\(\int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx\) [354]

Optimal result

Integrand size = 30, antiderivative size = 71 \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=-\frac {(c x)^{-1-2 p} \left (1-x^2\right )^{\frac {1}{2}+p}}{c (1+2 p)}-\frac {(c x)^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,1-p,x^2\right )}{2 c^2 p} \] Output:

-(c*x)^(-1-2*p)*(-x^2+1)^(1/2+p)/c/(1+2*p)-1/2*hypergeom([-p, 1/2-p],[1-p] 
,x^2)/c^2/p/((c*x)^(2*p))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.15 \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=-\frac {4^{1+p} (1-x)^{\frac {1}{2}+p} (c x)^{-2 p} \left (\frac {x}{1+x}\right )^{2 p} (1+x)^{-\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+p,2+2 p,\frac {3}{2}+p,\frac {1-x}{1+x}\right )}{c^2 (1+2 p)} \] Input:

Integrate[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]
 

Output:

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*p)*(1 + x)^(-1/2 + p)*Hyperg 
eometric2F1[1/2 + p, 2 + 2*p, 3/2 + p, (1 - x)/(1 + x)])/(c^2*(1 + 2*p)*(c 
*x)^(2*p)))
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {142}

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.

\(\displaystyle \int (1-x)^{p-\frac {1}{2}} (x+1)^{p+\frac {1}{2}} (c x)^{-2 (p+1)} \, dx\)

\(\Big \downarrow \) 142

\(\displaystyle -\frac {4^{p+1} (1-x)^{p+\frac {1}{2}} \left (\frac {x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac {3}{2}} (c x)^{-2 (p+1)} \operatorname {Hypergeometric2F1}\left (p+\frac {1}{2},2 (p+1),p+\frac {3}{2},\frac {1-x}{x+1}\right )}{2 p+1}\)

Input:

Int[((1 - x)^(-1/2 + p)*(1 + x)^(1/2 + p))/(c*x)^(2*(1 + p)),x]
 

Output:

-((4^(1 + p)*(1 - x)^(1/2 + p)*(x/(1 + x))^(2*(1 + p))*(1 + x)^(3/2 + p)*H 
ypergeometric2F1[1/2 + p, 2*(1 + p), 3/2 + p, (1 - x)/(1 + x)])/((1 + 2*p) 
*(c*x)^(2*(1 + p))))
 

Defintions of rubi rules used

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \left (1-x \right )^{-\frac {1}{2}+p} \left (1+x \right )^{\frac {1}{2}+p} \left (c x \right )^{-2 p -2}d x\]

Input:

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x)
 

Output:

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x)
 

Fricas [F]

\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \] Input:

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x, algorithm="frica 
s")
 

Output:

integral((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)
 

Sympy [F]

\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int \left (c x\right )^{- 2 p - 2} \left (1 - x\right )^{p - \frac {1}{2}} \left (x + 1\right )^{p + \frac {1}{2}}\, dx \] Input:

integrate((1-x)**(-1/2+p)*(1+x)**(1/2+p)/((c*x)**(2*p+2)),x)
 

Output:

Integral((c*x)**(-2*p - 2)*(1 - x)**(p - 1/2)*(x + 1)**(p + 1/2), x)
 

Maxima [F]

\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \] Input:

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x, algorithm="maxim 
a")
 

Output:

integrate((c*x)^(-2*p - 2)*(x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2), x)
 

Giac [F]

\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \] Input:

integrate((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x, algorithm="giac" 
)
 

Output:

integrate((x + 1)^(p + 1/2)*(-x + 1)^(p - 1/2)/(c*x)^(2*p + 2), x)
 

Mupad [F(-1)]

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

int(((1 - x)^(p - 1/2)*(x + 1)^(p + 1/2))/(c*x)^(2*p + 2),x)
 

Output:

int(((1 - x)^(p - 1/2)*(x + 1)^(p + 1/2))/(c*x)^(2*p + 2), x)
 

Reduce [F]

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

int((1-x)^(-1/2+p)*(1+x)^(1/2+p)/((c*x)^(2*p+2)),x)
 

Output:

int(((x + 1)**((2*p + 1)/2)*( - x + 1)**p)/(x**(2*p)*sqrt( - x + 1)*x**2), 
x)/(c**(2*p)*c**2)