Integrand size = 21, antiderivative size = 74 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=-\frac {2 (c x)^{-n/2}}{a c n}+\frac {2 \sqrt {b} x^{n/2} (c x)^{-n/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2} c n} \]
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Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {354, 352, 199, 327, 211} \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\frac {2 \sqrt {b} x^{n/2} (c x)^{-n/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2} c n}-\frac {2 (c x)^{-n/2}}{a c n} \]
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Rule 199
Rule 211
Rule 327
Rule 352
Rule 354
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{n/2} (c x)^{-n/2}\right ) \int \frac {x^{-1-\frac {n}{2}}}{a+b x^n} \, dx}{c} \\ & = -\frac {\left (2 x^{n/2} (c x)^{-n/2}\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b}{x^2}} \, dx,x,x^{-n/2}\right )}{c n} \\ & = -\frac {\left (2 x^{n/2} (c x)^{-n/2}\right ) \text {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{c n} \\ & = -\frac {2 (c x)^{-n/2}}{a c n}+\frac {\left (2 b x^{n/2} (c x)^{-n/2}\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a c n} \\ & = -\frac {2 (c x)^{-n/2}}{a c n}+\frac {2 \sqrt {b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2} c n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=-\frac {2 x (c x)^{-1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b x^n}{a}\right )}{a n} \]
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\[\int \frac {\left (c x \right )^{-1-\frac {n}{2}}}{a +b \,x^{n}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.04 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\left [-\frac {2 \, x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} - \sqrt {-\frac {b c^{-n - 2}}{a}} \log \left (\frac {a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) - {\left (n + 2\right )} \log \left (x\right )\right )} + 2 \, a \sqrt {-\frac {b c^{-n - 2}}{a}} x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} - b c^{-n - 2}}{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) - {\left (n + 2\right )} \log \left (x\right )\right )} + b c^{-n - 2}}\right )}{a n}, -\frac {2 \, {\left (x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} + \sqrt {\frac {b c^{-n - 2}}{a}} \arctan \left (\frac {\sqrt {\frac {b c^{-n - 2}}{a}} e^{\left (\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) + \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )}}{x}\right )\right )}}{a n}\right ] \]
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Time = 0.68 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=- \frac {2 c^{- \frac {n}{2} - 1} x^{- \frac {n}{2}}}{a n} - \frac {2 \sqrt {b} c^{- \frac {n}{2} - 1} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {3}{2}} n} \]
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\[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{b x^{n} + a} \,d x } \]
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\[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \]
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