Integrand size = 18, antiderivative size = 22 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x) \]
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Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 29} \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=x^{m+1} \log (x) \left (a x^n\right )^{-\frac {m+1}{n}} \]
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Rule 15
Rule 29
Rubi steps \begin{align*} \text {integral}& = \left (x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \int \frac {1}{x} \, dx \\ & = x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x) \]
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\[\int x^{m} \left (a \,x^{n}\right )^{-\frac {1+m}{n}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\frac {\log \left (x\right )}{a^{\frac {m + 1}{n}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (17) = 34\).
Time = 1.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\begin {cases} \frac {x x^{m} \left (a x^{n}\right )^{- \frac {m}{n} - \frac {1}{n}}}{m + n \left (- \frac {m}{n} - \frac {1}{n}\right ) + 1} & \text {for}\: m + n \left (- \frac {m}{n} - \frac {1}{n}\right ) \neq -1 \\x x^{m} \left (a x^{n}\right )^{- \frac {m}{n} - \frac {1}{n}} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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\[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int { \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}} \,d x } \]
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\[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int { \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}} \,d x } \]
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Timed out. \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int \frac {x^m}{{\left (a\,x^n\right )}^{\frac {m+1}{n}}} \,d x \]
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