Integrand size = 11, antiderivative size = 29 \[ \int \frac {x^m}{a+b x} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (1+m)} \]
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Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {66} \[ \int \frac {x^m}{a+b x} \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a (m+1)} \]
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Rule 66
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a (1+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{a+b x} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (1+m)} \]
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\[\int \frac {x^{m}}{b x +a}d x\]
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\[ \int \frac {x^m}{a+b x} \, dx=\int { \frac {x^{m}}{b x + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {x^m}{a+b x} \, dx=\frac {m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} \]
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\[ \int \frac {x^m}{a+b x} \, dx=\int { \frac {x^{m}}{b x + a} \,d x } \]
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\[ \int \frac {x^m}{a+b x} \, dx=\int { \frac {x^{m}}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {x^m}{a+b x} \, dx=\int \frac {x^m}{a+b\,x} \,d x \]
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