Integrand size = 22, antiderivative size = 69 \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=-\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^{n-q}}{a}\right )}{a (1+p) (n-q)} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2057, 272, 67} \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=-\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^{n-q}}{a}+1\right )}{a (p+1) (n-q)} \]
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Rule 67
Rule 272
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int \frac {\left (a+b x^{n-q}\right )^p}{x} \, dx \\ & = \frac {\left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^{n-q}\right )}{n-q} \\ & = -\frac {x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (1,1+p;2+p;1+\frac {b x^{n-q}}{a}\right )}{a (1+p) (n-q)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\frac {x^{-p q} \left (b x^n+a x^q\right )^p \left (1+\frac {a x^{-n+q}}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {a x^{-n+q}}{b}\right )}{p (n-q)} \]
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\[\int x^{-p q -1} \left (b \,x^{n}+a \,x^{q}\right )^{p}d x\]
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\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \]
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\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int x^{- p q - 1} \left (a x^{q} + b x^{n}\right )^{p}\, dx \]
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\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \]
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\[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int { {\left (b x^{n} + a x^{q}\right )}^{p} x^{-p q - 1} \,d x } \]
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Timed out. \[ \int x^{-1-p q} \left (b x^n+a x^q\right )^p \, dx=\int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{p\,q+1}} \,d x \]
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