Integrand size = 20, antiderivative size = 117 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {10}{17} a^2 b^2 (A b+a B) x^{17}+\frac {1}{4} a b^3 (A b+2 a B) x^{20}+\frac {1}{23} b^4 (A b+5 a B) x^{23}+\frac {1}{26} b^5 B x^{26} \]
1/8*a^5*A*x^8+1/11*a^4*(5*A*b+B*a)*x^11+5/14*a^3*b*(2*A*b+B*a)*x^14+10/17* a^2*b^2*(A*b+B*a)*x^17+1/4*a*b^3*(A*b+2*B*a)*x^20+1/23*b^4*(A*b+5*B*a)*x^2 3+1/26*b^5*B*x^26
Time = 0.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {10}{17} a^2 b^2 (A b+a B) x^{17}+\frac {1}{4} a b^3 (A b+2 a B) x^{20}+\frac {1}{23} b^4 (A b+5 a B) x^{23}+\frac {1}{26} b^5 B x^{26} \]
(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14) /14 + (10*a^2*b^2*(A*b + a*B)*x^17)/17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b ^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26
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.
\(\displaystyle \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^5 A x^7+a^4 x^{10} (a B+5 A b)+5 a^3 b x^{13} (a B+2 A b)+10 a^2 b^2 x^{16} (a B+A b)+b^4 x^{22} (5 a B+A b)+5 a b^3 x^{19} (2 a B+A b)+b^5 B x^{25}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {10}{17} a^2 b^2 x^{17} (a B+A b)+\frac {1}{23} b^4 x^{23} (5 a B+A b)+\frac {1}{4} a b^3 x^{20} (2 a B+A b)+\frac {1}{26} b^5 B x^{26}\) |
(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14) /14 + (10*a^2*b^2*(A*b + a*B)*x^17)/17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b ^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26
3.1.25.3.1 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 = 4.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{8}}{8}+\left (\frac {5}{11} a^{4} b A +\frac {1}{11} a^{5} B \right ) x^{11}+\left (\frac {5}{7} a^{3} b^{2} A +\frac {5}{14} a^{4} b B \right ) x^{14}+\left (\frac {10}{17} a^{2} b^{3} A +\frac {10}{17} a^{3} b^{2} B \right ) x^{17}+\left (\frac {1}{4} a \,b^{4} A +\frac {1}{2} a^{2} b^{3} B \right ) x^{20}+\left (\frac {1}{23} b^{5} A +\frac {5}{23} a \,b^{4} B \right ) x^{23}+\frac {b^{5} B \,x^{26}}{26}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{26}}{26}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{23}}{23}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{20}}{20}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{17}}{17}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{14}}{14}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{11}}{11}+\frac {a^{5} A \,x^{8}}{8}\) | \(124\) |
gosper | \(\frac {1}{8} a^{5} A \,x^{8}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {10}{17} x^{17} a^{2} b^{3} A +\frac {10}{17} x^{17} a^{3} b^{2} B +\frac {1}{4} x^{20} a \,b^{4} A +\frac {1}{2} x^{20} a^{2} b^{3} B +\frac {1}{23} x^{23} b^{5} A +\frac {5}{23} x^{23} a \,b^{4} B +\frac {1}{26} b^{5} B \,x^{26}\) | \(126\) |
risch | \(\frac {1}{8} a^{5} A \,x^{8}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {10}{17} x^{17} a^{2} b^{3} A +\frac {10}{17} x^{17} a^{3} b^{2} B +\frac {1}{4} x^{20} a \,b^{4} A +\frac {1}{2} x^{20} a^{2} b^{3} B +\frac {1}{23} x^{23} b^{5} A +\frac {5}{23} x^{23} a \,b^{4} B +\frac {1}{26} b^{5} B \,x^{26}\) | \(126\) |
parallelrisch | \(\frac {1}{8} a^{5} A \,x^{8}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {10}{17} x^{17} a^{2} b^{3} A +\frac {10}{17} x^{17} a^{3} b^{2} B +\frac {1}{4} x^{20} a \,b^{4} A +\frac {1}{2} x^{20} a^{2} b^{3} B +\frac {1}{23} x^{23} b^{5} A +\frac {5}{23} x^{23} a \,b^{4} B +\frac {1}{26} b^{5} B \,x^{26}\) | \(126\) |
1/8*a^5*A*x^8+(5/11*a^4*b*A+1/11*a^5*B)*x^11+(5/7*a^3*b^2*A+5/14*a^4*b*B)* x^14+(10/17*a^2*b^3*A+10/17*a^3*b^2*B)*x^17+(1/4*a*b^4*A+1/2*a^2*b^3*B)*x^ 20+(1/23*b^5*A+5/23*a*b^4*B)*x^23+1/26*b^5*B*x^26
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{26} \, B b^{5} x^{26} + \frac {1}{23} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{23} + \frac {1}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{20} + \frac {10}{17} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{17} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{8} \, A a^{5} x^{8} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]
1/26*B*b^5*x^26 + 1/23*(5*B*a*b^4 + A*b^5)*x^23 + 1/4*(2*B*a^2*b^3 + A*a*b ^4)*x^20 + 10/17*(B*a^3*b^2 + A*a^2*b^3)*x^17 + 5/14*(B*a^4*b + 2*A*a^3*b^ 2)*x^14 + 1/8*A*a^5*x^8 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11
Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.15 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {A a^{5} x^{8}}{8} + \frac {B b^{5} x^{26}}{26} + x^{23} \left (\frac {A b^{5}}{23} + \frac {5 B a b^{4}}{23}\right ) + x^{20} \left (\frac {A a b^{4}}{4} + \frac {B a^{2} b^{3}}{2}\right ) + x^{17} \cdot \left (\frac {10 A a^{2} b^{3}}{17} + \frac {10 B a^{3} b^{2}}{17}\right ) + x^{14} \cdot \left (\frac {5 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{14}\right ) + x^{11} \cdot \left (\frac {5 A a^{4} b}{11} + \frac {B a^{5}}{11}\right ) \]
A*a**5*x**8/8 + B*b**5*x**26/26 + x**23*(A*b**5/23 + 5*B*a*b**4/23) + x**2 0*(A*a*b**4/4 + B*a**2*b**3/2) + x**17*(10*A*a**2*b**3/17 + 10*B*a**3*b**2 /17) + x**14*(5*A*a**3*b**2/7 + 5*B*a**4*b/14) + x**11*(5*A*a**4*b/11 + B* a**5/11)
Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{26} \, B b^{5} x^{26} + \frac {1}{23} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{23} + \frac {1}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{20} + \frac {10}{17} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{17} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{8} \, A a^{5} x^{8} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]
1/26*B*b^5*x^26 + 1/23*(5*B*a*b^4 + A*b^5)*x^23 + 1/4*(2*B*a^2*b^3 + A*a*b ^4)*x^20 + 10/17*(B*a^3*b^2 + A*a^2*b^3)*x^17 + 5/14*(B*a^4*b + 2*A*a^3*b^ 2)*x^14 + 1/8*A*a^5*x^8 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=\frac {1}{26} \, B b^{5} x^{26} + \frac {5}{23} \, B a b^{4} x^{23} + \frac {1}{23} \, A b^{5} x^{23} + \frac {1}{2} \, B a^{2} b^{3} x^{20} + \frac {1}{4} \, A a b^{4} x^{20} + \frac {10}{17} \, B a^{3} b^{2} x^{17} + \frac {10}{17} \, A a^{2} b^{3} x^{17} + \frac {5}{14} \, B a^{4} b x^{14} + \frac {5}{7} \, A a^{3} b^{2} x^{14} + \frac {1}{11} \, B a^{5} x^{11} + \frac {5}{11} \, A a^{4} b x^{11} + \frac {1}{8} \, A a^{5} x^{8} \]
1/26*B*b^5*x^26 + 5/23*B*a*b^4*x^23 + 1/23*A*b^5*x^23 + 1/2*B*a^2*b^3*x^20 + 1/4*A*a*b^4*x^20 + 10/17*B*a^3*b^2*x^17 + 10/17*A*a^2*b^3*x^17 + 5/14*B *a^4*b*x^14 + 5/7*A*a^3*b^2*x^14 + 1/11*B*a^5*x^11 + 5/11*A*a^4*b*x^11 + 1 /8*A*a^5*x^8
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^7 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx=x^{11}\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^{23}\,\left (\frac {A\,b^5}{23}+\frac {5\,B\,a\,b^4}{23}\right )+\frac {A\,a^5\,x^8}{8}+\frac {B\,b^5\,x^{26}}{26}+\frac {10\,a^2\,b^2\,x^{17}\,\left (A\,b+B\,a\right )}{17}+\frac {5\,a^3\,b\,x^{14}\,\left (2\,A\,b+B\,a\right )}{14}+\frac {a\,b^3\,x^{20}\,\left (A\,b+2\,B\,a\right )}{4} \]