Integrand size = 20, antiderivative size = 77 \[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\frac {F^a (d x)^{1+m} \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-b \log (F),\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{d (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2306, 12, 372, 371} \[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\frac {F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-b \log (F),\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{d (m+1)} \]
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Rule 12
Rule 371
Rule 372
Rule 2306
Rubi steps \begin{align*} \text {integral}& = \int F^a (d x)^m \left (c+d x^n\right )^{b \log (F)} \, dx \\ & = F^a \int (d x)^m \left (c+d x^n\right )^{b \log (F)} \, 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 (d x)^m \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx \\ & = \frac {F^a (d x)^{1+m} \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+m}{n},-b \log (F);\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{d (1+m)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22 \[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} x (d x)^m \left (-\frac {d x^n}{c}\right )^{-\frac {1+m}{n}} \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (1-\frac {1+m}{n},1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \]
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\[\int F^{a +b \ln \left (c +d \,x^{n}\right )} \left (d x \right )^{m}d x\]
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\[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\int { \left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a} \,d x } \]
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\[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\int F^{a + b \log {\left (c + d x^{n} \right )}} \left (d x\right )^{m}\, dx \]
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\[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\int { \left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a} \,d x } \]
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\[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\int { \left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a} \,d x } \]
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Timed out. \[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx=\int F^{a+b\,\ln \left (c+d\,x^n\right )}\,{\left (d\,x\right )}^m \,d x \]
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