Integrand size = 20, antiderivative size = 111 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{7} b^5 B x^7 \] Output:
-1/5*a^5*A/x^5-1/3*a^4*(5*A*b+B*a)/x^3-5*a^3*b*(2*A*b+B*a)/x+10*a^2*b^2*(A *b+B*a)*x+5/3*a*b^3*(A*b+2*B*a)*x^3+1/5*b^4*(A*b+5*B*a)*x^5+1/7*b^5*B*x^7
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{7} b^5 B x^7 \] Input:
Integrate[((a + b*x^2)^5*(A + B*x^2))/x^6,x]
Output:
-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(3*x^3) - (5*a^3*b*(2*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^7)/7
Time = 0.24 (sec) , antiderivative size = 111, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (\frac {a^5 A}{x^6}+\frac {a^4 (a B+5 A b)}{x^4}+\frac {5 a^3 b (a B+2 A b)}{x^2}+10 a^2 b^2 (a B+A b)+b^4 x^4 (5 a B+A b)+5 a b^3 x^2 (2 a B+A b)+b^5 B x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 A}{5 x^5}-\frac {a^4 (a B+5 A b)}{3 x^3}-\frac {5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 x (a B+A b)+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{3} a b^3 x^3 (2 a B+A b)+\frac {1}{7} b^5 B x^7\) |
Input:
Int[((a + b*x^2)^5*(A + B*x^2))/x^6,x]
Output:
-1/5*(a^5*A)/x^5 - (a^4*(5*A*b + a*B))/(3*x^3) - (5*a^3*b*(2*A*b + a*B))/x + 10*a^2*b^2*(A*b + a*B)*x + (5*a*b^3*(A*b + 2*a*B)*x^3)/3 + (b^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^7)/7
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 = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x -\frac {a^{4} \left (5 A b +B a \right )}{3 x^{3}}-\frac {a^{5} A}{5 x^{5}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{x}\) | \(113\) |
norman | \(\frac {\frac {b^{5} B \,x^{12}}{7}+\left (\frac {1}{5} b^{5} A +a \,b^{4} B \right ) x^{10}+\left (\frac {5}{3} a \,b^{4} A +\frac {10}{3} a^{2} b^{3} B \right ) x^{8}+\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{6}+\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{4}+\left (-\frac {5}{3} a^{4} b A -\frac {1}{3} a^{5} B \right ) x^{2}-\frac {a^{5} A}{5}}{x^{5}}\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x +\frac {\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{4}+\left (-\frac {5}{3} a^{4} b A -\frac {1}{3} a^{5} B \right ) x^{2}-\frac {a^{5} A}{5}}{x^{5}}\) | \(121\) |
gosper | \(-\frac {-15 b^{5} B \,x^{12}-21 A \,b^{5} x^{10}-105 B a \,b^{4} x^{10}-175 a A \,b^{4} x^{8}-350 B \,a^{2} b^{3} x^{8}-1050 a^{2} A \,b^{3} x^{6}-1050 B \,a^{3} b^{2} x^{6}+1050 a^{3} A \,b^{2} x^{4}+525 B \,a^{4} b \,x^{4}+175 a^{4} A b \,x^{2}+35 B \,a^{5} x^{2}+21 a^{5} A}{105 x^{5}}\) | \(128\) |
parallelrisch | \(\frac {15 b^{5} B \,x^{12}+21 A \,b^{5} x^{10}+105 B a \,b^{4} x^{10}+175 a A \,b^{4} x^{8}+350 B \,a^{2} b^{3} x^{8}+1050 a^{2} A \,b^{3} x^{6}+1050 B \,a^{3} b^{2} x^{6}-1050 a^{3} A \,b^{2} x^{4}-525 B \,a^{4} b \,x^{4}-175 a^{4} A b \,x^{2}-35 B \,a^{5} x^{2}-21 a^{5} A}{105 x^{5}}\) | \(128\) |
orering | \(-\frac {-15 b^{5} B \,x^{12}-21 A \,b^{5} x^{10}-105 B a \,b^{4} x^{10}-175 a A \,b^{4} x^{8}-350 B \,a^{2} b^{3} x^{8}-1050 a^{2} A \,b^{3} x^{6}-1050 B \,a^{3} b^{2} x^{6}+1050 a^{3} A \,b^{2} x^{4}+525 B \,a^{4} b \,x^{4}+175 a^{4} A b \,x^{2}+35 B \,a^{5} x^{2}+21 a^{5} A}{105 x^{5}}\) | \(128\) |
Input:
int((b*x^2+a)^5*(B*x^2+A)/x^6,x,method=_RETURNVERBOSE)
Output:
1/7*b^5*B*x^7+1/5*A*b^5*x^5+B*a*b^4*x^5+5/3*A*a*b^4*x^3+10/3*B*a^2*b^3*x^3 +10*A*a^2*b^3*x+10*B*a^3*b^2*x-1/3*a^4*(5*A*b+B*a)/x^3-1/5*a^5*A/x^5-5*a^3 *b*(2*A*b+B*a)/x
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {15 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 21 \, A a^{5} - 525 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 35 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{5}} \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="fricas")
Output:
1/105*(15*B*b^5*x^12 + 21*(5*B*a*b^4 + A*b^5)*x^10 + 175*(2*B*a^2*b^3 + A* a*b^4)*x^8 + 1050*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 21*A*a^5 - 525*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 35*(B*a^5 + 5*A*a^4*b)*x^2)/x^5
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {B b^{5} x^{7}}{7} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{3} \cdot \left (\frac {5 A a b^{4}}{3} + \frac {10 B a^{2} b^{3}}{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) + \frac {- 3 A a^{5} + x^{4} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x^{2} \left (- 25 A a^{4} b - 5 B a^{5}\right )}{15 x^{5}} \] Input:
integrate((b*x**2+a)**5*(B*x**2+A)/x**6,x)
Output:
B*b**5*x**7/7 + x**5*(A*b**5/5 + B*a*b**4) + x**3*(5*A*a*b**4/3 + 10*B*a** 2*b**3/3) + x*(10*A*a**2*b**3 + 10*B*a**3*b**2) + (-3*A*a**5 + x**4*(-150* A*a**3*b**2 - 75*B*a**4*b) + x**2*(-25*A*a**4*b - 5*B*a**5))/(15*x**5)
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {1}{7} \, B b^{5} x^{7} + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{3} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x - \frac {3 \, A a^{5} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 5 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{15 \, x^{5}} \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="maxima")
Output:
1/7*B*b^5*x^7 + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/3*(2*B*a^2*b^3 + A*a*b^4)* x^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*x - 1/15*(3*A*a^5 + 75*(B*a^4*b + 2*A*a^3 *b^2)*x^4 + 5*(B*a^5 + 5*A*a^4*b)*x^2)/x^5
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {1}{7} \, B b^{5} x^{7} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + \frac {10}{3} \, B a^{2} b^{3} x^{3} + \frac {5}{3} \, A a b^{4} x^{3} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x - \frac {75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 5 \, B a^{5} x^{2} + 25 \, A a^{4} b x^{2} + 3 \, A a^{5}}{15 \, x^{5}} \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="giac")
Output:
1/7*B*b^5*x^7 + B*a*b^4*x^5 + 1/5*A*b^5*x^5 + 10/3*B*a^2*b^3*x^3 + 5/3*A*a *b^4*x^3 + 10*B*a^3*b^2*x + 10*A*a^2*b^3*x - 1/15*(75*B*a^4*b*x^4 + 150*A* a^3*b^2*x^4 + 5*B*a^5*x^2 + 25*A*a^4*b*x^2 + 3*A*a^5)/x^5
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )-\frac {\frac {A\,a^5}{5}+x^4\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )+x^2\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )}{x^5}+\frac {B\,b^5\,x^7}{7}+10\,a^2\,b^2\,x\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^3\,\left (A\,b+2\,B\,a\right )}{3} \] Input:
int(((A + B*x^2)*(a + b*x^2)^5)/x^6,x)
Output:
x^5*((A*b^5)/5 + B*a*b^4) - ((A*a^5)/5 + x^4*(10*A*a^3*b^2 + 5*B*a^4*b) + x^2*((B*a^5)/3 + (5*A*a^4*b)/3))/x^5 + (B*b^5*x^7)/7 + 10*a^2*b^2*x*(A*b + B*a) + (5*a*b^3*x^3*(A*b + 2*B*a))/3
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {5 b^{6} x^{12}+42 a \,b^{5} x^{10}+175 a^{2} b^{4} x^{8}+700 a^{3} b^{3} x^{6}-525 a^{4} b^{2} x^{4}-70 a^{5} b \,x^{2}-7 a^{6}}{35 x^{5}} \] Input:
int((b*x^2+a)^5*(B*x^2+A)/x^6,x)
Output:
( - 7*a**6 - 70*a**5*b*x**2 - 525*a**4*b**2*x**4 + 700*a**3*b**3*x**6 + 17 5*a**2*b**4*x**8 + 42*a*b**5*x**10 + 5*b**6*x**12)/(35*x**5)