Integrand size = 20, antiderivative size = 108 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^3}{3 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^4}{4 e^4}-\frac {c (3 B d-A e) (d+e x)^5}{5 e^4}+\frac {B c (d+e x)^6}{6 e^4} \]
-1/3*(-A*e+B*d)*(a*e^2+c*d^2)*(e*x+d)^3/e^4+1/4*(-2*A*c*d*e+B*a*e^2+3*B*c* d^2)*(e*x+d)^4/e^4-1/5*c*(-A*e+3*B*d)*(e*x+d)^5/e^4+1/6*B*c*(e*x+d)^6/e^4
Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=a A d^2 x+\frac {1}{2} a d (B d+2 A e) x^2+\frac {1}{3} \left (A c d^2+2 a B d e+a A e^2\right ) x^3+\frac {1}{4} \left (B c d^2+2 A c d e+a B e^2\right ) x^4+\frac {1}{5} c e (2 B d+A e) x^5+\frac {1}{6} B c e^2 x^6 \]
a*A*d^2*x + (a*d*(B*d + 2*A*e)*x^2)/2 + ((A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x ^3)/3 + ((B*c*d^2 + 2*A*c*d*e + a*B*e^2)*x^4)/4 + (c*e*(2*B*d + A*e)*x^5)/ 5 + (B*c*e^2*x^6)/6
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {652, 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.
\(\displaystyle \int \left (a+c x^2\right ) (A+B x) (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 652 |
\(\displaystyle \int \left (\frac {(d+e x)^3 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^3}+\frac {(d+e x)^2 \left (a e^2+c d^2\right ) (A e-B d)}{e^3}+\frac {c (d+e x)^4 (A e-3 B d)}{e^3}+\frac {B c (d+e x)^5}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^4 \left (a B e^2-2 A c d e+3 B c d^2\right )}{4 e^4}-\frac {(d+e x)^3 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4}-\frac {c (d+e x)^5 (3 B d-A e)}{5 e^4}+\frac {B c (d+e x)^6}{6 e^4}\) |
-1/3*((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^3)/e^4 + ((3*B*c*d^2 - 2*A*c*d *e + a*B*e^2)*(d + e*x)^4)/(4*e^4) - (c*(3*B*d - A*e)*(d + e*x)^5)/(5*e^4) + (B*c*(d + e*x)^6)/(6*e^4)
3.13.87.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {B c \,e^{2} x^{6}}{6}+\frac {\left (A \,e^{2}+2 B d e \right ) c \,x^{5}}{5}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c +B a \,e^{2}\right ) x^{4}}{4}+\frac {\left (A c \,d^{2}+\left (A \,e^{2}+2 B d e \right ) a \right ) x^{3}}{3}+\frac {\left (2 A d e +B \,d^{2}\right ) a \,x^{2}}{2}+A \,d^{2} a x\) | \(103\) |
norman | \(\frac {B c \,e^{2} x^{6}}{6}+\left (\frac {1}{5} A c \,e^{2}+\frac {2}{5} B c d e \right ) x^{5}+\left (\frac {1}{2} A c d e +\frac {1}{4} B a \,e^{2}+\frac {1}{4} B c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} A a \,e^{2}+\frac {1}{3} A c \,d^{2}+\frac {2}{3} B a d e \right ) x^{3}+\left (A a d e +\frac {1}{2} B a \,d^{2}\right ) x^{2}+A \,d^{2} a x\) | \(104\) |
gosper | \(\frac {1}{6} B c \,e^{2} x^{6}+\frac {1}{5} x^{5} A c \,e^{2}+\frac {2}{5} x^{5} B c d e +\frac {1}{2} x^{4} A c d e +\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{4} x^{4} B c \,d^{2}+\frac {1}{3} x^{3} A a \,e^{2}+\frac {1}{3} x^{3} A c \,d^{2}+\frac {2}{3} x^{3} B a d e +x^{2} A a d e +\frac {1}{2} x^{2} B a \,d^{2}+A \,d^{2} a x\) | \(114\) |
risch | \(\frac {1}{6} B c \,e^{2} x^{6}+\frac {1}{5} x^{5} A c \,e^{2}+\frac {2}{5} x^{5} B c d e +\frac {1}{2} x^{4} A c d e +\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{4} x^{4} B c \,d^{2}+\frac {1}{3} x^{3} A a \,e^{2}+\frac {1}{3} x^{3} A c \,d^{2}+\frac {2}{3} x^{3} B a d e +x^{2} A a d e +\frac {1}{2} x^{2} B a \,d^{2}+A \,d^{2} a x\) | \(114\) |
parallelrisch | \(\frac {1}{6} B c \,e^{2} x^{6}+\frac {1}{5} x^{5} A c \,e^{2}+\frac {2}{5} x^{5} B c d e +\frac {1}{2} x^{4} A c d e +\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{4} x^{4} B c \,d^{2}+\frac {1}{3} x^{3} A a \,e^{2}+\frac {1}{3} x^{3} A c \,d^{2}+\frac {2}{3} x^{3} B a d e +x^{2} A a d e +\frac {1}{2} x^{2} B a \,d^{2}+A \,d^{2} a x\) | \(114\) |
1/6*B*c*e^2*x^6+1/5*(A*e^2+2*B*d*e)*c*x^5+1/4*((2*A*d*e+B*d^2)*c+B*a*e^2)* x^4+1/3*(A*c*d^2+(A*e^2+2*B*d*e)*a)*x^3+1/2*(2*A*d*e+B*d^2)*a*x^2+A*d^2*a* x
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B c d e + A c e^{2}\right )} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (B c d^{2} + 2 \, A c d e + B a e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{2} + 2 \, B a d e + A a e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{2} + 2 \, A a d e\right )} x^{2} \]
1/6*B*c*e^2*x^6 + 1/5*(2*B*c*d*e + A*c*e^2)*x^5 + A*a*d^2*x + 1/4*(B*c*d^2 + 2*A*c*d*e + B*a*e^2)*x^4 + 1/3*(A*c*d^2 + 2*B*a*d*e + A*a*e^2)*x^3 + 1/ 2*(B*a*d^2 + 2*A*a*d*e)*x^2
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=A a d^{2} x + \frac {B c e^{2} x^{6}}{6} + x^{5} \left (\frac {A c e^{2}}{5} + \frac {2 B c d e}{5}\right ) + x^{4} \left (\frac {A c d e}{2} + \frac {B a e^{2}}{4} + \frac {B c d^{2}}{4}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {A c d^{2}}{3} + \frac {2 B a d e}{3}\right ) + x^{2} \left (A a d e + \frac {B a d^{2}}{2}\right ) \]
A*a*d**2*x + B*c*e**2*x**6/6 + x**5*(A*c*e**2/5 + 2*B*c*d*e/5) + x**4*(A*c *d*e/2 + B*a*e**2/4 + B*c*d**2/4) + x**3*(A*a*e**2/3 + A*c*d**2/3 + 2*B*a* d*e/3) + x**2*(A*a*d*e + B*a*d**2/2)
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B c d e + A c e^{2}\right )} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (B c d^{2} + 2 \, A c d e + B a e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{2} + 2 \, B a d e + A a e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{2} + 2 \, A a d e\right )} x^{2} \]
1/6*B*c*e^2*x^6 + 1/5*(2*B*c*d*e + A*c*e^2)*x^5 + A*a*d^2*x + 1/4*(B*c*d^2 + 2*A*c*d*e + B*a*e^2)*x^4 + 1/3*(A*c*d^2 + 2*B*a*d*e + A*a*e^2)*x^3 + 1/ 2*(B*a*d^2 + 2*A*a*d*e)*x^2
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {2}{5} \, B c d e x^{5} + \frac {1}{5} \, A c e^{2} x^{5} + \frac {1}{4} \, B c d^{2} x^{4} + \frac {1}{2} \, A c d e x^{4} + \frac {1}{4} \, B a e^{2} x^{4} + \frac {1}{3} \, A c d^{2} x^{3} + \frac {2}{3} \, B a d e x^{3} + \frac {1}{3} \, A a e^{2} x^{3} + \frac {1}{2} \, B a d^{2} x^{2} + A a d e x^{2} + A a d^{2} x \]
1/6*B*c*e^2*x^6 + 2/5*B*c*d*e*x^5 + 1/5*A*c*e^2*x^5 + 1/4*B*c*d^2*x^4 + 1/ 2*A*c*d*e*x^4 + 1/4*B*a*e^2*x^4 + 1/3*A*c*d^2*x^3 + 2/3*B*a*d*e*x^3 + 1/3* A*a*e^2*x^3 + 1/2*B*a*d^2*x^2 + A*a*d*e*x^2 + A*a*d^2*x
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.91 \[ \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^2}{3}+\frac {2\,B\,a\,d\,e}{3}+\frac {A\,a\,e^2}{3}\right )+x^4\,\left (\frac {B\,c\,d^2}{4}+\frac {A\,c\,d\,e}{2}+\frac {B\,a\,e^2}{4}\right )+A\,a\,d^2\,x+\frac {a\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c\,e\,x^5\,\left (A\,e+2\,B\,d\right )}{5}+\frac {B\,c\,e^2\,x^6}{6} \]