Integrand size = 28, antiderivative size = 40 \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=-\frac {x^{-((n+p) (1+q))} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)} \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2005, 2039} \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=-\frac {x^{-((q+1) (n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \]
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Rule 2005
Rule 2039
Rubi steps \begin{align*} \text {integral}& = \int x^{-1-n q-p (1+q)} \left (a x^n+b x^{n+p}\right )^q \, dx \\ & = -\frac {x^{-((n+p) (1+q))} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=-\frac {x^{-((n+p) (1+q))} \left (x^n \left (a+b x^p\right )\right )^{1+q}}{a p (1+q)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(41)=82\).
Time = 2.92 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.15
method | result | size |
parallelrisch | \(-\frac {x \,x^{p} x^{-q n -p q -p -1} \left (x^{n} \left (a +b \,x^{p}\right )\right )^{q} b^{2}+x \,x^{-q n -p q -p -1} \left (x^{n} \left (a +b \,x^{p}\right )\right )^{q} a b}{b p \left (q +1\right ) a}\) | \(86\) |
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none
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.60 \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=-\frac {{\left (b x x^{-{\left (n + p\right )} q - p - 1} x^{p} + a x x^{-{\left (n + p\right )} q - p - 1}\right )} {\left (b x^{n} x^{p} + a x^{n}\right )}^{q}}{a p q + a p} \]
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Timed out. \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=\text {Timed out} \]
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\[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=\int { \left ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p {\left (q + 1\right )} - n q - 1} \,d x } \]
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\[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=\int { \left ({\left (b x^{p} + a\right )} x^{n}\right )^{q} x^{-p {\left (q + 1\right )} - n q - 1} \,d x } \]
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Timed out. \[ \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx=\int \frac {{\left (x^n\,\left (a+b\,x^p\right )\right )}^q}{x^{n\,q+p\,\left (q+1\right )+1}} \,d x \]
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