3.1001 \(\int \frac{(1-a x)^{1-n} (1+a x)^{1+n}}{x^2} \, dx\)

Optimal. Leaf size=106 \[ \frac{a 2^n (1-a x)^{1-n} \, _2F_1\left (1-n,-n;2-n;\frac{1}{2} (1-a x)\right )}{1-n}-\frac{2 a (1-a x)^{1-n} (a x+1)^{n-1} \, _2F_1\left (2,1-n;2-n;\frac{1-a x}{a x+1}\right )}{1-n} \]

[Out]

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

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Rubi [C]  time = 0.0188463, antiderivative size = 48, normalized size of antiderivative = 0.45, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {136} \[ \frac{a 2^{1-n} (a x+1)^{n+2} F_1\left (n+2;n-1,2;n+3;\frac{1}{2} (a x+1),a x+1\right )}{n+2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(1 - n)*a*(1 + a*x)^(2 + n)*AppellF1[2 + n, -1 + n, 2, 3 + n, (1 + a*x)/2, 1 + a*x])/(2 + n)

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{(1-a x)^{1-n} (1+a x)^{1+n}}{x^2} \, dx &=\frac{2^{1-n} a (1+a x)^{2+n} F_1\left (2+n;-1+n,2;3+n;\frac{1}{2} (1+a x),1+a x\right )}{2+n}\\ \end{align*}

Mathematica [C]  time = 0.0998953, size = 128, normalized size = 1.21 \[ -\frac{\left (1-\frac{1}{a x}\right )^n \left (\frac{1}{a x}+1\right )^{-n} (a x+1)^n (1-a x)^{-n} F_1\left (1;n,-n;2;\frac{1}{a x},-\frac{1}{a x}\right )}{x}-\frac{a (a x+1)^{n+1} \left (\frac{1}{2} (-a x-1)+1\right )^n (1-a x)^{-n} \, _2F_1\left (n,n+1;n+2;\frac{1}{2} (a x+1)\right )}{n+1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((1 - 1/(a*x))^n*(1 + a*x)^n*AppellF1[1, n, -n, 2, 1/(a*x), -(1/(a*x))])/((1 + 1/(a*x))^n*x*(1 - a*x)^n)) -
(a*(1 + a*x)^(1 + n)*(1 + (-1 - a*x)/2)^n*Hypergeometric2F1[n, 1 + n, 2 + n, (1 + a*x)/2])/((1 + n)*(1 - a*x)^
n)

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{1+n} \left ( -ax+1 \right ) ^{1-n}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{n + 1}{\left (-a x + 1\right )}^{-n + 1}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x+1)^(1-n)*(a*x+1)^(1+n)/x^2,x, algorithm="maxima")

[Out]

integrate((a*x + 1)^(n + 1)*(-a*x + 1)^(-n + 1)/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a x + 1\right )}^{n + 1}{\left (-a x + 1\right )}^{-n + 1}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*x+1)^(1-n)*(a*x+1)^(1+n)/x^2,x, algorithm="fricas")

[Out]

integral((a*x + 1)^(n + 1)*(-a*x + 1)^(-n + 1)/x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{n + 1}{\left (-a x + 1\right )}^{-n + 1}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x + 1)^(n + 1)*(-a*x + 1)^(-n + 1)/x^2, x)