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]
________________________________________________________________________________________
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]
[Out]
Rule 136
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]