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 = 0.107359, 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 \[ \frac{(1-a x)^{-n} (a x+1)^n \, _2F_1\left (1,-n;1-n;\frac{1-a x}{a x+1}\right )}{n}-\frac{2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac{1}{2} (1-a x)\right )}{n} \]
Antiderivative was successfully verified.
[In] Int[(1 + a*x)^n/(x*(1 - a*x)^n),x]
[Out]
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Rubi in Sympy [A] time = 12.6793, size = 70, normalized size = 0.81 \[ - \frac{2^{n - 1} \left (- a x + 1\right )^{- n + 1}{{}_{2}F_{1}\left (\begin{matrix} - n + 1, - n + 1 \\ - n + 2 \end{matrix}\middle |{- \frac{a x}{2} + \frac{1}{2}} \right )}}{- n + 1} - \frac{\left (- a x + 1\right )^{- n + 1} \left (a x + 1\right )^{n - 1}{{}_{2}F_{1}\left (\begin{matrix} - n + 1, 1 \\ - n + 2 \end{matrix}\middle |{\frac{a x - 1}{- a x - 1}} \right )}}{- n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x+1)**n/x/((-a*x+1)**n),x)
[Out]
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Mathematica [C] time = 0.353238, size = 182, normalized size = 2.12 \[ \frac{2 (n-2) (1-a x)^{1-n} (a x+1)^n F_1\left (1-n;-n,1;2-n;\frac{1}{2} (1-a x),1-a x\right )}{a (1-n) x \left ((a x-1) \left (n F_1\left (2-n;1-n,1;3-n;\frac{1}{2} (1-a x),1-a x\right )-2 F_1\left (2-n;-n,2;3-n;\frac{1}{2} (1-a x),1-a x\right )\right )-2 (n-2) F_1\left (1-n;-n,1;2-n;\frac{1}{2} (1-a x),1-a x\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + a*x)^n/(x*(1 - a*x)^n),x]
[Out]
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Maple [F] time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ax+1 \right ) ^{n}}{x \left ( -ax+1 \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x+1)^n/x/((-a*x+1)^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{-n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^n/((-a*x + 1)^n*x),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^n/((-a*x + 1)^n*x),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x+1)**n/x/((-a*x+1)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x + 1\right )}^{n}}{{\left (-a x + 1\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + 1)^n/((-a*x + 1)^n*x),x, algorithm="giac")
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