Integrand size = 15, antiderivative size = 39 \[ \int x^{-1+n} (a+b x)^{-n} \, dx=\frac {x^n (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,n,1+n,-\frac {b x}{a}\right )}{n} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {68, 66} \[ \int x^{-1+n} (a+b x)^{-n} \, dx=\frac {x^n (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \operatorname {Hypergeometric2F1}\left (n,n,n+1,-\frac {b x}{a}\right )}{n} \]
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Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \left ((a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n\right ) \int x^{-1+n} \left (1+\frac {b x}{a}\right )^{-n} \, dx \\ & = \frac {x^n (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,n;1+n;-\frac {b x}{a}\right )}{n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int x^{-1+n} (a+b x)^{-n} \, dx=\frac {x^n (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,n,1+n,-\frac {b x}{a}\right )}{n} \]
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\[\int x^{-1+n} \left (b x +a \right )^{-n}d x\]
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\[ \int x^{-1+n} (a+b x)^{-n} \, dx=\int { \frac {x^{n - 1}}{{\left (b x + a\right )}^{n}} \,d x } \]
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Result contains complex when optimal does not.
Time = 10.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int x^{-1+n} (a+b x)^{-n} \, dx=\frac {a^{- n} x^{n} \Gamma \left (n\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n \\ n + 1 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 1\right )} \]
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\[ \int x^{-1+n} (a+b x)^{-n} \, dx=\int { \frac {x^{n - 1}}{{\left (b x + a\right )}^{n}} \,d x } \]
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\[ \int x^{-1+n} (a+b x)^{-n} \, dx=\int { \frac {x^{n - 1}}{{\left (b x + a\right )}^{n}} \,d x } \]
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Timed out. \[ \int x^{-1+n} (a+b x)^{-n} \, dx=\int \frac {x^{n-1}}{{\left (a+b\,x\right )}^n} \,d x \]
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