Integrand size = 15, antiderivative size = 25 \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=-\frac {\left (a \left (b x^n\right )^p\right )^q}{(2-n p q) x^2} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1971, 30} \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=-\frac {\left (a \left (b x^n\right )^p\right )^q}{x^2 (2-n p q)} \]
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Rule 30
Rule 1971
Rubi steps \begin{align*} \text {integral}& = \left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x^{-3+n p q} \, dx \\ & = -\frac {\left (a \left (b x^n\right )^p\right )^q}{(2-n p q) x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\frac {\left (a \left (b x^n\right )^p\right )^q}{(-2+n p q) x^2} \]
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Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {{\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}}{x^{2} \left (n p q -2\right )}\) | \(24\) |
parallelrisch | \(\frac {{\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}}{x^{2} \left (n p q -2\right )}\) | \(24\) |
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none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\frac {e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{{\left (n p q - 2\right )} x^{2}} \]
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Time = 0.91 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\begin {cases} \frac {\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2} \left (n p q - 2\right )} & \text {for}\: n p q \neq 2 \\\frac {\left (a \left (b x^{n}\right )^{p}\right )^{q} \log {\left (x \right )}}{x^{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\frac {a^{q} b^{p q} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 2\right )} x^{2}} \]
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\[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\int { \frac {\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a \left (b x^n\right )^p\right )^q}{x^3} \, dx=\int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^3} \,d x \]
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