\(\int \frac {x^m}{(1-\frac {\sqrt {a} x}{\sqrt {-b}})^2 (1+\frac {\sqrt {a} x}{\sqrt {-b}})^2} \, dx\) [536]

Optimal result

Integrand size = 41, antiderivative size = 36 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {a x^2}{b}\right )}{1+m} \] Output:

x^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],-a*x^2/b)/(1+m)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {a x^2}{b}\right )}{1+m} \] Input:

Integrate[x^m/((1 - (Sqrt[a]*x)/Sqrt[-b])^2*(1 + (Sqrt[a]*x)/Sqrt[-b])^2), 
x]
 

Output:

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((a*x^2)/b)])/( 
1 + m)
 

Rubi [A] (verified)

Time = 0.16 (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

\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {a x^2}{b}\right )}{m+1}\)

Input:

Int[x^m/((1 - (Sqrt[a]*x)/Sqrt[-b])^2*(1 + (Sqrt[a]*x)/Sqrt[-b])^2),x]
 

Output:

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((a*x^2)/b)])/(1 + 
m)
 

Defintions of rubi rules used

rule 82

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]
 

rule 278

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])
 

Maple [F]

\[\int \frac {x^{m}}{\left (1-\frac {\sqrt {a}\, x}{\sqrt {-b}}\right )^{2} \left (1+\frac {\sqrt {a}\, x}{\sqrt {-b}}\right )^{2}}d x\]

Input:

int(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x)
 

Output:

int(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x)
 

Fricas [F]

\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \] Input:

integrate(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x, alg 
orithm="fricas")
 

Output:

integral(b^2*x^m/(a^2*x^4 + 2*a*b*x^2 + b^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.49 (sec) , antiderivative size = 552, normalized size of antiderivative = 15.33 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(x**m/(1-a**(1/2)*x/(-b)**(1/2))**2/(1+a**(1/2)*x/(-b)**(1/2))**2 
,x)
 

Output:

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))
 

Maxima [F]

\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \] Input:

integrate(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x, alg 
orithm="maxima")
 

Output:

integrate(x^m/((sqrt(a)*x/sqrt(-b) + 1)^2*(sqrt(a)*x/sqrt(-b) - 1)^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x, alg 
orithm="giac")
 

Output:

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] 
%%%}]%%},
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int \frac {x^m}{{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}-1\right )}^2\,{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}+1\right )}^2} \,d x \] Input:

int(x^m/(((a^(1/2)*x)/(-b)^(1/2) - 1)^2*((a^(1/2)*x)/(-b)^(1/2) + 1)^2),x)
 

Output:

int(x^m/(((a^(1/2)*x)/(-b)^(1/2) - 1)^2*((a^(1/2)*x)/(-b)^(1/2) + 1)^2), x 
)
 

Reduce [F]

\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\left (\int \frac {x^{m}}{a^{2} x^{4}+2 a b \,x^{2}+b^{2}}d x \right ) b^{2} \] Input:

int(x^m/(1-a^(1/2)*x/(-b)^(1/2))^2/(1+a^(1/2)*x/(-b)^(1/2))^2,x)
 

Output:

int(x**m/(a**2*x**4 + 2*a*b*x**2 + b**2),x)*b**2