Integrand size = 16, antiderivative size = 117 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{7} a^3 b (2 A b+a B) x^7+\frac {5}{4} a^2 b^2 (A b+a B) x^8+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{11} b^5 B x^{11} \]
1/5*a^5*A*x^5+1/6*a^4*(5*A*b+B*a)*x^6+5/7*a^3*b*(2*A*b+B*a)*x^7+5/4*a^2*b^ 2*(A*b+B*a)*x^8+5/9*a*b^3*(A*b+2*B*a)*x^9+1/10*b^4*(A*b+5*B*a)*x^10+1/11*b ^5*B*x^11
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{7} a^3 b (2 A b+a B) x^7+\frac {5}{4} a^2 b^2 (A b+a B) x^8+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{11} b^5 B x^{11} \]
(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^6)/6 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + (5*a^2*b^2*(A*b + a*B)*x^8)/4 + (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A* b + 5*a*B)*x^10)/10 + (b^5*B*x^11)/11
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {85, 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.
\(\displaystyle \int x^4 (a+b x)^5 (A+B x) \, dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (a^5 A x^4+a^4 x^5 (a B+5 A b)+5 a^3 b x^6 (a B+2 A b)+10 a^2 b^2 x^7 (a B+A b)+b^4 x^9 (5 a B+A b)+5 a b^3 x^8 (2 a B+A b)+b^5 B x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 x^6 (a B+5 A b)+\frac {5}{7} a^3 b x^7 (a B+2 A b)+\frac {5}{4} a^2 b^2 x^8 (a B+A b)+\frac {1}{10} b^4 x^{10} (5 a B+A b)+\frac {5}{9} a b^3 x^9 (2 a B+A b)+\frac {1}{11} b^5 B x^{11}\) |
(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^6)/6 + (5*a^3*b*(2*A*b + a*B)*x^7)/7 + (5*a^2*b^2*(A*b + a*B)*x^8)/4 + (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A* b + 5*a*B)*x^10)/10 + (b^5*B*x^11)/11
3.2.20.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {b^{5} B \,x^{11}}{11}+\left (\frac {1}{10} b^{5} A +\frac {1}{2} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{9} a \,b^{4} A +\frac {10}{9} a^{2} b^{3} B \right ) x^{9}+\left (\frac {5}{4} a^{2} b^{3} A +\frac {5}{4} a^{3} b^{2} B \right ) x^{8}+\left (\frac {10}{7} a^{3} b^{2} A +\frac {5}{7} a^{4} b B \right ) x^{7}+\left (\frac {5}{6} a^{4} b A +\frac {1}{6} a^{5} B \right ) x^{6}+\frac {a^{5} A \,x^{5}}{5}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{11}}{11}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{10}}{10}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{9}}{9}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{8}}{8}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{7}}{7}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{6}}{6}+\frac {a^{5} A \,x^{5}}{5}\) | \(124\) |
gosper | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
risch | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
parallelrisch | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
1/11*b^5*B*x^11+(1/10*b^5*A+1/2*a*b^4*B)*x^10+(5/9*a*b^4*A+10/9*a^2*b^3*B) *x^9+(5/4*a^2*b^3*A+5/4*a^3*b^2*B)*x^8+(10/7*a^3*b^2*A+5/7*a^4*b*B)*x^7+(5 /6*a^4*b*A+1/6*a^5*B)*x^6+1/5*a^5*A*x^5
Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {5}{4} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \]
1/11*B*b^5*x^11 + 1/5*A*a^5*x^5 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/9*(2*B *a^2*b^3 + A*a*b^4)*x^9 + 5/4*(B*a^3*b^2 + A*a^2*b^3)*x^8 + 5/7*(B*a^4*b + 2*A*a^3*b^2)*x^7 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{5}}{5} + \frac {B b^{5} x^{11}}{11} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{9} \cdot \left (\frac {5 A a b^{4}}{9} + \frac {10 B a^{2} b^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a^{2} b^{3}}{4} + \frac {5 B a^{3} b^{2}}{4}\right ) + x^{7} \cdot \left (\frac {10 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b}{6} + \frac {B a^{5}}{6}\right ) \]
A*a**5*x**5/5 + B*b**5*x**11/11 + x**10*(A*b**5/10 + B*a*b**4/2) + x**9*(5 *A*a*b**4/9 + 10*B*a**2*b**3/9) + x**8*(5*A*a**2*b**3/4 + 5*B*a**3*b**2/4) + x**7*(10*A*a**3*b**2/7 + 5*B*a**4*b/7) + x**6*(5*A*a**4*b/6 + B*a**5/6)
Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {5}{4} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \]
1/11*B*b^5*x^11 + 1/5*A*a^5*x^5 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/9*(2*B *a^2*b^3 + A*a*b^4)*x^9 + 5/4*(B*a^3*b^2 + A*a^2*b^3)*x^8 + 5/7*(B*a^4*b + 2*A*a^3*b^2)*x^7 + 1/6*(B*a^5 + 5*A*a^4*b)*x^6
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {10}{9} \, B a^{2} b^{3} x^{9} + \frac {5}{9} \, A a b^{4} x^{9} + \frac {5}{4} \, B a^{3} b^{2} x^{8} + \frac {5}{4} \, A a^{2} b^{3} x^{8} + \frac {5}{7} \, B a^{4} b x^{7} + \frac {10}{7} \, A a^{3} b^{2} x^{7} + \frac {1}{6} \, B a^{5} x^{6} + \frac {5}{6} \, A a^{4} b x^{6} + \frac {1}{5} \, A a^{5} x^{5} \]
1/11*B*b^5*x^11 + 1/2*B*a*b^4*x^10 + 1/10*A*b^5*x^10 + 10/9*B*a^2*b^3*x^9 + 5/9*A*a*b^4*x^9 + 5/4*B*a^3*b^2*x^8 + 5/4*A*a^2*b^3*x^8 + 5/7*B*a^4*b*x^ 7 + 10/7*A*a^3*b^2*x^7 + 1/6*B*a^5*x^6 + 5/6*A*a^4*b*x^6 + 1/5*A*a^5*x^5
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=x^6\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+\frac {A\,a^5\,x^5}{5}+\frac {B\,b^5\,x^{11}}{11}+\frac {5\,a^2\,b^2\,x^8\,\left (A\,b+B\,a\right )}{4}+\frac {5\,a^3\,b\,x^7\,\left (2\,A\,b+B\,a\right )}{7}+\frac {5\,a\,b^3\,x^9\,\left (A\,b+2\,B\,a\right )}{9} \]