Integrand size = 20, antiderivative size = 117 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{4} a^5 A x^4+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {1}{2} a^3 b (2 A b+a B) x^{10}+\frac {10}{13} a^2 b^2 (A b+a B) x^{13}+\frac {5}{16} a b^3 (A b+2 a B) x^{16}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{22} b^5 B x^{22} \]
1/4*a^5*A*x^4+1/7*a^4*(5*A*b+B*a)*x^7+1/2*a^3*b*(2*A*b+B*a)*x^10+10/13*a^2 *b^2*(A*b+B*a)*x^13+5/16*a*b^3*(A*b+2*B*a)*x^16+1/19*b^4*(A*b+5*B*a)*x^19+ 1/22*b^5*B*x^22
Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{4} a^5 A x^4+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {1}{2} a^3 b (2 A b+a B) x^{10}+\frac {10}{13} a^2 b^2 (A b+a B) x^{13}+\frac {5}{16} a b^3 (A b+2 a B) x^{16}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{22} b^5 B x^{22} \]
(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4 *(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^22)/22
Time = 0.26 (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.
\(\displaystyle \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^5 A x^3+a^4 x^6 (a B+5 A b)+5 a^3 b x^9 (a B+2 A b)+10 a^2 b^2 x^{12} (a B+A b)+b^4 x^{18} (5 a B+A b)+5 a b^3 x^{15} (2 a B+A b)+b^5 B x^{21}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} a^5 A x^4+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {1}{2} a^3 b x^{10} (a B+2 A b)+\frac {10}{13} a^2 b^2 x^{13} (a B+A b)+\frac {1}{19} b^4 x^{19} (5 a B+A b)+\frac {5}{16} a b^3 x^{16} (2 a B+A b)+\frac {1}{22} b^5 B x^{22}\) |
(a^5*A*x^4)/4 + (a^4*(5*A*b + a*B)*x^7)/7 + (a^3*b*(2*A*b + a*B)*x^10)/2 + (10*a^2*b^2*(A*b + a*B)*x^13)/13 + (5*a*b^3*(A*b + 2*a*B)*x^16)/16 + (b^4 *(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^22)/22
3.1.29.3.1 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 = 4.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{4}}{4}+\left (\frac {5}{7} a^{4} b A +\frac {1}{7} a^{5} B \right ) x^{7}+\left (a^{3} b^{2} A +\frac {1}{2} a^{4} b B \right ) x^{10}+\left (\frac {10}{13} a^{2} b^{3} A +\frac {10}{13} a^{3} b^{2} B \right ) x^{13}+\left (\frac {5}{16} a \,b^{4} A +\frac {5}{8} a^{2} b^{3} B \right ) x^{16}+\left (\frac {1}{19} b^{5} A +\frac {5}{19} a \,b^{4} B \right ) x^{19}+\frac {b^{5} B \,x^{22}}{22}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{22}}{22}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{19}}{19}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{16}}{16}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{13}}{13}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{10}}{10}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{7}}{7}+\frac {a^{5} A \,x^{4}}{4}\) | \(124\) |
gosper | \(\frac {1}{4} a^{5} A \,x^{4}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {10}{13} x^{13} a^{2} b^{3} A +\frac {10}{13} x^{13} a^{3} b^{2} B +\frac {5}{16} x^{16} a \,b^{4} A +\frac {5}{8} x^{16} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(125\) |
risch | \(\frac {1}{4} a^{5} A \,x^{4}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {10}{13} x^{13} a^{2} b^{3} A +\frac {10}{13} x^{13} a^{3} b^{2} B +\frac {5}{16} x^{16} a \,b^{4} A +\frac {5}{8} x^{16} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(125\) |
parallelrisch | \(\frac {1}{4} a^{5} A \,x^{4}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {10}{13} x^{13} a^{2} b^{3} A +\frac {10}{13} x^{13} a^{3} b^{2} B +\frac {5}{16} x^{16} a \,b^{4} A +\frac {5}{8} x^{16} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(125\) |
1/4*a^5*A*x^4+(5/7*a^4*b*A+1/7*a^5*B)*x^7+(a^3*b^2*A+1/2*a^4*b*B)*x^10+(10 /13*a^2*b^3*A+10/13*a^3*b^2*B)*x^13+(5/16*a*b^4*A+5/8*a^2*b^3*B)*x^16+(1/1 9*b^5*A+5/19*a*b^4*B)*x^19+1/22*b^5*B*x^22
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{19} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac {5}{16} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{16} + \frac {10}{13} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{13} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
1/22*B*b^5*x^22 + 1/19*(5*B*a*b^4 + A*b^5)*x^19 + 5/16*(2*B*a^2*b^3 + A*a* b^4)*x^16 + 10/13*(B*a^3*b^2 + A*a^2*b^3)*x^13 + 1/2*(B*a^4*b + 2*A*a^3*b^ 2)*x^10 + 1/4*A*a^5*x^4 + 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 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {A a^{5} x^{4}}{4} + \frac {B b^{5} x^{22}}{22} + x^{19} \left (\frac {A b^{5}}{19} + \frac {5 B a b^{4}}{19}\right ) + x^{16} \cdot \left (\frac {5 A a b^{4}}{16} + \frac {5 B a^{2} b^{3}}{8}\right ) + x^{13} \cdot \left (\frac {10 A a^{2} b^{3}}{13} + \frac {10 B a^{3} b^{2}}{13}\right ) + x^{10} \left (A a^{3} b^{2} + \frac {B a^{4} b}{2}\right ) + x^{7} \cdot \left (\frac {5 A a^{4} b}{7} + \frac {B a^{5}}{7}\right ) \]
A*a**5*x**4/4 + B*b**5*x**22/22 + x**19*(A*b**5/19 + 5*B*a*b**4/19) + x**1 6*(5*A*a*b**4/16 + 5*B*a**2*b**3/8) + x**13*(10*A*a**2*b**3/13 + 10*B*a**3 *b**2/13) + x**10*(A*a**3*b**2 + B*a**4*b/2) + x**7*(5*A*a**4*b/7 + B*a**5 /7)
Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{19} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac {5}{16} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{16} + \frac {10}{13} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{13} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
1/22*B*b^5*x^22 + 1/19*(5*B*a*b^4 + A*b^5)*x^19 + 5/16*(2*B*a^2*b^3 + A*a* b^4)*x^16 + 10/13*(B*a^3*b^2 + A*a^2*b^3)*x^13 + 1/2*(B*a^4*b + 2*A*a^3*b^ 2)*x^10 + 1/4*A*a^5*x^4 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {5}{19} \, B a b^{4} x^{19} + \frac {1}{19} \, A b^{5} x^{19} + \frac {5}{8} \, B a^{2} b^{3} x^{16} + \frac {5}{16} \, A a b^{4} x^{16} + \frac {10}{13} \, B a^{3} b^{2} x^{13} + \frac {10}{13} \, A a^{2} b^{3} x^{13} + \frac {1}{2} \, B a^{4} b x^{10} + A a^{3} b^{2} x^{10} + \frac {1}{7} \, B a^{5} x^{7} + \frac {5}{7} \, A a^{4} b x^{7} + \frac {1}{4} \, A a^{5} x^{4} \]
1/22*B*b^5*x^22 + 5/19*B*a*b^4*x^19 + 1/19*A*b^5*x^19 + 5/8*B*a^2*b^3*x^16 + 5/16*A*a*b^4*x^16 + 10/13*B*a^3*b^2*x^13 + 10/13*A*a^2*b^3*x^13 + 1/2*B *a^4*b*x^10 + A*a^3*b^2*x^10 + 1/7*B*a^5*x^7 + 5/7*A*a^4*b*x^7 + 1/4*A*a^5 *x^4
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=x^7\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^{19}\,\left (\frac {A\,b^5}{19}+\frac {5\,B\,a\,b^4}{19}\right )+\frac {A\,a^5\,x^4}{4}+\frac {B\,b^5\,x^{22}}{22}+\frac {10\,a^2\,b^2\,x^{13}\,\left (A\,b+B\,a\right )}{13}+\frac {a^3\,b\,x^{10}\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^{16}\,\left (A\,b+2\,B\,a\right )}{16} \]