Integrand size = 20, antiderivative size = 117 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \]
1/10*a^5*A*x^10+1/12*a^4*(5*A*b+B*a)*x^12+5/14*a^3*b*(2*A*b+B*a)*x^14+5/8* a^2*b^2*(A*b+B*a)*x^16+5/18*a*b^3*(A*b+2*B*a)*x^18+1/20*b^4*(A*b+5*B*a)*x^ 20+1/22*b^5*B*x^22
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \]
(a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^12)/12 + (5*a^3*b*(2*A*b + a*B)*x^1 4)/14 + (5*a^2*b^2*(A*b + a*B)*x^16)/8 + (5*a*b^3*(A*b + 2*a*B)*x^18)/18 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^22)/22
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {354, 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.
\(\displaystyle \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int x^8 \left (b x^2+a\right )^5 \left (B x^2+A\right )dx^2\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \frac {1}{2} \int \left (b^5 B x^{20}+b^4 (A b+5 a B) x^{18}+5 a b^3 (A b+2 a B) x^{16}+10 a^2 b^2 (A b+a B) x^{14}+5 a^3 b (2 A b+a B) x^{12}+a^4 (5 A b+a B) x^{10}+a^5 A x^8\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{5} a^5 A x^{10}+\frac {1}{6} a^4 x^{12} (a B+5 A b)+\frac {5}{7} a^3 b x^{14} (a B+2 A b)+\frac {5}{4} a^2 b^2 x^{16} (a B+A b)+\frac {1}{10} b^4 x^{20} (5 a B+A b)+\frac {5}{9} a b^3 x^{18} (2 a B+A b)+\frac {1}{11} b^5 B x^{22}\right )\) |
((a^5*A*x^10)/5 + (a^4*(5*A*b + a*B)*x^12)/6 + (5*a^3*b*(2*A*b + a*B)*x^14 )/7 + (5*a^2*b^2*(A*b + a*B)*x^16)/4 + (5*a*b^3*(A*b + 2*a*B)*x^18)/9 + (b ^4*(A*b + 5*a*B)*x^20)/10 + (b^5*B*x^22)/11)/2
3.1.23.3.1 Defintions of rubi rules used
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{10}}{10}+\left (\frac {5}{12} a^{4} b A +\frac {1}{12} a^{5} B \right ) x^{12}+\left (\frac {5}{7} a^{3} b^{2} A +\frac {5}{14} a^{4} b B \right ) x^{14}+\left (\frac {5}{8} a^{2} b^{3} A +\frac {5}{8} a^{3} b^{2} B \right ) x^{16}+\left (\frac {5}{18} a \,b^{4} A +\frac {5}{9} a^{2} b^{3} B \right ) x^{18}+\left (\frac {1}{20} b^{5} A +\frac {1}{4} a \,b^{4} B \right ) x^{20}+\frac {b^{5} B \,x^{22}}{22}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{22}}{22}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{20}}{20}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{18}}{18}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{16}}{16}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{14}}{14}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{12}}{12}+\frac {a^{5} A \,x^{10}}{10}\) | \(124\) |
gosper | \(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(126\) |
risch | \(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(126\) |
parallelrisch | \(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(126\) |
1/10*a^5*A*x^10+(5/12*a^4*b*A+1/12*a^5*B)*x^12+(5/7*a^3*b^2*A+5/14*a^4*b*B )*x^14+(5/8*a^2*b^3*A+5/8*a^3*b^2*B)*x^16+(5/18*a*b^4*A+5/9*a^2*b^3*B)*x^1 8+(1/20*b^5*A+1/4*a*b^4*B)*x^20+1/22*b^5*B*x^22
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \]
1/22*B*b^5*x^22 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/18*(2*B*a^2*b^3 + A*a* b^4)*x^18 + 5/8*(B*a^3*b^2 + A*a^2*b^3)*x^16 + 1/10*A*a^5*x^10 + 5/14*(B*a ^4*b + 2*A*a^3*b^2)*x^14 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{10}}{10} + \frac {B b^{5} x^{22}}{22} + x^{20} \left (\frac {A b^{5}}{20} + \frac {B a b^{4}}{4}\right ) + x^{18} \cdot \left (\frac {5 A a b^{4}}{18} + \frac {5 B a^{2} b^{3}}{9}\right ) + x^{16} \cdot \left (\frac {5 A a^{2} b^{3}}{8} + \frac {5 B a^{3} b^{2}}{8}\right ) + x^{14} \cdot \left (\frac {5 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{14}\right ) + x^{12} \cdot \left (\frac {5 A a^{4} b}{12} + \frac {B a^{5}}{12}\right ) \]
A*a**5*x**10/10 + B*b**5*x**22/22 + x**20*(A*b**5/20 + B*a*b**4/4) + x**18 *(5*A*a*b**4/18 + 5*B*a**2*b**3/9) + x**16*(5*A*a**2*b**3/8 + 5*B*a**3*b** 2/8) + x**14*(5*A*a**3*b**2/7 + 5*B*a**4*b/14) + x**12*(5*A*a**4*b/12 + B* a**5/12)
Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \]
1/22*B*b^5*x^22 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/18*(2*B*a^2*b^3 + A*a* b^4)*x^18 + 5/8*(B*a^3*b^2 + A*a^2*b^3)*x^16 + 1/10*A*a^5*x^10 + 5/14*(B*a ^4*b + 2*A*a^3*b^2)*x^14 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12
Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{4} \, B a b^{4} x^{20} + \frac {1}{20} \, A b^{5} x^{20} + \frac {5}{9} \, B a^{2} b^{3} x^{18} + \frac {5}{18} \, A a b^{4} x^{18} + \frac {5}{8} \, B a^{3} b^{2} x^{16} + \frac {5}{8} \, A a^{2} b^{3} x^{16} + \frac {5}{14} \, B a^{4} b x^{14} + \frac {5}{7} \, A a^{3} b^{2} x^{14} + \frac {1}{12} \, B a^{5} x^{12} + \frac {5}{12} \, A a^{4} b x^{12} + \frac {1}{10} \, A a^{5} x^{10} \]
1/22*B*b^5*x^22 + 1/4*B*a*b^4*x^20 + 1/20*A*b^5*x^20 + 5/9*B*a^2*b^3*x^18 + 5/18*A*a*b^4*x^18 + 5/8*B*a^3*b^2*x^16 + 5/8*A*a^2*b^3*x^16 + 5/14*B*a^4 *b*x^14 + 5/7*A*a^3*b^2*x^14 + 1/12*B*a^5*x^12 + 5/12*A*a^4*b*x^12 + 1/10* A*a^5*x^10
Time = 4.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^{12}\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{20}\,\left (\frac {A\,b^5}{20}+\frac {B\,a\,b^4}{4}\right )+\frac {A\,a^5\,x^{10}}{10}+\frac {B\,b^5\,x^{22}}{22}+\frac {5\,a^2\,b^2\,x^{16}\,\left (A\,b+B\,a\right )}{8}+\frac {5\,a^3\,b\,x^{14}\,\left (2\,A\,b+B\,a\right )}{14}+\frac {5\,a\,b^3\,x^{18}\,\left (A\,b+2\,B\,a\right )}{18} \]