3.1000 \(\int \frac{(1-a x)^{-n} (1+a x)^n}{x} \, dx\)

Optimal. Leaf size=86 \[ \frac{(1-a x)^{-n} (a x+1)^n \, _2F_1\left (1,-n;1-n;\frac{1-a x}{a x+1}\right )}{n}-\frac{2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac{1}{2} (1-a x)\right )}{n} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0296242, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {105, 69, 131} \[ \frac{(1-a x)^{-n} (a x+1)^n \, _2F_1\left (1,-n;1-n;\frac{1-a x}{a x+1}\right )}{n}-\frac{2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac{1}{2} (1-a x)\right )}{n} \]

Antiderivative was successfully verified.

[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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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 131

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)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{(1-a x)^{-n} (1+a x)^n}{x} \, dx &=-\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]  time = 0.0237632, size = 74, normalized size = 0.86 \[ \frac{(1-a x)^{-n} \left ((a x+1)^n \, _2F_1\left (1,-n;1-n;\frac{1-a x}{a x+1}\right )-2^n \, _2F_1\left (-n,-n;1-n;\frac{1}{2} (1-a x)\right )\right )}{n} \]

Antiderivative was successfully verified.

[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]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{n}}{x \left ( -ax+1 \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- a x + 1\right )^{- n} \left (a x + 1\right )^{n}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)