Integrand size = 26, antiderivative size = 102 \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=-b n x \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {e x^{-\frac {1}{1+q}}}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (-1-q,-1-q,-q,-\frac {e x^{-\frac {1}{1+q}}}{d}\right )+\frac {x \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2351, 252, 251} \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (d+e x^{-\frac {1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-b n x \left (d+e x^{-\frac {1}{q+1}}\right )^q \left (\frac {e x^{-\frac {1}{q+1}}}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (-q-1,-q-1,-q,-\frac {e x^{-\frac {1}{q+1}}}{d}\right ) \]
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Rule 251
Rule 252
Rule 2351
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {(b n) \int \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \, dx}{d} \\ & = \frac {x \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\left (b n \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {e x^{-\frac {1}{1+q}}}{d}\right )^{-q}\right ) \int \left (1+\frac {e x^{-\frac {1}{1+q}}}{d}\right )^{1+q} \, dx \\ & = -b n x \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {e x^{-\frac {1}{1+q}}}{d}\right )^{-q} \, _2F_1\left (-1-q,-1-q;-q;-\frac {e x^{-\frac {1}{1+q}}}{d}\right )+\frac {x \left (d+e x^{-\frac {1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{-\frac {1}{1+q}} \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^{-q} \left (-b d n (1+q)^2 x^{\frac {2+q}{1+q}} \, _3F_2\left (1,1,-q;2,2;-\frac {d x^{\frac {1}{1+q}}}{e}\right )-b e n x \log (x)+\left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^q \left (e x+d x^{\frac {2+q}{1+q}}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d} \]
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\[\int \left (d +e \,x^{-\frac {1}{1+q}}\right )^{q} \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
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\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
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\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
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\[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q} \,d x } \]
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Timed out. \[ \int \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d+\frac {e}{x^{\frac {1}{q+1}}}\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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