Integrand size = 25, antiderivative size = 175 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right ) (d+e x)^4}{4 e^5}-\frac {\left (a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right ) (d+e x)^5}{5 e^5}+\frac {\left (a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right ) (d+e x)^6}{6 e^5}-\frac {c (4 C d-B e) (d+e x)^7}{7 e^5}+\frac {c C (d+e x)^8}{8 e^5} \]
1/4*(a*e^2+c*d^2)*(A*e^2-B*d*e+C*d^2)*(e*x+d)^4/e^5-1/5*(a*e^2*(-B*e+2*C*d )+c*d*(4*C*d^2-e*(-2*A*e+3*B*d)))*(e*x+d)^5/e^5+1/6*(a*C*e^2+c*(6*C*d^2-e* (-A*e+3*B*d)))*(e*x+d)^6/e^5-1/7*c*(-B*e+4*C*d)*(e*x+d)^7/e^5+1/8*c*C*(e*x +d)^8/e^5
Time = 0.05 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=a A d^3 x+\frac {1}{2} a d^2 (B d+3 A e) x^2+\frac {1}{3} d \left (a d (C d+3 B e)+A \left (c d^2+3 a e^2\right )\right ) x^3+\frac {1}{4} \left (B c d^3+3 A c d^2 e+3 a C d^2 e+3 a B d e^2+a A e^3\right ) x^4+\frac {1}{5} \left (c C d^3+3 c d e (B d+A e)+a e^2 (3 C d+B e)\right ) x^5+\frac {1}{6} e \left (3 c C d^2+a C e^2+c e (3 B d+A e)\right ) x^6+\frac {1}{7} c e^2 (3 C d+B e) x^7+\frac {1}{8} c C e^3 x^8 \]
a*A*d^3*x + (a*d^2*(B*d + 3*A*e)*x^2)/2 + (d*(a*d*(C*d + 3*B*e) + A*(c*d^2 + 3*a*e^2))*x^3)/3 + ((B*c*d^3 + 3*A*c*d^2*e + 3*a*C*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + ((c*C*d^3 + 3*c*d*e*(B*d + A*e) + a*e^2*(3*C*d + B*e)) *x^5)/5 + (e*(3*c*C*d^2 + a*C*e^2 + c*e*(3*B*d + A*e))*x^6)/6 + (c*e^2*(3* C*d + B*e)*x^7)/7 + (c*C*e^3*x^8)/8
Time = 0.50 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \left (a+c x^2\right ) (d+e x)^3 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {(d+e x)^4 \left (-a e^2 (2 C d-B e)+c d e (3 B d-2 A e)-4 c C d^3\right )}{e^4}+\frac {(d+e x)^5 \left (a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{e^4}+\frac {(d+e x)^3 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^4}+\frac {c (d+e x)^6 (B e-4 C d)}{e^4}+\frac {c C (d+e x)^7}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(d+e x)^5 \left (a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{5 e^5}+\frac {(d+e x)^6 \left (a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{6 e^5}+\frac {(d+e x)^4 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{4 e^5}-\frac {c (d+e x)^7 (4 C d-B e)}{7 e^5}+\frac {c C (d+e x)^8}{8 e^5}\) |
((c*d^2 + a*e^2)*(C*d^2 - B*d*e + A*e^2)*(d + e*x)^4)/(4*e^5) - ((4*c*C*d^ 3 - c*d*e*(3*B*d - 2*A*e) + a*e^2*(2*C*d - B*e))*(d + e*x)^5)/(5*e^5) + (( 6*c*C*d^2 + a*C*e^2 - c*e*(3*B*d - A*e))*(d + e*x)^6)/(6*e^5) - (c*(4*C*d - B*e)*(d + e*x)^7)/(7*e^5) + (c*C*(d + e*x)^8)/(8*e^5)
3.1.18.3.1 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 = 0.52 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.22
| method | result | size |
| norman | \(\frac {c \,e^{3} C \,x^{8}}{8}+\left (\frac {1}{7} B c \,e^{3}+\frac {3}{7} c d \,e^{2} C \right ) x^{7}+\left (\frac {1}{6} A c \,e^{3}+\frac {1}{2} B c d \,e^{2}+\frac {1}{6} C a \,e^{3}+\frac {1}{2} C c \,d^{2} e \right ) x^{6}+\left (\frac {3}{5} A c d \,e^{2}+\frac {1}{5} B \,e^{3} a +\frac {3}{5} B c \,d^{2} e +\frac {3}{5} C a d \,e^{2}+\frac {1}{5} C c \,d^{3}\right ) x^{5}+\left (\frac {1}{4} A a \,e^{3}+\frac {3}{4} A c \,d^{2} e +\frac {3}{4} B a d \,e^{2}+\frac {1}{4} B c \,d^{3}+\frac {3}{4} a \,d^{2} e C \right ) x^{4}+\left (A a d \,e^{2}+\frac {1}{3} A c \,d^{3}+B a \,d^{2} e +\frac {1}{3} a \,d^{3} C \right ) x^{3}+\left (\frac {3}{2} a A \,d^{2} e +\frac {1}{2} B a \,d^{3}\right ) x^{2}+A \,d^{3} a x\) | \(213\) |
| default | \(\frac {c \,e^{3} C \,x^{8}}{8}+\frac {\left (B c \,e^{3}+3 c d \,e^{2} C \right ) x^{7}}{7}+\frac {\left (\left (a \,e^{3}+3 c \,d^{2} e \right ) C +3 B c d \,e^{2}+A c \,e^{3}\right ) x^{6}}{6}+\frac {\left (\left (3 a d \,e^{2}+c \,d^{3}\right ) C +\left (a \,e^{3}+3 c \,d^{2} e \right ) B +3 A c d \,e^{2}\right ) x^{5}}{5}+\frac {\left (3 a \,d^{2} e C +\left (3 a d \,e^{2}+c \,d^{3}\right ) B +\left (a \,e^{3}+3 c \,d^{2} e \right ) A \right ) x^{4}}{4}+\frac {\left (a \,d^{3} C +3 B a \,d^{2} e +\left (3 a d \,e^{2}+c \,d^{3}\right ) A \right ) x^{3}}{3}+\frac {\left (3 a A \,d^{2} e +B a \,d^{3}\right ) x^{2}}{2}+A \,d^{3} a x\) | \(217\) |
| gosper | \(\frac {1}{8} c \,e^{3} C \,x^{8}+\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {3}{7} x^{7} c d \,e^{2} C +\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{6} x^{6} C a \,e^{3}+\frac {1}{2} x^{6} C c \,d^{2} e +\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {3}{5} x^{5} C a d \,e^{2}+\frac {1}{5} x^{5} C c \,d^{3}+\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+\frac {3}{4} x^{4} a \,d^{2} e C +x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} x^{3} a \,d^{3} C +\frac {3}{2} x^{2} a A \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(249\) |
| risch | \(\frac {1}{8} c \,e^{3} C \,x^{8}+\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {3}{7} x^{7} c d \,e^{2} C +\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{6} x^{6} C a \,e^{3}+\frac {1}{2} x^{6} C c \,d^{2} e +\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {3}{5} x^{5} C a d \,e^{2}+\frac {1}{5} x^{5} C c \,d^{3}+\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+\frac {3}{4} x^{4} a \,d^{2} e C +x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} x^{3} a \,d^{3} C +\frac {3}{2} x^{2} a A \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(249\) |
| parallelrisch | \(\frac {1}{8} c \,e^{3} C \,x^{8}+\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {3}{7} x^{7} c d \,e^{2} C +\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{6} x^{6} C a \,e^{3}+\frac {1}{2} x^{6} C c \,d^{2} e +\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {3}{5} x^{5} C a d \,e^{2}+\frac {1}{5} x^{5} C c \,d^{3}+\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+\frac {3}{4} x^{4} a \,d^{2} e C +x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} x^{3} a \,d^{3} C +\frac {3}{2} x^{2} a A \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(249\) |
1/8*c*e^3*C*x^8+(1/7*B*c*e^3+3/7*c*d*e^2*C)*x^7+(1/6*A*c*e^3+1/2*B*c*d*e^2 +1/6*C*a*e^3+1/2*C*c*d^2*e)*x^6+(3/5*A*c*d*e^2+1/5*B*e^3*a+3/5*B*c*d^2*e+3 /5*C*a*d*e^2+1/5*C*c*d^3)*x^5+(1/4*A*a*e^3+3/4*A*c*d^2*e+3/4*B*a*d*e^2+1/4 *B*c*d^3+3/4*a*d^2*e*C)*x^4+(A*a*d*e^2+1/3*A*c*d^3+B*a*d^2*e+1/3*a*d^3*C)* x^3+(3/2*a*A*d^2*e+1/2*B*a*d^3)*x^2+A*d^3*a*x
Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{8} \, C c e^{3} x^{8} + \frac {1}{7} \, {\left (3 \, C c d e^{2} + B c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, C c d^{2} e + 3 \, B c d e^{2} + {\left (C a + A c\right )} e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (C c d^{3} + 3 \, B c d^{2} e + B a e^{3} + 3 \, {\left (C a + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, B a d e^{2} + A a e^{3} + 3 \, {\left (C a + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a d^{2} e + 3 \, A a d e^{2} + {\left (C a + A c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \]
1/8*C*c*e^3*x^8 + 1/7*(3*C*c*d*e^2 + B*c*e^3)*x^7 + 1/6*(3*C*c*d^2*e + 3*B *c*d*e^2 + (C*a + A*c)*e^3)*x^6 + A*a*d^3*x + 1/5*(C*c*d^3 + 3*B*c*d^2*e + B*a*e^3 + 3*(C*a + A*c)*d*e^2)*x^5 + 1/4*(B*c*d^3 + 3*B*a*d*e^2 + A*a*e^3 + 3*(C*a + A*c)*d^2*e)*x^4 + 1/3*(3*B*a*d^2*e + 3*A*a*d*e^2 + (C*a + A*c) *d^3)*x^3 + 1/2*(B*a*d^3 + 3*A*a*d^2*e)*x^2
Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.47 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=A a d^{3} x + \frac {C c e^{3} x^{8}}{8} + x^{7} \left (\frac {B c e^{3}}{7} + \frac {3 C c d e^{2}}{7}\right ) + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B c d e^{2}}{2} + \frac {C a e^{3}}{6} + \frac {C c d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 A c d e^{2}}{5} + \frac {B a e^{3}}{5} + \frac {3 B c d^{2} e}{5} + \frac {3 C a d e^{2}}{5} + \frac {C c d^{3}}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B a d e^{2}}{4} + \frac {B c d^{3}}{4} + \frac {3 C a d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + \frac {A c d^{3}}{3} + B a d^{2} e + \frac {C a d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a d^{2} e}{2} + \frac {B a d^{3}}{2}\right ) \]
A*a*d**3*x + C*c*e**3*x**8/8 + x**7*(B*c*e**3/7 + 3*C*c*d*e**2/7) + x**6*( A*c*e**3/6 + B*c*d*e**2/2 + C*a*e**3/6 + C*c*d**2*e/2) + x**5*(3*A*c*d*e** 2/5 + B*a*e**3/5 + 3*B*c*d**2*e/5 + 3*C*a*d*e**2/5 + C*c*d**3/5) + x**4*(A *a*e**3/4 + 3*A*c*d**2*e/4 + 3*B*a*d*e**2/4 + B*c*d**3/4 + 3*C*a*d**2*e/4) + x**3*(A*a*d*e**2 + A*c*d**3/3 + B*a*d**2*e + C*a*d**3/3) + x**2*(3*A*a* d**2*e/2 + B*a*d**3/2)
Time = 0.19 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{8} \, C c e^{3} x^{8} + \frac {1}{7} \, {\left (3 \, C c d e^{2} + B c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, C c d^{2} e + 3 \, B c d e^{2} + {\left (C a + A c\right )} e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (C c d^{3} + 3 \, B c d^{2} e + B a e^{3} + 3 \, {\left (C a + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, B a d e^{2} + A a e^{3} + 3 \, {\left (C a + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a d^{2} e + 3 \, A a d e^{2} + {\left (C a + A c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \]
1/8*C*c*e^3*x^8 + 1/7*(3*C*c*d*e^2 + B*c*e^3)*x^7 + 1/6*(3*C*c*d^2*e + 3*B *c*d*e^2 + (C*a + A*c)*e^3)*x^6 + A*a*d^3*x + 1/5*(C*c*d^3 + 3*B*c*d^2*e + B*a*e^3 + 3*(C*a + A*c)*d*e^2)*x^5 + 1/4*(B*c*d^3 + 3*B*a*d*e^2 + A*a*e^3 + 3*(C*a + A*c)*d^2*e)*x^4 + 1/3*(3*B*a*d^2*e + 3*A*a*d*e^2 + (C*a + A*c) *d^3)*x^3 + 1/2*(B*a*d^3 + 3*A*a*d^2*e)*x^2
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.42 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{8} \, C c e^{3} x^{8} + \frac {3}{7} \, C c d e^{2} x^{7} + \frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, C c d^{2} e x^{6} + \frac {1}{2} \, B c d e^{2} x^{6} + \frac {1}{6} \, C a e^{3} x^{6} + \frac {1}{6} \, A c e^{3} x^{6} + \frac {1}{5} \, C c d^{3} x^{5} + \frac {3}{5} \, B c d^{2} e x^{5} + \frac {3}{5} \, C a d e^{2} x^{5} + \frac {3}{5} \, A c d e^{2} x^{5} + \frac {1}{5} \, B a e^{3} x^{5} + \frac {1}{4} \, B c d^{3} x^{4} + \frac {3}{4} \, C a d^{2} e x^{4} + \frac {3}{4} \, A c d^{2} e x^{4} + \frac {3}{4} \, B a d e^{2} x^{4} + \frac {1}{4} \, A a e^{3} x^{4} + \frac {1}{3} \, C a d^{3} x^{3} + \frac {1}{3} \, A c d^{3} x^{3} + B a d^{2} e x^{3} + A a d e^{2} x^{3} + \frac {1}{2} \, B a d^{3} x^{2} + \frac {3}{2} \, A a d^{2} e x^{2} + A a d^{3} x \]
1/8*C*c*e^3*x^8 + 3/7*C*c*d*e^2*x^7 + 1/7*B*c*e^3*x^7 + 1/2*C*c*d^2*e*x^6 + 1/2*B*c*d*e^2*x^6 + 1/6*C*a*e^3*x^6 + 1/6*A*c*e^3*x^6 + 1/5*C*c*d^3*x^5 + 3/5*B*c*d^2*e*x^5 + 3/5*C*a*d*e^2*x^5 + 3/5*A*c*d*e^2*x^5 + 1/5*B*a*e^3* x^5 + 1/4*B*c*d^3*x^4 + 3/4*C*a*d^2*e*x^4 + 3/4*A*c*d^2*e*x^4 + 3/4*B*a*d* e^2*x^4 + 1/4*A*a*e^3*x^4 + 1/3*C*a*d^3*x^3 + 1/3*A*c*d^3*x^3 + B*a*d^2*e* x^3 + A*a*d*e^2*x^3 + 1/2*B*a*d^3*x^2 + 3/2*A*a*d^2*e*x^2 + A*a*d^3*x
Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.18 \[ \int (d+e x)^3 \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^3}{3}+\frac {C\,a\,d^3}{3}+A\,a\,d\,e^2+B\,a\,d^2\,e\right )+x^6\,\left (\frac {A\,c\,e^3}{6}+\frac {C\,a\,e^3}{6}+\frac {B\,c\,d\,e^2}{2}+\frac {C\,c\,d^2\,e}{2}\right )+x^4\,\left (\frac {A\,a\,e^3}{4}+\frac {B\,c\,d^3}{4}+\frac {3\,B\,a\,d\,e^2}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,C\,a\,d^2\,e}{4}\right )+x^5\,\left (\frac {B\,a\,e^3}{5}+\frac {C\,c\,d^3}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {3\,C\,a\,d\,e^2}{5}+\frac {3\,B\,c\,d^2\,e}{5}\right )+A\,a\,d^3\,x+\frac {C\,c\,e^3\,x^8}{8}+\frac {a\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c\,e^2\,x^7\,\left (B\,e+3\,C\,d\right )}{7} \]
x^3*((A*c*d^3)/3 + (C*a*d^3)/3 + A*a*d*e^2 + B*a*d^2*e) + x^6*((A*c*e^3)/6 + (C*a*e^3)/6 + (B*c*d*e^2)/2 + (C*c*d^2*e)/2) + x^4*((A*a*e^3)/4 + (B*c* d^3)/4 + (3*B*a*d*e^2)/4 + (3*A*c*d^2*e)/4 + (3*C*a*d^2*e)/4) + x^5*((B*a* e^3)/5 + (C*c*d^3)/5 + (3*A*c*d*e^2)/5 + (3*C*a*d*e^2)/5 + (3*B*c*d^2*e)/5 ) + A*a*d^3*x + (C*c*e^3*x^8)/8 + (a*d^2*x^2*(3*A*e + B*d))/2 + (c*e^2*x^7 *(B*e + 3*C*d))/7