Integrand size = 41, antiderivative size = 36 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {a x^2}{b}\right )}{1+m} \]
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {a x^2}{b}\right )}{1+m} \]
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {82, 278}
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.
\(\displaystyle \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (\frac {\sqrt {a} x}{\sqrt {-b}}+1\right )^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {x^m}{\left (\frac {a x^2}{b}+1\right )^2}dx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {a x^2}{b}\right )}{m+1}\) |
3.4.86.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
\[\int \frac {x^{m}}{\left (1-\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2} \left (1+\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2}}d x\]
\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 3.66 (sec) , antiderivative size = 552, normalized size of antiderivative = 15.33 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {a b^{2} m^{2} x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {4 a b^{2} m x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {2 a b^{2} m x^{2} x^{m - 3} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {3 a b^{2} x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {6 a b^{2} x^{2} x^{m - 3} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {b^{3} m^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {4 b^{3} m x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {3 b^{3} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} \]
a*b**2*m**2*x**2*x**(m - 3)*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)) - 4*a*b**2*m*x**2*x**(m - 3)*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1 , 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gam ma(5/2 - m/2)) + 2*a*b**2*m*x**2*x**(m - 3)*gamma(3/2 - m/2)/(8*a**3*x**2* gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)) + 3*a*b**2*x**2*x**(m - 3)*l erchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3 *x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)) - 6*a*b**2*x**2*x**(m - 3)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)) + b**3*m**2*x**(m - 3)*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2)) - 4*b**3*m*x**(m - 3)*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/ 2 - m/2)) + 3*b**3*x**(m - 3)*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, 3/2 - m/2)*gamma(3/2 - m/2)/(8*a**3*x**2*gamma(5/2 - m/2) + 8*a**2*b*gamma(5/2 - m/2))
\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[1]%%%},[2]%%%}+%%%{%%{[%%%{%%{[-2,0]:[1,0,%%%{1,[1] %%%}]%%},
Timed out. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int \frac {x^m}{{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}-1\right )}^2\,{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}+1\right )}^2} \,d x \]