Integrand size = 18, antiderivative size = 66 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=-\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-b \log (F),-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{x} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2306, 12, 372, 371} \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=-\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-b \log (F),-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{x} \]
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Rule 12
Rule 371
Rule 372
Rule 2306
Rubi steps \begin{align*} \text {integral}& = \int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x^2} \, dx \\ & = F^a \int \frac {\left (c+d x^n\right )^{b \log (F)}}{x^2} \, dx \\ & = \left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)}\right ) \int \frac {\left (1+\frac {d x^n}{c}\right )^{b \log (F)}}{x^2} \, dx \\ & = -\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (-\frac {1}{n},-b \log (F);-\frac {1-n}{n};-\frac {d x^n}{c}\right )}{x} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} \left (-\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (1+\frac {1}{n},1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n x (1+b \log (F))} \]
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\[\int \frac {F^{a +b \ln \left (c +d \,x^{n}\right )}}{x^{2}}d x\]
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\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \]
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\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int \frac {F^{a + b \log {\left (c + d x^{n} \right )}}}{x^{2}}\, dx \]
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\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \]
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\[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int { \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^2} \, dx=\int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x^2} \,d x \]
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