Integrand size = 20, antiderivative size = 108 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=-\frac {a^5 A}{x}+a^4 (5 A b+a B) x+\frac {5}{3} a^3 b (2 A b+a B) x^3+2 a^2 b^2 (A b+a B) x^5+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{9} b^4 (A b+5 a B) x^9+\frac {1}{11} b^5 B x^{11} \]
-a^5*A/x+a^4*(5*A*b+B*a)*x+5/3*a^3*b*(2*A*b+B*a)*x^3+2*a^2*b^2*(A*b+B*a)*x ^5+5/7*a*b^3*(A*b+2*B*a)*x^7+1/9*b^4*(A*b+5*B*a)*x^9+1/11*b^5*B*x^11
Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=-\frac {a^5 A}{x}+a^4 (5 A b+a B) x+\frac {5}{3} a^3 b (2 A b+a B) x^3+2 a^2 b^2 (A b+a B) x^5+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{9} b^4 (A b+5 a B) x^9+\frac {1}{11} b^5 B x^{11} \]
-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2 *b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)* x^9)/9 + (b^5*B*x^11)/11
Time = 0.24 (sec) , antiderivative size = 108, 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 = {355, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (\frac {a^5 A}{x^2}+a^4 (a B+5 A b)+5 a^3 b x^2 (a B+2 A b)+10 a^2 b^2 x^4 (a B+A b)+b^4 x^8 (5 a B+A b)+5 a b^3 x^6 (2 a B+A b)+b^5 B x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{x}+a^4 x (a B+5 A b)+\frac {5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac {1}{9} b^4 x^9 (5 a B+A b)+\frac {5}{7} a b^3 x^7 (2 a B+A b)+\frac {1}{11} b^5 B x^{11}\) |
-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2 *b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)* x^9)/9 + (b^5*B*x^11)/11
3.1.34.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Time = 2.47 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {b^{5} B \,x^{11}}{11}+\frac {A \,b^{5} x^{9}}{9}+\frac {5 B a \,b^{4} x^{9}}{9}+\frac {5 A a \,b^{4} x^{7}}{7}+\frac {10 B \,a^{2} b^{3} x^{7}}{7}+2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {10 a^{3} A \,b^{2} x^{3}}{3}+\frac {5 B \,a^{4} b \,x^{3}}{3}+5 a^{4} A b x +a^{5} B x -\frac {a^{5} A}{x}\) | \(121\) |
norman | \(\frac {\frac {b^{5} B \,x^{12}}{11}+\left (\frac {1}{9} b^{5} A +\frac {5}{9} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{7} a \,b^{4} A +\frac {10}{7} a^{2} b^{3} B \right ) x^{8}+\left (2 a^{2} b^{3} A +2 a^{3} b^{2} B \right ) x^{6}+\left (\frac {10}{3} a^{3} b^{2} A +\frac {5}{3} a^{4} b B \right ) x^{4}+\left (5 a^{4} b A +a^{5} B \right ) x^{2}-a^{5} A}{x}\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{11}}{11}+\frac {A \,b^{5} x^{9}}{9}+\frac {5 B a \,b^{4} x^{9}}{9}+\frac {5 A a \,b^{4} x^{7}}{7}+\frac {10 B \,a^{2} b^{3} x^{7}}{7}+2 A \,a^{2} b^{3} x^{5}+2 B \,a^{3} b^{2} x^{5}+\frac {10 a^{3} A \,b^{2} x^{3}}{3}+\frac {5 B \,a^{4} b \,x^{3}}{3}+5 a^{4} A b x +a^{5} B x -\frac {a^{5} A}{x}\) | \(121\) |
gosper | \(-\frac {-63 b^{5} B \,x^{12}-77 A \,b^{5} x^{10}-385 B a \,b^{4} x^{10}-495 a A \,b^{4} x^{8}-990 B \,a^{2} b^{3} x^{8}-1386 a^{2} A \,b^{3} x^{6}-1386 B \,a^{3} b^{2} x^{6}-2310 a^{3} A \,b^{2} x^{4}-1155 B \,a^{4} b \,x^{4}-3465 a^{4} A b \,x^{2}-693 a^{5} B \,x^{2}+693 a^{5} A}{693 x}\) | \(128\) |
parallelrisch | \(\frac {63 b^{5} B \,x^{12}+77 A \,b^{5} x^{10}+385 B a \,b^{4} x^{10}+495 a A \,b^{4} x^{8}+990 B \,a^{2} b^{3} x^{8}+1386 a^{2} A \,b^{3} x^{6}+1386 B \,a^{3} b^{2} x^{6}+2310 a^{3} A \,b^{2} x^{4}+1155 B \,a^{4} b \,x^{4}+3465 a^{4} A b \,x^{2}+693 a^{5} B \,x^{2}-693 a^{5} A}{693 x}\) | \(128\) |
1/11*b^5*B*x^11+1/9*A*b^5*x^9+5/9*B*a*b^4*x^9+5/7*A*a*b^4*x^7+10/7*B*a^2*b ^3*x^7+2*A*a^2*b^3*x^5+2*B*a^3*b^2*x^5+10/3*a^3*A*b^2*x^3+5/3*B*a^4*b*x^3+ 5*a^4*A*b*x+a^5*B*x-a^5*A/x
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=\frac {63 \, B b^{5} x^{12} + 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 495 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 693 \, A a^{5} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x} \]
1/693*(63*B*b^5*x^12 + 77*(5*B*a*b^4 + A*b^5)*x^10 + 495*(2*B*a^2*b^3 + A* a*b^4)*x^8 + 1386*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 693*A*a^5 + 1155*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 693*(B*a^5 + 5*A*a^4*b)*x^2)/x
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=- \frac {A a^{5}}{x} + \frac {B b^{5} x^{11}}{11} + x^{9} \left (\frac {A b^{5}}{9} + \frac {5 B a b^{4}}{9}\right ) + x^{7} \cdot \left (\frac {5 A a b^{4}}{7} + \frac {10 B a^{2} b^{3}}{7}\right ) + x^{5} \cdot \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{3} \cdot \left (\frac {10 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{3}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]
-A*a**5/x + B*b**5*x**11/11 + x**9*(A*b**5/9 + 5*B*a*b**4/9) + x**7*(5*A*a *b**4/7 + 10*B*a**2*b**3/7) + x**5*(2*A*a**2*b**3 + 2*B*a**3*b**2) + x**3* (10*A*a**3*b**2/3 + 5*B*a**4*b/3) + x*(5*A*a**4*b + B*a**5)
Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{9} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} - \frac {A a^{5}}{x} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + {\left (B a^{5} + 5 \, A a^{4} b\right )} x \]
1/11*B*b^5*x^11 + 1/9*(5*B*a*b^4 + A*b^5)*x^9 + 5/7*(2*B*a^2*b^3 + A*a*b^4 )*x^7 + 2*(B*a^3*b^2 + A*a^2*b^3)*x^5 - A*a^5/x + 5/3*(B*a^4*b + 2*A*a^3*b ^2)*x^3 + (B*a^5 + 5*A*a^4*b)*x
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {5}{9} \, B a b^{4} x^{9} + \frac {1}{9} \, A b^{5} x^{9} + \frac {10}{7} \, B a^{2} b^{3} x^{7} + \frac {5}{7} \, A a b^{4} x^{7} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac {5}{3} \, B a^{4} b x^{3} + \frac {10}{3} \, A a^{3} b^{2} x^{3} + B a^{5} x + 5 \, A a^{4} b x - \frac {A a^{5}}{x} \]
1/11*B*b^5*x^11 + 5/9*B*a*b^4*x^9 + 1/9*A*b^5*x^9 + 10/7*B*a^2*b^3*x^7 + 5 /7*A*a*b^4*x^7 + 2*B*a^3*b^2*x^5 + 2*A*a^2*b^3*x^5 + 5/3*B*a^4*b*x^3 + 10/ 3*A*a^3*b^2*x^3 + B*a^5*x + 5*A*a^4*b*x - A*a^5/x
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx=x\,\left (B\,a^5+5\,A\,b\,a^4\right )+x^9\,\left (\frac {A\,b^5}{9}+\frac {5\,B\,a\,b^4}{9}\right )-\frac {A\,a^5}{x}+\frac {B\,b^5\,x^{11}}{11}+2\,a^2\,b^2\,x^5\,\left (A\,b+B\,a\right )+\frac {5\,a^3\,b\,x^3\,\left (2\,A\,b+B\,a\right )}{3}+\frac {5\,a\,b^3\,x^7\,\left (A\,b+2\,B\,a\right )}{7} \]