Integrand size = 13, antiderivative size = 45 \[ \int x^n (a+b x)^{-n} \, dx=\frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,-\frac {b x}{a}\right )}{1+n} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {68, 66} \[ \int x^n (a+b x)^{-n} \, dx=\frac {x^{n+1} (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {b x}{a}\right )}{n+1} \]
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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^n \left (1+\frac {b x}{a}\right )^{-n} \, dx \\ & = \frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {b x}{a}\right )}{1+n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^n (a+b x)^{-n} \, dx=\frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,-\frac {b x}{a}\right )}{1+n} \]
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\[\int x^{n} \left (b x +a \right )^{-n}d x\]
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\[ \int x^n (a+b x)^{-n} \, dx=\int { \frac {x^{n}}{{\left (b x + a\right )}^{n}} \,d x } \]
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Result contains complex when optimal does not.
Time = 21.91 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71 \[ \int x^n (a+b x)^{-n} \, dx=\frac {a^{- n} x^{n + 1} \Gamma \left (n + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 1 \\ n + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 2\right )} \]
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\[ \int x^n (a+b x)^{-n} \, dx=\int { \frac {x^{n}}{{\left (b x + a\right )}^{n}} \,d x } \]
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\[ \int x^n (a+b x)^{-n} \, dx=\int { \frac {x^{n}}{{\left (b x + a\right )}^{n}} \,d x } \]
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Timed out. \[ \int x^n (a+b x)^{-n} \, dx=\int \frac {x^n}{{\left (a+b\,x\right )}^n} \,d x \]
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