Integrand size = 16, antiderivative size = 117 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{8} a^3 b (2 A b+a B) x^8+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {1}{2} a b^3 (A b+2 a B) x^{10}+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{12} b^5 B x^{12} \] Output:
1/6*a^5*A*x^6+1/7*a^4*(5*A*b+B*a)*x^7+5/8*a^3*b*(2*A*b+B*a)*x^8+10/9*a^2*b ^2*(A*b+B*a)*x^9+1/2*a*b^3*(A*b+2*B*a)*x^10+1/11*b^4*(A*b+5*B*a)*x^11+1/12 *b^5*B*x^12
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{8} a^3 b (2 A b+a B) x^8+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {1}{2} a b^3 (A b+2 a B) x^{10}+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{12} b^5 B x^{12} \] Input:
Integrate[x^5*(a + b*x)^5*(A + B*x),x]
Output:
(a^5*A*x^6)/6 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (a*b^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A* b + 5*a*B)*x^11)/11 + (b^5*B*x^12)/12
Time = 0.27 (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.
\(\displaystyle \int x^5 (a+b x)^5 (A+B x) \, dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (a^5 A x^5+a^4 x^6 (a B+5 A b)+5 a^3 b x^7 (a B+2 A b)+10 a^2 b^2 x^8 (a B+A b)+b^4 x^{10} (5 a B+A b)+5 a b^3 x^9 (2 a B+A b)+b^5 B x^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {5}{8} a^3 b x^8 (a B+2 A b)+\frac {10}{9} a^2 b^2 x^9 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {1}{2} a b^3 x^{10} (2 a B+A b)+\frac {1}{12} b^5 B x^{12}\) |
Input:
Int[x^5*(a + b*x)^5*(A + B*x),x]
Output:
(a^5*A*x^6)/6 + (a^4*(5*A*b + a*B)*x^7)/7 + (5*a^3*b*(2*A*b + a*B)*x^8)/8 + (10*a^2*b^2*(A*b + a*B)*x^9)/9 + (a*b^3*(A*b + 2*a*B)*x^10)/2 + (b^4*(A* b + 5*a*B)*x^11)/11 + (b^5*B*x^12)/12
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 = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {b^{5} B \,x^{12}}{12}+\left (\frac {1}{11} b^{5} A +\frac {5}{11} a \,b^{4} B \right ) x^{11}+\left (\frac {1}{2} a \,b^{4} A +a^{2} b^{3} B \right ) x^{10}+\left (\frac {10}{9} a^{2} b^{3} A +\frac {10}{9} a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{4} a^{3} b^{2} A +\frac {5}{8} a^{4} b B \right ) x^{8}+\left (\frac {5}{7} a^{4} b A +\frac {1}{7} a^{5} B \right ) x^{7}+\frac {a^{5} A \,x^{6}}{6}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{12}}{12}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{11}}{11}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{10}}{10}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{9}}{9}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{8}}{8}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{7}}{7}+\frac {a^{5} A \,x^{6}}{6}\) | \(124\) |
orering | \(\frac {x^{6} \left (462 B \,b^{5} x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2772 a A \,b^{4} x^{4}+5544 B \,a^{2} b^{3} x^{4}+6160 a^{2} A \,b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+6930 a^{3} A \,b^{2} x^{2}+3465 B \,a^{4} b \,x^{2}+3960 a^{4} A b x +792 B \,a^{5} x +924 a^{5} A \right )}{5544}\) | \(124\) |
gosper | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
risch | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
parallelrisch | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
Input:
int(x^5*(b*x+a)^5*(B*x+A),x,method=_RETURNVERBOSE)
Output:
1/12*b^5*B*x^12+(1/11*b^5*A+5/11*a*b^4*B)*x^11+(1/2*a*b^4*A+a^2*b^3*B)*x^1 0+(10/9*a^2*b^3*A+10/9*a^3*b^2*B)*x^9+(5/4*a^3*b^2*A+5/8*a^4*b*B)*x^8+(5/7 *a^4*b*A+1/7*a^5*B)*x^7+1/6*a^5*A*x^6
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \] Input:
integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="fricas")
Output:
1/12*B*b^5*x^12 + 1/6*A*a^5*x^6 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 1/2*(2*B *a^2*b^3 + A*a*b^4)*x^10 + 10/9*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{12}}{12} + x^{11} \left (\frac {A b^{5}}{11} + \frac {5 B a b^{4}}{11}\right ) + x^{10} \left (\frac {A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{9} \cdot \left (\frac {10 A a^{2} b^{3}}{9} + \frac {10 B a^{3} b^{2}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a^{3} b^{2}}{4} + \frac {5 B a^{4} b}{8}\right ) + x^{7} \cdot \left (\frac {5 A a^{4} b}{7} + \frac {B a^{5}}{7}\right ) \] Input:
integrate(x**5*(b*x+a)**5*(B*x+A),x)
Output:
A*a**5*x**6/6 + B*b**5*x**12/12 + x**11*(A*b**5/11 + 5*B*a*b**4/11) + x**1 0*(A*a*b**4/2 + B*a**2*b**3) + x**9*(10*A*a**2*b**3/9 + 10*B*a**3*b**2/9) + x**8*(5*A*a**3*b**2/4 + 5*B*a**4*b/8) + x**7*(5*A*a**4*b/7 + B*a**5/7)
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \] Input:
integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="maxima")
Output:
1/12*B*b^5*x^12 + 1/6*A*a^5*x^6 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 1/2*(2*B *a^2*b^3 + A*a*b^4)*x^10 + 10/9*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {5}{11} \, B a b^{4} x^{11} + \frac {1}{11} \, A b^{5} x^{11} + B a^{2} b^{3} x^{10} + \frac {1}{2} \, A a b^{4} x^{10} + \frac {10}{9} \, B a^{3} b^{2} x^{9} + \frac {10}{9} \, A a^{2} b^{3} x^{9} + \frac {5}{8} \, B a^{4} b x^{8} + \frac {5}{4} \, A a^{3} b^{2} x^{8} + \frac {1}{7} \, B a^{5} x^{7} + \frac {5}{7} \, A a^{4} b x^{7} + \frac {1}{6} \, A a^{5} x^{6} \] Input:
integrate(x^5*(b*x+a)^5*(B*x+A),x, algorithm="giac")
Output:
1/12*B*b^5*x^12 + 5/11*B*a*b^4*x^11 + 1/11*A*b^5*x^11 + B*a^2*b^3*x^10 + 1 /2*A*a*b^4*x^10 + 10/9*B*a^3*b^2*x^9 + 10/9*A*a^2*b^3*x^9 + 5/8*B*a^4*b*x^ 8 + 5/4*A*a^3*b^2*x^8 + 1/7*B*a^5*x^7 + 5/7*A*a^4*b*x^7 + 1/6*A*a^5*x^6
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=x^7\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^{11}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{12}}{12}+\frac {10\,a^2\,b^2\,x^9\,\left (A\,b+B\,a\right )}{9}+\frac {5\,a^3\,b\,x^8\,\left (2\,A\,b+B\,a\right )}{8}+\frac {a\,b^3\,x^{10}\,\left (A\,b+2\,B\,a\right )}{2} \] Input:
int(x^5*(A + B*x)*(a + b*x)^5,x)
Output:
x^7*((B*a^5)/7 + (5*A*a^4*b)/7) + x^11*((A*b^5)/11 + (5*B*a*b^4)/11) + (A* a^5*x^6)/6 + (B*b^5*x^12)/12 + (10*a^2*b^2*x^9*(A*b + B*a))/9 + (5*a^3*b*x ^8*(2*A*b + B*a))/8 + (a*b^3*x^10*(A*b + 2*B*a))/2
Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.58 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {x^{6} \left (462 b^{6} x^{6}+3024 a \,b^{5} x^{5}+8316 a^{2} b^{4} x^{4}+12320 a^{3} b^{3} x^{3}+10395 a^{4} b^{2} x^{2}+4752 a^{5} b x +924 a^{6}\right )}{5544} \] Input:
int(x^5*(b*x+a)^5*(B*x+A),x)
Output:
(x**6*(924*a**6 + 4752*a**5*b*x + 10395*a**4*b**2*x**2 + 12320*a**3*b**3*x **3 + 8316*a**2*b**4*x**4 + 3024*a*b**5*x**5 + 462*b**6*x**6))/5544