Integrand size = 20, antiderivative size = 117 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{7} a^5 A x^7+\frac {1}{10} a^4 (5 A b+a B) x^{10}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{19} a b^3 (A b+2 a B) x^{19}+\frac {1}{22} b^4 (A b+5 a B) x^{22}+\frac {1}{25} b^5 B x^{25} \]
1/7*a^5*A*x^7+1/10*a^4*(5*A*b+B*a)*x^10+5/13*a^3*b*(2*A*b+B*a)*x^13+5/8*a^ 2*b^2*(A*b+B*a)*x^16+5/19*a*b^3*(A*b+2*B*a)*x^19+1/22*b^4*(A*b+5*B*a)*x^22 +1/25*b^5*B*x^25
Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{7} a^5 A x^7+\frac {1}{10} a^4 (5 A b+a B) x^{10}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{19} a b^3 (A b+2 a B) x^{19}+\frac {1}{22} b^4 (A b+5 a B) x^{22}+\frac {1}{25} b^5 B x^{25} \]
(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13) /13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + ( b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {950, 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.
\(\displaystyle \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^5 A x^6+a^4 x^9 (a B+5 A b)+5 a^3 b x^{12} (a B+2 A b)+10 a^2 b^2 x^{15} (a B+A b)+b^4 x^{21} (5 a B+A b)+5 a b^3 x^{18} (2 a B+A b)+b^5 B x^{24}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} a^5 A x^7+\frac {1}{10} a^4 x^{10} (a B+5 A b)+\frac {5}{13} a^3 b x^{13} (a B+2 A b)+\frac {5}{8} a^2 b^2 x^{16} (a B+A b)+\frac {1}{22} b^4 x^{22} (5 a B+A b)+\frac {5}{19} a b^3 x^{19} (2 a B+A b)+\frac {1}{25} b^5 B x^{25}\) |
(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^10)/10 + (5*a^3*b*(2*A*b + a*B)*x^13) /13 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^19)/19 + ( b^4*(A*b + 5*a*B)*x^22)/22 + (b^5*B*x^25)/25
3.1.26.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Time = 4.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{7}}{7}+\left (\frac {1}{2} a^{4} b A +\frac {1}{10} a^{5} B \right ) x^{10}+\left (\frac {10}{13} a^{3} b^{2} A +\frac {5}{13} a^{4} b B \right ) x^{13}+\left (\frac {5}{8} a^{2} b^{3} A +\frac {5}{8} a^{3} b^{2} B \right ) x^{16}+\left (\frac {5}{19} a \,b^{4} A +\frac {10}{19} a^{2} b^{3} B \right ) x^{19}+\left (\frac {1}{22} b^{5} A +\frac {5}{22} a \,b^{4} B \right ) x^{22}+\frac {b^{5} B \,x^{25}}{25}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{25}}{25}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{22}}{22}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{19}}{19}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{16}}{16}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{13}}{13}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{10}}{10}+\frac {a^{5} A \,x^{7}}{7}\) | \(124\) |
gosper | \(\frac {1}{7} a^{5} A \,x^{7}+\frac {1}{2} x^{10} a^{4} b A +\frac {1}{10} x^{10} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{19} x^{19} a \,b^{4} A +\frac {10}{19} x^{19} a^{2} b^{3} B +\frac {1}{22} x^{22} b^{5} A +\frac {5}{22} x^{22} a \,b^{4} B +\frac {1}{25} b^{5} B \,x^{25}\) | \(126\) |
risch | \(\frac {1}{7} a^{5} A \,x^{7}+\frac {1}{2} x^{10} a^{4} b A +\frac {1}{10} x^{10} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{19} x^{19} a \,b^{4} A +\frac {10}{19} x^{19} a^{2} b^{3} B +\frac {1}{22} x^{22} b^{5} A +\frac {5}{22} x^{22} a \,b^{4} B +\frac {1}{25} b^{5} B \,x^{25}\) | \(126\) |
parallelrisch | \(\frac {1}{7} a^{5} A \,x^{7}+\frac {1}{2} x^{10} a^{4} b A +\frac {1}{10} x^{10} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{19} x^{19} a \,b^{4} A +\frac {10}{19} x^{19} a^{2} b^{3} B +\frac {1}{22} x^{22} b^{5} A +\frac {5}{22} x^{22} a \,b^{4} B +\frac {1}{25} b^{5} B \,x^{25}\) | \(126\) |
1/7*a^5*A*x^7+(1/2*a^4*b*A+1/10*a^5*B)*x^10+(10/13*a^3*b^2*A+5/13*a^4*b*B) *x^13+(5/8*a^2*b^3*A+5/8*a^3*b^2*B)*x^16+(5/19*a*b^4*A+10/19*a^2*b^3*B)*x^ 19+(1/22*b^5*A+5/22*a*b^4*B)*x^22+1/25*b^5*B*x^25
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{25} \, B b^{5} x^{25} + \frac {1}{22} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{22} + \frac {5}{19} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{19} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {5}{13} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac {1}{7} \, A a^{5} x^{7} + \frac {1}{10} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{10} \]
1/25*B*b^5*x^25 + 1/22*(5*B*a*b^4 + A*b^5)*x^22 + 5/19*(2*B*a^2*b^3 + A*a* b^4)*x^19 + 5/8*(B*a^3*b^2 + A*a^2*b^3)*x^16 + 5/13*(B*a^4*b + 2*A*a^3*b^2 )*x^13 + 1/7*A*a^5*x^7 + 1/10*(B*a^5 + 5*A*a^4*b)*x^10
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {A a^{5} x^{7}}{7} + \frac {B b^{5} x^{25}}{25} + x^{22} \left (\frac {A b^{5}}{22} + \frac {5 B a b^{4}}{22}\right ) + x^{19} \cdot \left (\frac {5 A a b^{4}}{19} + \frac {10 B a^{2} b^{3}}{19}\right ) + x^{16} \cdot \left (\frac {5 A a^{2} b^{3}}{8} + \frac {5 B a^{3} b^{2}}{8}\right ) + x^{13} \cdot \left (\frac {10 A a^{3} b^{2}}{13} + \frac {5 B a^{4} b}{13}\right ) + x^{10} \left (\frac {A a^{4} b}{2} + \frac {B a^{5}}{10}\right ) \]
A*a**5*x**7/7 + B*b**5*x**25/25 + x**22*(A*b**5/22 + 5*B*a*b**4/22) + x**1 9*(5*A*a*b**4/19 + 10*B*a**2*b**3/19) + x**16*(5*A*a**2*b**3/8 + 5*B*a**3* b**2/8) + x**13*(10*A*a**3*b**2/13 + 5*B*a**4*b/13) + x**10*(A*a**4*b/2 + B*a**5/10)
Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{25} \, B b^{5} x^{25} + \frac {1}{22} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{22} + \frac {5}{19} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{19} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {5}{13} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac {1}{7} \, A a^{5} x^{7} + \frac {1}{10} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{10} \]
1/25*B*b^5*x^25 + 1/22*(5*B*a*b^4 + A*b^5)*x^22 + 5/19*(2*B*a^2*b^3 + A*a* b^4)*x^19 + 5/8*(B*a^3*b^2 + A*a^2*b^3)*x^16 + 5/13*(B*a^4*b + 2*A*a^3*b^2 )*x^13 + 1/7*A*a^5*x^7 + 1/10*(B*a^5 + 5*A*a^4*b)*x^10
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{25} \, B b^{5} x^{25} + \frac {5}{22} \, B a b^{4} x^{22} + \frac {1}{22} \, A b^{5} x^{22} + \frac {10}{19} \, B a^{2} b^{3} x^{19} + \frac {5}{19} \, A a b^{4} x^{19} + \frac {5}{8} \, B a^{3} b^{2} x^{16} + \frac {5}{8} \, A a^{2} b^{3} x^{16} + \frac {5}{13} \, B a^{4} b x^{13} + \frac {10}{13} \, A a^{3} b^{2} x^{13} + \frac {1}{10} \, B a^{5} x^{10} + \frac {1}{2} \, A a^{4} b x^{10} + \frac {1}{7} \, A a^{5} x^{7} \]
1/25*B*b^5*x^25 + 5/22*B*a*b^4*x^22 + 1/22*A*b^5*x^22 + 10/19*B*a^2*b^3*x^ 19 + 5/19*A*a*b^4*x^19 + 5/8*B*a^3*b^2*x^16 + 5/8*A*a^2*b^3*x^16 + 5/13*B* a^4*b*x^13 + 10/13*A*a^3*b^2*x^13 + 1/10*B*a^5*x^10 + 1/2*A*a^4*b*x^10 + 1 /7*A*a^5*x^7
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^6 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=x^{10}\,\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )+x^{22}\,\left (\frac {A\,b^5}{22}+\frac {5\,B\,a\,b^4}{22}\right )+\frac {A\,a^5\,x^7}{7}+\frac {B\,b^5\,x^{25}}{25}+\frac {5\,a^2\,b^2\,x^{16}\,\left (A\,b+B\,a\right )}{8}+\frac {5\,a^3\,b\,x^{13}\,\left (2\,A\,b+B\,a\right )}{13}+\frac {5\,a\,b^3\,x^{19}\,\left (A\,b+2\,B\,a\right )}{19} \]