c^2 \left (56 C d^5-5 d^3 e (7 B d-4 A e)\right )\right )}{3 e^9}-\frac {c^2 (d+e x)^5 \left (3 a e^2 (6 C d-B e)-3 c d e (7 B d-2 A e)+56 c C d^3\right )}{5 e^9}+\frac {c^2 (d+e x)^6 \left (3 a C e^2-c e (7 B d-A e)+28 c C d^2\right )}{6 e^9}+\frac {\left (a e^2+c d^2\right )^3 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^9}-\frac {x \left (a e^2+c d^2\right )^2 \left (a e^2 (2 C d-B e)-c d e (7 B d-6 A e)+8 c C d^3\right )}{e^8}-\frac {c^3 (d+e x)^7 (8 C d-B e)}{7 e^9}+\frac {c^3 C (d+e x)^8}{8 e^9}\) |
-(((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))*x)/e^8) + ((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)))*(d + e*x)^2)/(2*e^9) - (c*(3*a^2*e^4*(4*C*d - B*e) + c^2*(56*C*d^5 - 5*d^3*e*(7*B*d - 4*A*e)) + 6*a*c*d*e^2*(10*C*d^2 - e*(5*B*d - 2*A*e)))*(d + e*x)^3)/(3*e^9) + (c*(3 *a^2*C*e^4 + 5*c^2*(14*C*d^4 - d^2*e*(7*B*d - 3*A*e)) + 3*a*c*e^2*(15*C*d^ 2 - e*(5*B*d - A*e)))*(d + e*x)^4)/(4*e^9) - (c^2*(56*c*C*d^3 - 3*c*d*e*(7 *B*d - 2*A*e) + 3*a*e^2*(6*C*d - B*e))*(d + e*x)^5)/(5*e^9) + (c^2*(28*c*C *d^2 + 3*a*C*e^2 - c*e*(7*B*d - A*e))*(d + e*x)^6)/(6*e^9) - (c^3*(8*C*d - B*e)*(d + e*x)^7)/(7*e^9) + (c^3*C*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2 )^3*(C*d^2 - B*d*e + A*e^2)*Log[d + e*x])/e^9
3.1.36.3.1 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 = 0.58 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.39
| method | result | size |
| norman | \(\frac {\left (3 A \,a^{2} c \,e^{6}+3 A a \,c^{2} d^{2} e^{4}+A \,c^{3} d^{4} e^{2}-3 B \,a^{2} c d \,e^{5}-3 B a \,c^{2} d^{3} e^{3}-B \,c^{3} d^{5} e +a^{3} C \,e^{6}+3 C \,a^{2} c \,d^{2} e^{4}+3 C a \,c^{2} d^{4} e^{2}+C \,c^{3} d^{6}\right ) x^{2}}{2 e^{7}}-\frac {\left (3 A \,a^{2} c d \,e^{6}+3 A a \,c^{2} d^{3} e^{4}+A \,c^{3} d^{5} e^{2}-B \,a^{3} e^{7}-3 B \,a^{2} c \,d^{2} e^{5}-3 B a \,c^{2} d^{4} e^{3}-B \,c^{3} d^{6} e +C \,a^{3} d \,e^{6}+3 C \,a^{2} c \,d^{3} e^{4}+3 C a \,c^{2} d^{5} e^{2}+C \,c^{3} d^{7}\right ) x}{e^{8}}+\frac {C \,c^{3} x^{8}}{8 e}-\frac {c \left (3 A a c d \,e^{4}+A \,c^{2} d^{3} e^{2}-3 B \,e^{5} a^{2}-3 B a c \,d^{2} e^{3}-B \,c^{2} d^{4} e +3 C \,a^{2} d \,e^{4}+3 C a c \,d^{3} e^{2}+C \,c^{2} d^{5}\right ) x^{3}}{3 e^{6}}+\frac {c \left (3 A a c \,e^{4}+A \,c^{2} d^{2} e^{2}-3 B a c d \,e^{3}-B \,c^{2} d^{3} e +3 a^{2} C \,e^{4}+3 C a c \,d^{2} e^{2}+C \,c^{2} d^{4}\right ) x^{4}}{4 e^{5}}-\frac {c^{2} \left (A c d \,e^{2}-3 B \,e^{3} a -B c \,d^{2} e +3 C a d \,e^{2}+C c \,d^{3}\right ) x^{5}}{5 e^{4}}+\frac {c^{2} \left (A c \,e^{2}-B c d e +3 a C \,e^{2}+C c \,d^{2}\right ) x^{6}}{6 e^{3}}+\frac {c^{3} \left (B e -C d \right ) x^{7}}{7 e^{2}}+\frac {\left (A \,a^{3} e^{8}+3 A \,a^{2} c \,d^{2} e^{6}+3 A a \,c^{2} d^{4} e^{4}+A \,c^{3} d^{6} e^{2}-B \,a^{3} d \,e^{7}-3 B \,a^{2} c \,d^{3} e^{5}-3 B a \,c^{2} d^{5} e^{3}-B \,c^{3} d^{7} e +C \,a^{3} d^{2} e^{6}+3 C \,a^{2} c \,d^{4} e^{4}+3 C a \,c^{2} d^{6} e^{2}+C \,c^{3} d^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(679\) |
| default | \(-\frac {C \,a^{3} d \,e^{6} x +C \,a^{2} c d \,e^{6} x^{3}+C a \,c^{2} d^{3} e^{4} x^{3}-\frac {3}{2} C a \,c^{2} d^{4} e^{3} x^{2}-\frac {3}{2} A \,x^{2} a \,c^{2} d^{2} e^{5}+\frac {3}{2} B \,x^{2} a^{2} c d \,e^{6}+\frac {3}{2} B \,x^{2} a \,c^{2} d^{3} e^{4}+3 A x \,a^{2} c d \,e^{6}+3 A x a \,c^{2} d^{3} e^{4}+\frac {3}{4} B \,x^{4} a \,c^{2} d \,e^{6}+A \,x^{3} a \,c^{2} d \,e^{6}-B \,x^{3} a \,c^{2} d^{2} e^{5}-3 B x \,a^{2} c \,d^{2} e^{5}-3 B x a \,c^{2} d^{4} e^{3}-\frac {3}{2} C \,a^{2} c \,d^{2} e^{5} x^{2}-\frac {3}{4} C a \,c^{2} d^{2} e^{5} x^{4}+\frac {3}{5} C a \,c^{2} d \,e^{6} x^{5}-\frac {1}{2} C \,a^{3} e^{7} x^{2}-\frac {1}{8} c^{3} C \,x^{8} e^{7}-B x \,a^{3} e^{7}+C \,c^{3} d^{7} x -\frac {1}{7} B \,x^{7} c^{3} e^{7}-\frac {1}{6} A \,x^{6} c^{3} e^{7}+3 C a \,c^{2} d^{5} e^{2} x +3 C \,a^{2} c \,d^{3} e^{4} x -\frac {1}{2} C \,c^{3} d^{6} e \,x^{2}+\frac {1}{7} C \,c^{3} d \,e^{6} x^{7}-\frac {1}{2} C a \,c^{2} e^{7} x^{6}-\frac {1}{6} C \,c^{3} d^{2} e^{5} x^{6}+A x \,c^{3} d^{5} e^{2}-B x \,c^{3} d^{6} e +\frac {1}{6} B \,x^{6} c^{3} d \,e^{6}+\frac {1}{5} A \,x^{5} c^{3} d \,e^{6}-\frac {1}{5} B \,x^{5} c^{3} d^{2} e^{5}-\frac {1}{4} A \,x^{4} c^{3} d^{2} e^{5}+\frac {1}{4} B \,x^{4} c^{3} d^{3} e^{4}+\frac {1}{3} A \,x^{3} c^{3} d^{3} e^{4}-\frac {1}{3} B \,x^{3} c^{3} d^{4} e^{3}-\frac {1}{2} A \,x^{2} c^{3} d^{4} e^{3}+\frac {1}{2} B \,x^{2} c^{3} d^{5} e^{2}-B \,x^{3} a^{2} c \,e^{7}-\frac {3}{2} A \,x^{2} a^{2} c \,e^{7}-\frac {3}{5} B \,x^{5} a \,c^{2} e^{7}-\frac {3}{4} A \,x^{4} a \,c^{2} e^{7}+\frac {1}{5} C \,c^{3} d^{3} e^{4} x^{5}-\frac {3}{4} C \,a^{2} c \,e^{7} x^{4}-\frac {1}{4} C \,c^{3} d^{4} e^{3} x^{4}+\frac {1}{3} C \,c^{3} d^{5} e^{2} x^{3}}{e^{8}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(811\) |
| risch | \(-\frac {C \,c^{3} d \,x^{7}}{7 e^{2}}+\frac {C a \,c^{2} x^{6}}{2 e}-\frac {3 C a \,c^{2} d^{5} x}{e^{6}}-\frac {3 C \,a^{2} c \,d^{3} x}{e^{4}}-\frac {C \,a^{2} c d \,x^{3}}{e^{2}}+\frac {3 A \,x^{2} a \,c^{2} d^{2}}{2 e^{3}}-\frac {3 B \,x^{2} a^{2} c d}{2 e^{2}}-\frac {3 B \,x^{2} a \,c^{2} d^{3}}{2 e^{4}}-\frac {3 A x \,a^{2} c d}{e^{2}}-\frac {3 A x a \,c^{2} d^{3}}{e^{4}}-\frac {3 B \,x^{4} a \,c^{2} d}{4 e^{2}}-\frac {A \,x^{3} a \,c^{2} d}{e^{2}}+\frac {B \,x^{3} a \,c^{2} d^{2}}{e^{3}}+\frac {B \,c^{3} x^{7}}{7 e}-\frac {C a \,c^{2} d^{3} x^{3}}{e^{4}}+\frac {3 C a \,c^{2} d^{4} x^{2}}{2 e^{5}}+\frac {C \,a^{3} x^{2}}{2 e}+\frac {B x \,a^{3}}{e}+\frac {\ln \left (e x +d \right ) A \,a^{3}}{e}+\frac {C \,c^{3} x^{8}}{8 e}+\frac {A \,c^{3} x^{6}}{6 e}+\frac {3 \ln \left (e x +d \right ) A \,a^{2} c \,d^{2}}{e^{3}}+\frac {3 \ln \left (e x +d \right ) A a \,c^{2} d^{4}}{e^{5}}-\frac {3 \ln \left (e x +d \right ) B \,a^{2} c \,d^{3}}{e^{4}}-\frac {3 \ln \left (e x +d \right ) B a \,c^{2} d^{5}}{e^{6}}+\frac {3 \ln \left (e x +d \right ) C \,a^{2} c \,d^{4}}{e^{5}}+\frac {3 \ln \left (e x +d \right ) C a \,c^{2} d^{6}}{e^{7}}-\frac {C \,a^{3} d x}{e^{2}}-\frac {C \,c^{3} d^{7} x}{e^{8}}+\frac {C \,c^{3} d^{6} x^{2}}{2 e^{7}}+\frac {C \,c^{3} d^{2} x^{6}}{6 e^{3}}+\frac {3 C \,a^{2} c \,x^{4}}{4 e}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {A \,c^{3} d^{5} x}{e^{6}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}+\frac {B \,x^{3} a^{2} c}{e}+\frac {3 A \,x^{2} a^{2} c}{2 e}+\frac {3 B \,x^{5} a \,c^{2}}{5 e}+\frac {3 A \,x^{4} a \,c^{2}}{4 e}-\frac {C \,c^{3} d^{3} x^{5}}{5 e^{4}}-\frac {\ln \left (e x +d \right ) B \,a^{3} d}{e^{2}}+\frac {3 B x \,a^{2} c \,d^{2}}{e^{3}}+\frac {3 B x a \,c^{2} d^{4}}{e^{5}}+\frac {3 C \,a^{2} c \,d^{2} x^{2}}{2 e^{3}}+\frac {3 C a \,c^{2} d^{2} x^{4}}{4 e^{3}}-\frac {3 C a \,c^{2} d \,x^{5}}{5 e^{2}}+\frac {\ln \left (e x +d \right ) C \,a^{3} d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) C \,c^{3} d^{8}}{e^{9}}+\frac {d^{6} \ln \left (e x +d \right ) A \,c^{3}}{e^{7}}-\frac {d^{7} \ln \left (e x +d \right ) B \,c^{3}}{e^{8}}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}+\frac {C \,c^{3} d^{4} x^{4}}{4 e^{5}}-\frac {C \,c^{3} d^{5} x^{3}}{3 e^{6}}+\frac {B \,c^{3} d^{6} x}{e^{7}}\) | \(880\) |
| parallelrisch | \(\frac {-280 C \,x^{3} c^{3} d^{5} e^{3}+1260 A \,x^{2} a^{2} c \,e^{8}+420 A \,x^{2} c^{3} d^{4} e^{4}-420 B \,x^{2} c^{3} d^{5} e^{3}+420 C \,x^{2} c^{3} d^{6} e^{2}-840 A x \,c^{3} d^{5} e^{3}+840 B x \,c^{3} d^{6} e^{2}-840 C x \,a^{3} d \,e^{7}-840 C x \,c^{3} d^{7} e -840 B \ln \left (e x +d \right ) c^{3} d^{7} e +840 C \ln \left (e x +d \right ) a^{3} d^{2} e^{6}-120 C \,x^{7} c^{3} d \,e^{7}-168 A \,x^{5} c^{3} d \,e^{7}+504 B \,x^{5} a \,c^{2} e^{8}+168 B \,x^{5} c^{3} d^{2} e^{6}-168 C \,x^{5} c^{3} d^{3} e^{5}+630 A \,x^{4} a \,c^{2} e^{8}+210 A \,x^{4} c^{3} d^{2} e^{6}-210 B \,x^{4} c^{3} d^{3} e^{5}+630 C \,x^{4} a^{2} c \,e^{8}+210 C \,x^{4} c^{3} d^{4} e^{4}-280 A \,x^{3} c^{3} d^{3} e^{5}+840 B \,x^{3} a^{2} c \,e^{8}+280 B \,x^{3} c^{3} d^{4} e^{4}+2520 A \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{6}+105 C \,x^{8} c^{3} e^{8}+420 C \,x^{2} a^{3} e^{8}+840 B x \,a^{3} e^{8}+840 A \ln \left (e x +d \right ) a^{3} e^{8}+840 C \ln \left (e x +d \right ) c^{3} d^{8}+120 B \,x^{7} c^{3} e^{8}+140 A \,x^{6} c^{3} e^{8}-140 B \,x^{6} c^{3} d \,e^{7}+420 C \,x^{6} a \,c^{2} e^{8}+140 C \,x^{6} c^{3} d^{2} e^{6}+2520 C \ln \left (e x +d \right ) a^{2} c \,d^{4} e^{4}+2520 C \ln \left (e x +d \right ) a \,c^{2} d^{6} e^{2}+2520 B x \,a^{2} c \,d^{2} e^{6}+2520 B x a \,c^{2} d^{4} e^{4}-2520 C x \,a^{2} c \,d^{3} e^{5}-2520 C x a \,c^{2} d^{5} e^{3}-1260 B \,x^{2} a \,c^{2} d^{3} e^{5}+1260 C \,x^{2} a^{2} c \,d^{2} e^{6}+1260 C \,x^{2} a \,c^{2} d^{4} e^{4}-2520 A x \,a^{2} c d \,e^{7}-2520 A x a \,c^{2} d^{3} e^{5}-504 C \,x^{5} a \,c^{2} d \,e^{7}-630 B \,x^{4} a \,c^{2} d \,e^{7}+630 C \,x^{4} a \,c^{2} d^{2} e^{6}-840 A \,x^{3} a \,c^{2} d \,e^{7}+840 B \,x^{3} a \,c^{2} d^{2} e^{6}+840 A \ln \left (e x +d \right ) c^{3} d^{6} e^{2}-840 B \ln \left (e x +d \right ) a^{3} d \,e^{7}+2520 A \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{4}-840 C \,x^{3} a^{2} c d \,e^{7}-840 C \,x^{3} a \,c^{2} d^{3} e^{5}+1260 A \,x^{2} a \,c^{2} d^{2} e^{6}-1260 B \,x^{2} a^{2} c d \,e^{7}-2520 B \ln \left (e x +d \right ) a^{2} c \,d^{3} e^{5}-2520 B \ln \left (e x +d \right ) a \,c^{2} d^{5} e^{3}}{840 e^{9}}\) | \(886\) |
1/2/e^7*(3*A*a^2*c*e^6+3*A*a*c^2*d^2*e^4+A*c^3*d^4*e^2-3*B*a^2*c*d*e^5-3*B *a*c^2*d^3*e^3-B*c^3*d^5*e+C*a^3*e^6+3*C*a^2*c*d^2*e^4+3*C*a*c^2*d^4*e^2+C *c^3*d^6)*x^2-(3*A*a^2*c*d*e^6+3*A*a*c^2*d^3*e^4+A*c^3*d^5*e^2-B*a^3*e^7-3 *B*a^2*c*d^2*e^5-3*B*a*c^2*d^4*e^3-B*c^3*d^6*e+C*a^3*d*e^6+3*C*a^2*c*d^3*e ^4+3*C*a*c^2*d^5*e^2+C*c^3*d^7)/e^8*x+1/8*C*c^3/e*x^8-1/3*c/e^6*(3*A*a*c*d *e^4+A*c^2*d^3*e^2-3*B*a^2*e^5-3*B*a*c*d^2*e^3-B*c^2*d^4*e+3*C*a^2*d*e^4+3 *C*a*c*d^3*e^2+C*c^2*d^5)*x^3+1/4*c/e^5*(3*A*a*c*e^4+A*c^2*d^2*e^2-3*B*a*c *d*e^3-B*c^2*d^3*e+3*C*a^2*e^4+3*C*a*c*d^2*e^2+C*c^2*d^4)*x^4-1/5*c^2/e^4* (A*c*d*e^2-3*B*a*e^3-B*c*d^2*e+3*C*a*d*e^2+C*c*d^3)*x^5+1/6*c^2/e^3*(A*c*e ^2-B*c*d*e+3*C*a*e^2+C*c*d^2)*x^6+1/7*c^3/e^2*(B*e-C*d)*x^7+(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)/e^9*ln(e*x+d)
Time = 0.27 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=\frac {105 \, C c^{3} e^{8} x^{8} - 120 \, {\left (C c^{3} d e^{7} - B c^{3} e^{8}\right )} x^{7} + 140 \, {\left (C c^{3} d^{2} e^{6} - B c^{3} d e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{8}\right )} x^{6} - 168 \, {\left (C c^{3} d^{3} e^{5} - B c^{3} d^{2} e^{6} - 3 \, B a c^{2} e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{7}\right )} x^{5} + 210 \, {\left (C c^{3} d^{4} e^{4} - B c^{3} d^{3} e^{5} - 3 \, B a c^{2} d e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{6} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{8}\right )} x^{4} - 280 \, {\left (C c^{3} d^{5} e^{3} - B c^{3} d^{4} e^{4} - 3 \, B a c^{2} d^{2} e^{6} - 3 \, B a^{2} c e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{5} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{7}\right )} x^{3} + 420 \, {\left (C c^{3} d^{6} e^{2} - B c^{3} d^{5} e^{3} - 3 \, B a c^{2} d^{3} e^{5} - 3 \, B a^{2} c d e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{4} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{6} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{8}\right )} x^{2} - 840 \, {\left (C c^{3} d^{7} e - B c^{3} d^{6} e^{2} - 3 \, B a c^{2} d^{4} e^{4} - 3 \, B a^{2} c d^{2} e^{6} - B a^{3} e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{3} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{5} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{7}\right )} x + 840 \, {\left (C c^{3} d^{8} - B c^{3} d^{7} e - 3 \, B a c^{2} d^{5} e^{3} - 3 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} + A a^{3} e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \]
1/840*(105*C*c^3*e^8*x^8 - 120*(C*c^3*d*e^7 - B*c^3*e^8)*x^7 + 140*(C*c^3* d^2*e^6 - B*c^3*d*e^7 + (3*C*a*c^2 + A*c^3)*e^8)*x^6 - 168*(C*c^3*d^3*e^5 - B*c^3*d^2*e^6 - 3*B*a*c^2*e^8 + (3*C*a*c^2 + A*c^3)*d*e^7)*x^5 + 210*(C* c^3*d^4*e^4 - B*c^3*d^3*e^5 - 3*B*a*c^2*d*e^7 + (3*C*a*c^2 + A*c^3)*d^2*e^ 6 + 3*(C*a^2*c + A*a*c^2)*e^8)*x^4 - 280*(C*c^3*d^5*e^3 - B*c^3*d^4*e^4 - 3*B*a*c^2*d^2*e^6 - 3*B*a^2*c*e^8 + (3*C*a*c^2 + A*c^3)*d^3*e^5 + 3*(C*a^2 *c + A*a*c^2)*d*e^7)*x^3 + 420*(C*c^3*d^6*e^2 - B*c^3*d^5*e^3 - 3*B*a*c^2* d^3*e^5 - 3*B*a^2*c*d*e^7 + (3*C*a*c^2 + A*c^3)*d^4*e^4 + 3*(C*a^2*c + A*a *c^2)*d^2*e^6 + (C*a^3 + 3*A*a^2*c)*e^8)*x^2 - 840*(C*c^3*d^7*e - B*c^3*d^ 6*e^2 - 3*B*a*c^2*d^4*e^4 - 3*B*a^2*c*d^2*e^6 - B*a^3*e^8 + (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 + (C*a^3 + 3*A*a^2*c)*d*e^7) *x + 840*(C*c^3*d^8 - B*c^3*d^7*e - 3*B*a*c^2*d^5*e^3 - 3*B*a^2*c*d^3*e^5 - B*a^3*d*e^7 + A*a^3*e^8 + (3*C*a*c^2 + A*c^3)*d^6*e^2 + 3*(C*a^2*c + A*a *c^2)*d^4*e^4 + (C*a^3 + 3*A*a^2*c)*d^2*e^6)*log(e*x + d))/e^9
Time = 0.74 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=\frac {C c^{3} x^{8}}{8 e} + x^{7} \left (\frac {B c^{3}}{7 e} - \frac {C c^{3} d}{7 e^{2}}\right ) + x^{6} \left (\frac {A c^{3}}{6 e} - \frac {B c^{3} d}{6 e^{2}} + \frac {C a c^{2}}{2 e} + \frac {C c^{3} d^{2}}{6 e^{3}}\right ) + x^{5} \left (- \frac {A c^{3} d}{5 e^{2}} + \frac {3 B a c^{2}}{5 e} + \frac {B c^{3} d^{2}}{5 e^{3}} - \frac {3 C a c^{2} d}{5 e^{2}} - \frac {C c^{3} d^{3}}{5 e^{4}}\right ) + x^{4} \cdot \left (\frac {3 A a c^{2}}{4 e} + \frac {A c^{3} d^{2}}{4 e^{3}} - \frac {3 B a c^{2} d}{4 e^{2}} - \frac {B c^{3} d^{3}}{4 e^{4}} + \frac {3 C a^{2} c}{4 e} + \frac {3 C a c^{2} d^{2}}{4 e^{3}} + \frac {C c^{3} d^{4}}{4 e^{5}}\right ) + x^{3} \left (- \frac {A a c^{2} d}{e^{2}} - \frac {A c^{3} d^{3}}{3 e^{4}} + \frac {B a^{2} c}{e} + \frac {B a c^{2} d^{2}}{e^{3}} + \frac {B c^{3} d^{4}}{3 e^{5}} - \frac {C a^{2} c d}{e^{2}} - \frac {C a c^{2} d^{3}}{e^{4}} - \frac {C c^{3} d^{5}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} c}{2 e} + \frac {3 A a c^{2} d^{2}}{2 e^{3}} + \frac {A c^{3} d^{4}}{2 e^{5}} - \frac {3 B a^{2} c d}{2 e^{2}} - \frac {3 B a c^{2} d^{3}}{2 e^{4}} - \frac {B c^{3} d^{5}}{2 e^{6}} + \frac {C a^{3}}{2 e} + \frac {3 C a^{2} c d^{2}}{2 e^{3}} + \frac {3 C a c^{2} d^{4}}{2 e^{5}} + \frac {C c^{3} d^{6}}{2 e^{7}}\right ) + x \left (- \frac {3 A a^{2} c d}{e^{2}} - \frac {3 A a c^{2} d^{3}}{e^{4}} - \frac {A c^{3} d^{5}}{e^{6}} + \frac {B a^{3}}{e} + \frac {3 B a^{2} c d^{2}}{e^{3}} + \frac {3 B a c^{2} d^{4}}{e^{5}} + \frac {B c^{3} d^{6}}{e^{7}} - \frac {C a^{3} d}{e^{2}} - \frac {3 C a^{2} c d^{3}}{e^{4}} - \frac {3 C a c^{2} d^{5}}{e^{6}} - \frac {C c^{3} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{3} \left (A e^{2} - B d e + C d^{2}\right ) \log {\left (d + e x \right )}}{e^{9}} \]
C*c**3*x**8/(8*e) + x**7*(B*c**3/(7*e) - C*c**3*d/(7*e**2)) + x**6*(A*c**3 /(6*e) - B*c**3*d/(6*e**2) + C*a*c**2/(2*e) + C*c**3*d**2/(6*e**3)) + x**5 *(-A*c**3*d/(5*e**2) + 3*B*a*c**2/(5*e) + B*c**3*d**2/(5*e**3) - 3*C*a*c** 2*d/(5*e**2) - C*c**3*d**3/(5*e**4)) + x**4*(3*A*a*c**2/(4*e) + A*c**3*d** 2/(4*e**3) - 3*B*a*c**2*d/(4*e**2) - B*c**3*d**3/(4*e**4) + 3*C*a**2*c/(4* e) + 3*C*a*c**2*d**2/(4*e**3) + C*c**3*d**4/(4*e**5)) + x**3*(-A*a*c**2*d/ e**2 - A*c**3*d**3/(3*e**4) + B*a**2*c/e + B*a*c**2*d**2/e**3 + B*c**3*d** 4/(3*e**5) - C*a**2*c*d/e**2 - C*a*c**2*d**3/e**4 - C*c**3*d**5/(3*e**6)) + x**2*(3*A*a**2*c/(2*e) + 3*A*a*c**2*d**2/(2*e**3) + A*c**3*d**4/(2*e**5) - 3*B*a**2*c*d/(2*e**2) - 3*B*a*c**2*d**3/(2*e**4) - B*c**3*d**5/(2*e**6) + C*a**3/(2*e) + 3*C*a**2*c*d**2/(2*e**3) + 3*C*a*c**2*d**4/(2*e**5) + C* c**3*d**6/(2*e**7)) + x*(-3*A*a**2*c*d/e**2 - 3*A*a*c**2*d**3/e**4 - A*c** 3*d**5/e**6 + B*a**3/e + 3*B*a**2*c*d**2/e**3 + 3*B*a*c**2*d**4/e**5 + B*c **3*d**6/e**7 - C*a**3*d/e**2 - 3*C*a**2*c*d**3/e**4 - 3*C*a*c**2*d**5/e** 6 - C*c**3*d**7/e**8) + (a*e**2 + c*d**2)**3*(A*e**2 - B*d*e + C*d**2)*log (d + e*x)/e**9
Time = 0.19 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=\frac {105 \, C c^{3} e^{7} x^{8} - 120 \, {\left (C c^{3} d e^{6} - B c^{3} e^{7}\right )} x^{7} + 140 \, {\left (C c^{3} d^{2} e^{5} - B c^{3} d e^{6} + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} - 168 \, {\left (C c^{3} d^{3} e^{4} - B c^{3} d^{2} e^{5} - 3 \, B a c^{2} e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} d e^{6}\right )} x^{5} + 210 \, {\left (C c^{3} d^{4} e^{3} - B c^{3} d^{3} e^{4} - 3 \, B a c^{2} d e^{6} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{7}\right )} x^{4} - 280 \, {\left (C c^{3} d^{5} e^{2} - B c^{3} d^{4} e^{3} - 3 \, B a c^{2} d^{2} e^{5} - 3 \, B a^{2} c e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d e^{6}\right )} x^{3} + 420 \, {\left (C c^{3} d^{6} e - B c^{3} d^{5} e^{2} - 3 \, B a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d e^{6} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{5} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{7}\right )} x^{2} - 840 \, {\left (C c^{3} d^{7} - B c^{3} d^{6} e - 3 \, B a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{2} e^{5} - B a^{3} e^{7} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{3} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d e^{6}\right )} x}{840 \, e^{8}} + \frac {{\left (C c^{3} d^{8} - B c^{3} d^{7} e - 3 \, B a c^{2} d^{5} e^{3} - 3 \, B a^{2} c d^{3} e^{5} - B a^{3} d e^{7} + A a^{3} e^{8} + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{6} e^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{4} e^{4} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2} e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \]
1/840*(105*C*c^3*e^7*x^8 - 120*(C*c^3*d*e^6 - B*c^3*e^7)*x^7 + 140*(C*c^3* d^2*e^5 - B*c^3*d*e^6 + (3*C*a*c^2 + A*c^3)*e^7)*x^6 - 168*(C*c^3*d^3*e^4 - B*c^3*d^2*e^5 - 3*B*a*c^2*e^7 + (3*C*a*c^2 + A*c^3)*d*e^6)*x^5 + 210*(C* c^3*d^4*e^3 - B*c^3*d^3*e^4 - 3*B*a*c^2*d*e^6 + (3*C*a*c^2 + A*c^3)*d^2*e^ 5 + 3*(C*a^2*c + A*a*c^2)*e^7)*x^4 - 280*(C*c^3*d^5*e^2 - B*c^3*d^4*e^3 - 3*B*a*c^2*d^2*e^5 - 3*B*a^2*c*e^7 + (3*C*a*c^2 + A*c^3)*d^3*e^4 + 3*(C*a^2 *c + A*a*c^2)*d*e^6)*x^3 + 420*(C*c^3*d^6*e - B*c^3*d^5*e^2 - 3*B*a*c^2*d^ 3*e^4 - 3*B*a^2*c*d*e^6 + (3*C*a*c^2 + A*c^3)*d^4*e^3 + 3*(C*a^2*c + A*a*c ^2)*d^2*e^5 + (C*a^3 + 3*A*a^2*c)*e^7)*x^2 - 840*(C*c^3*d^7 - B*c^3*d^6*e - 3*B*a*c^2*d^4*e^3 - 3*B*a^2*c*d^2*e^5 - B*a^3*e^7 + (3*C*a*c^2 + A*c^3)* d^5*e^2 + 3*(C*a^2*c + A*a*c^2)*d^3*e^4 + (C*a^3 + 3*A*a^2*c)*d*e^6)*x)/e^ 8 + (C*c^3*d^8 - B*c^3*d^7*e - 3*B*a*c^2*d^5*e^3 - 3*B*a^2*c*d^3*e^5 - B*a ^3*d*e^7 + A*a^3*e^8 + (3*C*a*c^2 + A*c^3)*d^6*e^2 + 3*(C*a^2*c + A*a*c^2) *d^4*e^4 + (C*a^3 + 3*A*a^2*c)*d^2*e^6)*log(e*x + d)/e^9
Time = 0.28 (sec) , antiderivative size = 817, 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} \, dx=\frac {105 \, C c^{3} e^{7} x^{8} - 120 \, C c^{3} d e^{6} x^{7} + 120 \, B c^{3} e^{7} x^{7} + 140 \, C c^{3} d^{2} e^{5} x^{6} - 140 \, B c^{3} d e^{6} x^{6} + 420 \, C a c^{2} e^{7} x^{6} + 140 \, A c^{3} e^{7} x^{6} - 168 \, C c^{3} d^{3} e^{4} x^{5} + 168 \, B c^{3} d^{2} e^{5} x^{5} - 504 \, C a c^{2} d e^{6} x^{5} - 168 \, A c^{3} d e^{6} x^{5} + 504 \, B a c^{2} e^{7} x^{5} + 210 \, C c^{3} d^{4} e^{3} x^{4} - 210 \, B c^{3} d^{3} e^{4} x^{4} + 630 \, C a c^{2} d^{2} e^{5} x^{4} + 210 \, A c^{3} d^{2} e^{5} x^{4} - 630 \, B a c^{2} d e^{6} x^{4} + 630 \, C a^{2} c e^{7} x^{4} + 630 \, A a c^{2} e^{7} x^{4} - 280 \, C c^{3} d^{5} e^{2} x^{3} + 280 \, B c^{3} d^{4} e^{3} x^{3} - 840 \, C a c^{2} d^{3} e^{4} x^{3} - 280 \, A c^{3} d^{3} e^{4} x^{3} + 840 \, B a c^{2} d^{2} e^{5} x^{3} - 840 \, C a^{2} c d e^{6} x^{3} - 840 \, A a c^{2} d e^{6} x^{3} + 840 \, B a^{2} c e^{7} x^{3} + 420 \, C c^{3} d^{6} e x^{2} - 420 \, B c^{3} d^{5} e^{2} x^{2} + 1260 \, C a c^{2} d^{4} e^{3} x^{2} + 420 \, A c^{3} d^{4} e^{3} x^{2} - 1260 \, B a c^{2} d^{3} e^{4} x^{2} + 1260 \, C a^{2} c d^{2} e^{5} x^{2} + 1260 \, A a c^{2} d^{2} e^{5} x^{2} - 1260 \, B a^{2} c d e^{6} x^{2} + 420 \, C a^{3} e^{7} x^{2} + 1260 \, A a^{2} c e^{7} x^{2} - 840 \, C c^{3} d^{7} x + 840 \, B c^{3} d^{6} e x - 2520 \, C a c^{2} d^{5} e^{2} x - 840 \, A c^{3} d^{5} e^{2} x + 2520 \, B a c^{2} d^{4} e^{3} x - 2520 \, C a^{2} c d^{3} e^{4} x - 2520 \, A a c^{2} d^{3} e^{4} x + 2520 \, B a^{2} c d^{2} e^{5} x - 840 \, C a^{3} d e^{6} x - 2520 \, A a^{2} c d e^{6} x + 840 \, B a^{3} e^{7} x}{840 \, e^{8}} + \frac {{\left (C c^{3} d^{8} - B c^{3} d^{7} e + 3 \, C a c^{2} d^{6} e^{2} + A c^{3} d^{6} e^{2} - 3 \, B a c^{2} d^{5} e^{3} + 3 \, C a^{2} c d^{4} e^{4} + 3 \, A a c^{2} d^{4} e^{4} - 3 \, B a^{2} c d^{3} e^{5} + C a^{3} d^{2} e^{6} + 3 \, A a^{2} c d^{2} e^{6} - B a^{3} d e^{7} + A a^{3} e^{8}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} \]
1/840*(105*C*c^3*e^7*x^8 - 120*C*c^3*d*e^6*x^7 + 120*B*c^3*e^7*x^7 + 140*C *c^3*d^2*e^5*x^6 - 140*B*c^3*d*e^6*x^6 + 420*C*a*c^2*e^7*x^6 + 140*A*c^3*e ^7*x^6 - 168*C*c^3*d^3*e^4*x^5 + 168*B*c^3*d^2*e^5*x^5 - 504*C*a*c^2*d*e^6 *x^5 - 168*A*c^3*d*e^6*x^5 + 504*B*a*c^2*e^7*x^5 + 210*C*c^3*d^4*e^3*x^4 - 210*B*c^3*d^3*e^4*x^4 + 630*C*a*c^2*d^2*e^5*x^4 + 210*A*c^3*d^2*e^5*x^4 - 630*B*a*c^2*d*e^6*x^4 + 630*C*a^2*c*e^7*x^4 + 630*A*a*c^2*e^7*x^4 - 280*C *c^3*d^5*e^2*x^3 + 280*B*c^3*d^4*e^3*x^3 - 840*C*a*c^2*d^3*e^4*x^3 - 280*A *c^3*d^3*e^4*x^3 + 840*B*a*c^2*d^2*e^5*x^3 - 840*C*a^2*c*d*e^6*x^3 - 840*A *a*c^2*d*e^6*x^3 + 840*B*a^2*c*e^7*x^3 + 420*C*c^3*d^6*e*x^2 - 420*B*c^3*d ^5*e^2*x^2 + 1260*C*a*c^2*d^4*e^3*x^2 + 420*A*c^3*d^4*e^3*x^2 - 1260*B*a*c ^2*d^3*e^4*x^2 + 1260*C*a^2*c*d^2*e^5*x^2 + 1260*A*a*c^2*d^2*e^5*x^2 - 126 0*B*a^2*c*d*e^6*x^2 + 420*C*a^3*e^7*x^2 + 1260*A*a^2*c*e^7*x^2 - 840*C*c^3 *d^7*x + 840*B*c^3*d^6*e*x - 2520*C*a*c^2*d^5*e^2*x - 840*A*c^3*d^5*e^2*x + 2520*B*a*c^2*d^4*e^3*x - 2520*C*a^2*c*d^3*e^4*x - 2520*A*a*c^2*d^3*e^4*x + 2520*B*a^2*c*d^2*e^5*x - 840*C*a^3*d*e^6*x - 2520*A*a^2*c*d*e^6*x + 840 *B*a^3*e^7*x)/e^8 + (C*c^3*d^8 - B*c^3*d^7*e + 3*C*a*c^2*d^6*e^2 + A*c^3*d ^6*e^2 - 3*B*a*c^2*d^5*e^3 + 3*C*a^2*c*d^4*e^4 + 3*A*a*c^2*d^4*e^4 - 3*B*a ^2*c*d^3*e^5 + C*a^3*d^2*e^6 + 3*A*a^2*c*d^2*e^6 - B*a^3*d*e^7 + A*a^3*e^8 )*log(abs(e*x + d))/e^9
Time = 12.38 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+c x^2\right )^3 \left (A+B x+C x^2\right )}{d+e x} \, dx=x\,\left (\frac {B\,a^3}{e}-\frac {d\,\left (\frac {C\,a^3+3\,A\,c\,a^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{e}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{e}\right )}{e}\right )+x^7\,\left (\frac {B\,c^3}{7\,e}-\frac {C\,c^3\,d}{7\,e^2}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{e}\right )}{5\,e}-\frac {3\,B\,a\,c^2}{5\,e}\right )+x^4\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{e}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{4\,e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{4\,e}\right )+x^2\,\left (\frac {C\,a^3+3\,A\,c\,a^2}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{e}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e}\right )}{e}-\frac {3\,B\,a^2\,c}{e}\right )}{2\,e}\right )+x^6\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{6\,e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{6\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,C\,a\,c^2}{e}-\frac {d\,\left (\frac {B\,c^3}{e}-\frac {C\,c^3\,d}{e^2}\right )}{e}\right )}{e}-\frac {3\,B\,a\,c^2}{e}\right )}{e}+\frac {3\,a\,c\,\left (A\,c+C\,a\right )}{e}\right )}{3\,e}-\frac {B\,a^2\,c}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (C\,a^3\,d^2\,e^6-B\,a^3\,d\,e^7+A\,a^3\,e^8+3\,C\,a^2\,c\,d^4\,e^4-3\,B\,a^2\,c\,d^3\,e^5+3\,A\,a^2\,c\,d^2\,e^6+3\,C\,a\,c^2\,d^6\,e^2-3\,B\,a\,c^2\,d^5\,e^3+3\,A\,a\,c^2\,d^4\,e^4+C\,c^3\,d^8-B\,c^3\,d^7\,e+A\,c^3\,d^6\,e^2\right )}{e^9}+\frac {C\,c^3\,x^8}{8\,e} \]
x*((B*a^3)/e - (d*((C*a^3 + 3*A*a^2*c)/e + (d*((d*((d*((d*((A*c^3 + 3*C*a* c^2)/e - (d*((B*c^3)/e - (C*c^3*d)/e^2))/e))/e - (3*B*a*c^2)/e))/e + (3*a* c*(A*c + C*a))/e))/e - (3*B*a^2*c)/e))/e))/e) + x^7*((B*c^3)/(7*e) - (C*c^ 3*d)/(7*e^2)) - x^5*((d*((A*c^3 + 3*C*a*c^2)/e - (d*((B*c^3)/e - (C*c^3*d) /e^2))/e))/(5*e) - (3*B*a*c^2)/(5*e)) + x^4*((d*((d*((A*c^3 + 3*C*a*c^2)/e - (d*((B*c^3)/e - (C*c^3*d)/e^2))/e))/e - (3*B*a*c^2)/e))/(4*e) + (3*a*c* (A*c + C*a))/(4*e)) + x^2*((C*a^3 + 3*A*a^2*c)/(2*e) + (d*((d*((d*((d*((A* c^3 + 3*C*a*c^2)/e - (d*((B*c^3)/e - (C*c^3*d)/e^2))/e))/e - (3*B*a*c^2)/e ))/e + (3*a*c*(A*c + C*a))/e))/e - (3*B*a^2*c)/e))/(2*e)) + x^6*((A*c^3 + 3*C*a*c^2)/(6*e) - (d*((B*c^3)/e - (C*c^3*d)/e^2))/(6*e)) - x^3*((d*((d*(( d*((A*c^3 + 3*C*a*c^2)/e - (d*((B*c^3)/e - (C*c^3*d)/e^2))/e))/e - (3*B*a* c^2)/e))/e + (3*a*c*(A*c + C*a))/e))/(3*e) - (B*a^2*c)/e) + (log(d + e*x)* (A*a^3*e^8 + C*c^3*d^8 - B*a^3*d*e^7 - B*c^3*d^7*e + A*c^3*d^6*e^2 + C*a^3 *d^2*e^6 + 3*A*a*c^2*d^4*e^4 + 3*A*a^2*c*d^2*e^6 - 3*B*a*c^2*d^5*e^3 - 3*B *a^2*c*d^3*e^5 + 3*C*a*c^2*d^6*e^2 + 3*C*a^2*c*d^4*e^4))/e^9 + (C*c^3*x^8) /(8*e)