Integrand size = 17, antiderivative size = 71 \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=-\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {1}{n q},1-\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {377, 372, 371} \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=-\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (\frac {b \left (c x^q\right )^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {1}{n q},1-\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]
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Rule 371
Rule 372
Rule 377
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (a+b c^n x^{n q}\right )^p}{x^2} \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right ) \\ & = \text {Subst}\left (\left (\left (a+b c^n x^{n q}\right )^p \left (1+\frac {b c^n x^{n q}}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b c^n x^{n q}}{a}\right )^p}{x^2} \, dx,x^{n q},c^{-n} \left (c x^q\right )^n\right ) \\ & = -\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \, _2F_1\left (-p,-\frac {1}{n q};1-\frac {1}{n q};-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=-\frac {\left (a+b \left (c x^q\right )^n\right )^p \left (1+\frac {b \left (c x^q\right )^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {1}{n q},1-\frac {1}{n q},-\frac {b \left (c x^q\right )^n}{a}\right )}{x} \]
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\[\int \frac {{\left (a +b \left (c \,x^{q}\right )^{n}\right )}^{p}}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=\int \frac {\left (a + b \left (c x^{q}\right )^{n}\right )^{p}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=\int { \frac {{\left (\left (c x^{q}\right )^{n} b + a\right )}^{p}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \left (c x^q\right )^n\right )^p}{x^2} \, dx=\int \frac {{\left (a+b\,{\left (c\,x^q\right )}^n\right )}^p}{x^2} \,d x \]
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