\(\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx\) [336]

Optimal result

Integrand size = 21, antiderivative size = 86 \[ \int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\frac {(1-a x)^{-n} (1+a x)^n \operatorname {Hypergeometric2F1}\left (1,-n,1-n,\frac {1-a x}{1+a x}\right )}{n}-\frac {2^n (1-a x)^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )}{n} \] Output:

(a*x+1)^n*hypergeom([1, -n],[1-n],(-a*x+1)/(a*x+1))/n/((-a*x+1)^n)-2^n*hyp 
ergeom([-n, -n],[1-n],-1/2*a*x+1/2)/n/((-a*x+1)^n)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\frac {(1-a x)^{-n} \left ((1+a x)^n \operatorname {Hypergeometric2F1}\left (1,-n,1-n,\frac {1-a x}{1+a x}\right )-2^n \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )\right )}{n} \] Input:

Integrate[(1 + a*x)^n/(x*(1 - a*x)^n),x]
 

Output:

((1 + a*x)^n*Hypergeometric2F1[1, -n, 1 - n, (1 - a*x)/(1 + a*x)] - 2^n*Hy 
pergeometric2F1[-n, -n, 1 - n, (1 - a*x)/2])/(n*(1 - a*x)^n)
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {140, 79, 141}

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.

\(\displaystyle \int \frac {(1-a x)^{-n} (a x+1)^n}{x} \, dx\)

\(\Big \downarrow \) 140

\(\displaystyle \int \frac {(1-a x)^{-n-1} (a x+1)^n}{x}dx-a \int (1-a x)^{-n-1} (a x+1)^ndx\)

\(\Big \downarrow \) 79

\(\displaystyle \int \frac {(1-a x)^{-n-1} (a x+1)^n}{x}dx-\frac {2^n (1-a x)^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )}{n}\)

\(\Big \downarrow \) 141

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

Input:

Int[(1 + a*x)^n/(x*(1 - a*x)^n),x]
 

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 140

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

rule 141

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

Maple [F]

\[\int \frac {\left (a x +1\right )^{n} \left (-a x +1\right )^{-n}}{x}d x\]

Input:

int((a*x+1)^n/x/((-a*x+1)^n),x)
 

Output:

int((a*x+1)^n/x/((-a*x+1)^n),x)
 

Fricas [F]

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

integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="fricas")
 

Output:

integral((a*x + 1)^n/((-a*x + 1)^n*x), x)
 

Sympy [F]

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

integrate((a*x+1)**n/x/((-a*x+1)**n),x)
 

Output:

Integral((a*x + 1)**n/(x*(-a*x + 1)**n), x)
 

Maxima [F]

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

integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="maxima")
 

Output:

integrate((a*x + 1)^n/((-a*x + 1)^n*x), x)
 

Giac [F]

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

integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="giac")
 

Output:

integrate((a*x + 1)^n/((-a*x + 1)^n*x), x)
 

Mupad [F(-1)]

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

int((a*x + 1)^n/(x*(1 - a*x)^n),x)
 

Output:

int((a*x + 1)^n/(x*(1 - a*x)^n), x)
 

Reduce [F]

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

int((a*x+1)^n/x/((-a*x+1)^n),x)
 

Output:

int((a*x + 1)**n/(( - a*x + 1)**n*x),x)