Integrand size = 20, antiderivative size = 117 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{8} a^4 (5 A b+a B) x^8+\frac {5}{11} a^3 b (2 A b+a B) x^{11}+\frac {5}{7} a^2 b^2 (A b+a B) x^{14}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{23} b^5 B x^{23} \]
1/5*a^5*A*x^5+1/8*a^4*(5*A*b+B*a)*x^8+5/11*a^3*b*(2*A*b+B*a)*x^11+5/7*a^2* b^2*(A*b+B*a)*x^14+5/17*a*b^3*(A*b+2*B*a)*x^17+1/20*b^4*(A*b+5*B*a)*x^20+1 /23*b^5*B*x^23
Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{8} a^4 (5 A b+a B) x^8+\frac {5}{11} a^3 b (2 A b+a B) x^{11}+\frac {5}{7} a^2 b^2 (A b+a B) x^{14}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{23} b^5 B x^{23} \]
(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^8)/8 + (5*a^3*b*(2*A*b + a*B)*x^11)/1 1 + (5*a^2*b^2*(A*b + a*B)*x^14)/7 + (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^ 4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^23)/23
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.
\(\displaystyle \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^5 A x^4+a^4 x^7 (a B+5 A b)+5 a^3 b x^{10} (a B+2 A b)+10 a^2 b^2 x^{13} (a B+A b)+b^4 x^{19} (5 a B+A b)+5 a b^3 x^{16} (2 a B+A b)+b^5 B x^{22}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} a^5 A x^5+\frac {1}{8} a^4 x^8 (a B+5 A b)+\frac {5}{11} a^3 b x^{11} (a B+2 A b)+\frac {5}{7} a^2 b^2 x^{14} (a B+A b)+\frac {1}{20} b^4 x^{20} (5 a B+A b)+\frac {5}{17} a b^3 x^{17} (2 a B+A b)+\frac {1}{23} b^5 B x^{23}\) |
(a^5*A*x^5)/5 + (a^4*(5*A*b + a*B)*x^8)/8 + (5*a^3*b*(2*A*b + a*B)*x^11)/1 1 + (5*a^2*b^2*(A*b + a*B)*x^14)/7 + (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^ 4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^23)/23
3.1.28.3.1 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 = 4.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{5}}{5}+\left (\frac {5}{8} a^{4} b A +\frac {1}{8} a^{5} B \right ) x^{8}+\left (\frac {10}{11} a^{3} b^{2} A +\frac {5}{11} a^{4} b B \right ) x^{11}+\left (\frac {5}{7} a^{2} b^{3} A +\frac {5}{7} a^{3} b^{2} B \right ) x^{14}+\left (\frac {5}{17} a \,b^{4} A +\frac {10}{17} a^{2} b^{3} B \right ) x^{17}+\left (\frac {1}{20} b^{5} A +\frac {1}{4} a \,b^{4} B \right ) x^{20}+\frac {b^{5} B \,x^{23}}{23}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{23}}{23}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{20}}{20}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{17}}{17}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{14}}{14}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{11}}{11}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{8}}{8}+\frac {a^{5} A \,x^{5}}{5}\) | \(124\) |
gosper | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +\frac {10}{11} x^{11} a^{3} b^{2} A +\frac {5}{11} x^{11} a^{4} b B +\frac {5}{7} x^{14} a^{2} b^{3} A +\frac {5}{7} x^{14} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{23} b^{5} B \,x^{23}\) | \(126\) |
risch | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +\frac {10}{11} x^{11} a^{3} b^{2} A +\frac {5}{11} x^{11} a^{4} b B +\frac {5}{7} x^{14} a^{2} b^{3} A +\frac {5}{7} x^{14} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{23} b^{5} B \,x^{23}\) | \(126\) |
parallelrisch | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +\frac {10}{11} x^{11} a^{3} b^{2} A +\frac {5}{11} x^{11} a^{4} b B +\frac {5}{7} x^{14} a^{2} b^{3} A +\frac {5}{7} x^{14} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{23} b^{5} B \,x^{23}\) | \(126\) |
1/5*a^5*A*x^5+(5/8*a^4*b*A+1/8*a^5*B)*x^8+(10/11*a^3*b^2*A+5/11*a^4*b*B)*x ^11+(5/7*a^2*b^3*A+5/7*a^3*b^2*B)*x^14+(5/17*a*b^4*A+10/17*a^2*b^3*B)*x^17 +(1/20*b^5*A+1/4*a*b^4*B)*x^20+1/23*b^5*B*x^23
Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{23} \, B b^{5} x^{23} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{17} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac {5}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{14} + \frac {5}{11} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \]
1/23*B*b^5*x^23 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/17*(2*B*a^2*b^3 + A*a* b^4)*x^17 + 5/7*(B*a^3*b^2 + A*a^2*b^3)*x^14 + 5/11*(B*a^4*b + 2*A*a^3*b^2 )*x^11 + 1/5*A*a^5*x^5 + 1/8*(B*a^5 + 5*A*a^4*b)*x^8
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {A a^{5} x^{5}}{5} + \frac {B b^{5} x^{23}}{23} + x^{20} \left (\frac {A b^{5}}{20} + \frac {B a b^{4}}{4}\right ) + x^{17} \cdot \left (\frac {5 A a b^{4}}{17} + \frac {10 B a^{2} b^{3}}{17}\right ) + x^{14} \cdot \left (\frac {5 A a^{2} b^{3}}{7} + \frac {5 B a^{3} b^{2}}{7}\right ) + x^{11} \cdot \left (\frac {10 A a^{3} b^{2}}{11} + \frac {5 B a^{4} b}{11}\right ) + x^{8} \cdot \left (\frac {5 A a^{4} b}{8} + \frac {B a^{5}}{8}\right ) \]
A*a**5*x**5/5 + B*b**5*x**23/23 + x**20*(A*b**5/20 + B*a*b**4/4) + x**17*( 5*A*a*b**4/17 + 10*B*a**2*b**3/17) + x**14*(5*A*a**2*b**3/7 + 5*B*a**3*b** 2/7) + x**11*(10*A*a**3*b**2/11 + 5*B*a**4*b/11) + x**8*(5*A*a**4*b/8 + B* a**5/8)
Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{23} \, B b^{5} x^{23} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{17} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac {5}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{14} + \frac {5}{11} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \]
1/23*B*b^5*x^23 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/17*(2*B*a^2*b^3 + A*a* b^4)*x^17 + 5/7*(B*a^3*b^2 + A*a^2*b^3)*x^14 + 5/11*(B*a^4*b + 2*A*a^3*b^2 )*x^11 + 1/5*A*a^5*x^5 + 1/8*(B*a^5 + 5*A*a^4*b)*x^8
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{23} \, B b^{5} x^{23} + \frac {1}{4} \, B a b^{4} x^{20} + \frac {1}{20} \, A b^{5} x^{20} + \frac {10}{17} \, B a^{2} b^{3} x^{17} + \frac {5}{17} \, A a b^{4} x^{17} + \frac {5}{7} \, B a^{3} b^{2} x^{14} + \frac {5}{7} \, A a^{2} b^{3} x^{14} + \frac {5}{11} \, B a^{4} b x^{11} + \frac {10}{11} \, A a^{3} b^{2} x^{11} + \frac {1}{8} \, B a^{5} x^{8} + \frac {5}{8} \, A a^{4} b x^{8} + \frac {1}{5} \, A a^{5} x^{5} \]
1/23*B*b^5*x^23 + 1/4*B*a*b^4*x^20 + 1/20*A*b^5*x^20 + 10/17*B*a^2*b^3*x^1 7 + 5/17*A*a*b^4*x^17 + 5/7*B*a^3*b^2*x^14 + 5/7*A*a^2*b^3*x^14 + 5/11*B*a ^4*b*x^11 + 10/11*A*a^3*b^2*x^11 + 1/8*B*a^5*x^8 + 5/8*A*a^4*b*x^8 + 1/5*A *a^5*x^5
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^4 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=x^8\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^{20}\,\left (\frac {A\,b^5}{20}+\frac {B\,a\,b^4}{4}\right )+\frac {A\,a^5\,x^5}{5}+\frac {B\,b^5\,x^{23}}{23}+\frac {5\,a^2\,b^2\,x^{14}\,\left (A\,b+B\,a\right )}{7}+\frac {5\,a^3\,b\,x^{11}\,\left (2\,A\,b+B\,a\right )}{11}+\frac {5\,a\,b^3\,x^{17}\,\left (A\,b+2\,B\,a\right )}{17} \]