Optimal. Leaf size=86 \[ \frac {(1-a x)^{-n} (a x+1)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{a x+1}\right )}{n}-\frac {2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )}{n} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, 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.
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Rule 69
Rule 105
Rule 131
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*}
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Mathematica [A] time = 0.03, 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.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a x +1\right )^{-n} \left (a x +1\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a\,x+1\right )}^n}{x\,{\left (1-a\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- a x + 1\right )^{- n} \left (a x + 1\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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