\(\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx\) [1000]
Optimal result
Integrand size = 21, antiderivative size = 86 \[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\frac {(1-a x)^{-n} (1+a x)^n \operatorname {Hypergeometric2F1}\left (1,-n,1-n,\frac {1-a x}{1+a x}\right )}{n}-\frac {2^n (1-a x)^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )}{n}
\]
[Out]
(a*x+1)^n*hypergeom([1, -n],[1-n],(-a*x+1)/(a*x+1))/n/((-a*x+1)^n)-2^n*hypergeom([-n, -n],[1-n],-1/2*a*x+1/2)/
n/((-a*x+1)^n)
Rubi [A] (verified)
Time = 0.02 (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 = {132, 71, 133}
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\frac {(1-a x)^{-n} (a x+1)^n \operatorname {Hypergeometric2F1}\left (1,-n,1-n,\frac {1-a x}{a x+1}\right )}{n}-\frac {2^n (1-a x)^{-n} \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )}{n}
\]
[In]
Int[(1 + a*x)^n/(x*(1 - a*x)^n),x]
[Out]
((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)
Rule 71
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] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Rule 132
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[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)*ExpandTo
Sum[(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 133
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> 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] || !SumSimplerQ[p, 1]) && !ILtQ[m, 0]
Rubi steps \begin{align*}
\text {integral}& = -\left (a \int (1-a x)^{-1-n} (1+a x)^n \, dx\right )+\int \frac {(1-a x)^{-1-n} (1+a x)^n}{x} \, dx \\ & = \frac {(1-a x)^{-n} (1+a x)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{1+a x}\right )}{n}-\frac {2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )}{n} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\frac {(1-a x)^{-n} \left ((1+a x)^n \operatorname {Hypergeometric2F1}\left (1,-n,1-n,\frac {1-a x}{1+a x}\right )-2^n \operatorname {Hypergeometric2F1}\left (-n,-n,1-n,\frac {1}{2} (1-a x)\right )\right )}{n}
\]
[In]
Integrate[(1 + a*x)^n/(x*(1 - a*x)^n),x]
[Out]
((1 + a*x)^n*Hypergeometric2F1[1, -n, 1 - n, (1 - a*x)/(1 + a*x)] - 2^n*Hypergeometric2F1[-n, -n, 1 - n, (1 -
a*x)/2])/(n*(1 - a*x)^n)
Maple [F]
\[\int \frac {\left (a x +1\right )^{n} \left (-a x +1\right )^{-n}}{x}d x\]
[In]
int((a*x+1)^n/x/((-a*x+1)^n),x)
[Out]
int((a*x+1)^n/x/((-a*x+1)^n),x)
Fricas [F]
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x} \,d x }
\]
[In]
integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="fricas")
[Out]
integral((a*x + 1)^n/((-a*x + 1)^n*x), x)
Sympy [F]
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\int \frac {\left (- a x + 1\right )^{- n} \left (a x + 1\right )^{n}}{x}\, dx
\]
[In]
integrate((a*x+1)**n/x/((-a*x+1)**n),x)
[Out]
Integral((a*x + 1)**n/(x*(-a*x + 1)**n), x)
Maxima [F]
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x} \,d x }
\]
[In]
integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="maxima")
[Out]
integrate((a*x + 1)^n/((-a*x + 1)^n*x), x)
Giac [F]
\[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\int { \frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x} \,d x }
\]
[In]
integrate((a*x+1)^n/x/((-a*x+1)^n),x, algorithm="giac")
[Out]
integrate((a*x + 1)^n/((-a*x + 1)^n*x), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx=\int \frac {{\left (a\,x+1\right )}^n}{x\,{\left (1-a\,x\right )}^n} \,d x
\]
[In]
int((a*x + 1)^n/(x*(1 - a*x)^n),x)
[Out]
int((a*x + 1)^n/(x*(1 - a*x)^n), x)