Integrand size = 22, antiderivative size = 118 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=-\frac {d (B d-A e) (c d-b e) (d+e x)^4}{4 e^4}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^5}{5 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^6}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]
-1/4*d*(-A*e+B*d)*(-b*e+c*d)*(e*x+d)^4/e^4+1/5*(B*d*(-2*b*e+3*c*d)-A*e*(-b *e+2*c*d))*(e*x+d)^5/e^4-1/6*(-A*c*e-B*b*e+3*B*c*d)*(e*x+d)^6/e^4+1/7*B*c* (e*x+d)^7/e^4
Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=\frac {1}{2} A b d^3 x^2+\frac {1}{3} d^2 (b B d+A c d+3 A b e) x^3+\frac {1}{4} d (3 A e (c d+b e)+B d (c d+3 b e)) x^4+\frac {1}{5} e (3 B d (c d+b e)+A e (3 c d+b e)) x^5+\frac {1}{6} e^2 (3 B c d+b B e+A c e) x^6+\frac {1}{7} B c e^3 x^7 \]
(A*b*d^3*x^2)/2 + (d^2*(b*B*d + A*c*d + 3*A*b*e)*x^3)/3 + (d*(3*A*e*(c*d + b*e) + B*d*(c*d + 3*b*e))*x^4)/4 + (e*(3*B*d*(c*d + b*e) + A*e*(3*c*d + b *e))*x^5)/5 + (e^2*(3*B*c*d + b*B*e + A*c*e)*x^6)/6 + (B*c*e^3*x^7)/7
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1195, 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.
\(\displaystyle \int (A+B x) \left (b x+c x^2\right ) (d+e x)^3 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {(d+e x)^5 (A c e+b B e-3 B c d)}{e^3}+\frac {(d+e x)^4 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^3}-\frac {d (d+e x)^3 (B d-A e) (c d-b e)}{e^3}+\frac {B c (d+e x)^6}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac {B c (d+e x)^7}{7 e^4}\) |
-1/4*(d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^4)/e^4 + ((B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*(d + e*x)^5)/(5*e^4) - ((3*B*c*d - b*B*e - A*c*e)*(d + e*x)^6)/(6*e^4) + (B*c*(d + e*x)^7)/(7*e^4)
3.12.4.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.25
method | result | size |
norman | \(\frac {B \,e^{3} c \,x^{7}}{7}+\left (\frac {1}{6} A c \,e^{3}+\frac {1}{6} B \,e^{3} b +\frac {1}{2} d B c \,e^{2}\right ) x^{6}+\left (\frac {1}{5} A b \,e^{3}+\frac {3}{5} A c d \,e^{2}+\frac {3}{5} B b d \,e^{2}+\frac {3}{5} B c \,d^{2} e \right ) x^{5}+\left (\frac {3}{4} A b d \,e^{2}+\frac {3}{4} A c \,d^{2} e +\frac {3}{4} B b \,d^{2} e +\frac {1}{4} B c \,d^{3}\right ) x^{4}+\left (A b \,d^{2} e +\frac {1}{3} A c \,d^{3}+\frac {1}{3} B b \,d^{3}\right ) x^{3}+\frac {A \,d^{3} b \,x^{2}}{2}\) | \(147\) |
default | \(\frac {B \,e^{3} c \,x^{7}}{7}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) c +B \,e^{3} b \right ) x^{6}}{6}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c +\left (A \,e^{3}+3 B d \,e^{2}\right ) b \right ) x^{5}}{5}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b \right ) x^{4}}{4}+\frac {\left (A c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b \right ) x^{3}}{3}+\frac {A \,d^{3} b \,x^{2}}{2}\) | \(152\) |
gosper | \(\frac {x^{2} \left (60 B \,e^{3} c \,x^{5}+70 x^{4} A c \,e^{3}+70 x^{4} B \,e^{3} b +210 x^{4} d B c \,e^{2}+84 x^{3} A b \,e^{3}+252 x^{3} A c d \,e^{2}+252 x^{3} B b d \,e^{2}+252 x^{3} B c \,d^{2} e +315 x^{2} A b d \,e^{2}+315 x^{2} A c \,d^{2} e +315 x^{2} B b \,d^{2} e +105 x^{2} B c \,d^{3}+420 x A b \,d^{2} e +140 x A c \,d^{3}+140 x B b \,d^{3}+210 A \,d^{3} b \right )}{420}\) | \(166\) |
risch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{6} x^{6} B \,e^{3} b +\frac {1}{2} x^{6} d B c \,e^{2}+\frac {1}{5} x^{5} A b \,e^{3}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {3}{5} x^{5} B b d \,e^{2}+\frac {3}{5} x^{5} B c \,d^{2} e +\frac {3}{4} x^{4} A b d \,e^{2}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B b \,d^{2} e +\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A b \,d^{2} e +\frac {1}{3} x^{3} A c \,d^{3}+\frac {1}{3} x^{3} B b \,d^{3}+\frac {1}{2} A \,d^{3} b \,x^{2}\) | \(169\) |
parallelrisch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{6} x^{6} B \,e^{3} b +\frac {1}{2} x^{6} d B c \,e^{2}+\frac {1}{5} x^{5} A b \,e^{3}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {3}{5} x^{5} B b d \,e^{2}+\frac {3}{5} x^{5} B c \,d^{2} e +\frac {3}{4} x^{4} A b d \,e^{2}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B b \,d^{2} e +\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A b \,d^{2} e +\frac {1}{3} x^{3} A c \,d^{3}+\frac {1}{3} x^{3} B b \,d^{3}+\frac {1}{2} A \,d^{3} b \,x^{2}\) | \(169\) |
1/7*B*e^3*c*x^7+(1/6*A*c*e^3+1/6*B*e^3*b+1/2*d*B*c*e^2)*x^6+(1/5*A*b*e^3+3 /5*A*c*d*e^2+3/5*B*b*d*e^2+3/5*B*c*d^2*e)*x^5+(3/4*A*b*d*e^2+3/4*A*c*d^2*e +3/4*B*b*d^2*e+1/4*B*c*d^3)*x^4+(A*b*d^2*e+1/3*A*c*d^3+1/3*B*b*d^3)*x^3+1/ 2*A*d^3*b*x^2
Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.16 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B c d^{2} e + A b e^{3} + 3 \, {\left (B b + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A b d e^{2} + 3 \, {\left (B b + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A b d^{2} e + {\left (B b + A c\right )} d^{3}\right )} x^{3} \]
1/7*B*c*e^3*x^7 + 1/2*A*b*d^3*x^2 + 1/6*(3*B*c*d*e^2 + (B*b + A*c)*e^3)*x^ 6 + 1/5*(3*B*c*d^2*e + A*b*e^3 + 3*(B*b + A*c)*d*e^2)*x^5 + 1/4*(B*c*d^3 + 3*A*b*d*e^2 + 3*(B*b + A*c)*d^2*e)*x^4 + 1/3*(3*A*b*d^2*e + (B*b + A*c)*d ^3)*x^3
Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.50 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=\frac {A b d^{3} x^{2}}{2} + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B b e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \left (\frac {A b e^{3}}{5} + \frac {3 A c d e^{2}}{5} + \frac {3 B b d e^{2}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \cdot \left (\frac {3 A b d e^{2}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B b d^{2} e}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A b d^{2} e + \frac {A c d^{3}}{3} + \frac {B b d^{3}}{3}\right ) \]
A*b*d**3*x**2/2 + B*c*e**3*x**7/7 + x**6*(A*c*e**3/6 + B*b*e**3/6 + B*c*d* e**2/2) + x**5*(A*b*e**3/5 + 3*A*c*d*e**2/5 + 3*B*b*d*e**2/5 + 3*B*c*d**2* e/5) + x**4*(3*A*b*d*e**2/4 + 3*A*c*d**2*e/4 + 3*B*b*d**2*e/4 + B*c*d**3/4 ) + x**3*(A*b*d**2*e + A*c*d**3/3 + B*b*d**3/3)
Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.16 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B c d^{2} e + A b e^{3} + 3 \, {\left (B b + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A b d e^{2} + 3 \, {\left (B b + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A b d^{2} e + {\left (B b + A c\right )} d^{3}\right )} x^{3} \]
1/7*B*c*e^3*x^7 + 1/2*A*b*d^3*x^2 + 1/6*(3*B*c*d*e^2 + (B*b + A*c)*e^3)*x^ 6 + 1/5*(3*B*c*d^2*e + A*b*e^3 + 3*(B*b + A*c)*d*e^2)*x^5 + 1/4*(B*c*d^3 + 3*A*b*d*e^2 + 3*(B*b + A*c)*d^2*e)*x^4 + 1/3*(3*A*b*d^2*e + (B*b + A*c)*d ^3)*x^3
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.42 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, B c d e^{2} x^{6} + \frac {1}{6} \, B b e^{3} x^{6} + \frac {1}{6} \, A c e^{3} x^{6} + \frac {3}{5} \, B c d^{2} e x^{5} + \frac {3}{5} \, B b d e^{2} x^{5} + \frac {3}{5} \, A c d e^{2} x^{5} + \frac {1}{5} \, A b e^{3} x^{5} + \frac {1}{4} \, B c d^{3} x^{4} + \frac {3}{4} \, B b d^{2} e x^{4} + \frac {3}{4} \, A c d^{2} e x^{4} + \frac {3}{4} \, A b d e^{2} x^{4} + \frac {1}{3} \, B b d^{3} x^{3} + \frac {1}{3} \, A c d^{3} x^{3} + A b d^{2} e x^{3} + \frac {1}{2} \, A b d^{3} x^{2} \]
1/7*B*c*e^3*x^7 + 1/2*B*c*d*e^2*x^6 + 1/6*B*b*e^3*x^6 + 1/6*A*c*e^3*x^6 + 3/5*B*c*d^2*e*x^5 + 3/5*B*b*d*e^2*x^5 + 3/5*A*c*d*e^2*x^5 + 1/5*A*b*e^3*x^ 5 + 1/4*B*c*d^3*x^4 + 3/4*B*b*d^2*e*x^4 + 3/4*A*c*d^2*e*x^4 + 3/4*A*b*d*e^ 2*x^4 + 1/3*B*b*d^3*x^3 + 1/3*A*c*d^3*x^3 + A*b*d^2*e*x^3 + 1/2*A*b*d^3*x^ 2
Time = 10.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.24 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx=x^3\,\left (\frac {A\,c\,d^3}{3}+\frac {B\,b\,d^3}{3}+A\,b\,d^2\,e\right )+x^6\,\left (\frac {A\,c\,e^3}{6}+\frac {B\,b\,e^3}{6}+\frac {B\,c\,d\,e^2}{2}\right )+x^4\,\left (\frac {B\,c\,d^3}{4}+\frac {3\,A\,b\,d\,e^2}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,b\,d^2\,e}{4}\right )+x^5\,\left (\frac {A\,b\,e^3}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {3\,B\,b\,d\,e^2}{5}+\frac {3\,B\,c\,d^2\,e}{5}\right )+\frac {A\,b\,d^3\,x^2}{2}+\frac {B\,c\,e^3\,x^7}{7} \]