Optimal. Leaf size=20 \[ -\frac{\left (b x^n\right )^p}{x (1-n p)} \]
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Rubi [A] time = 0.0076785, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ -\frac{\left (b x^n\right )^p}{x (1-n p)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (b x^n\right )^p}{x^2} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-2+n p} \, dx\\ &=-\frac{\left (b x^n\right )^p}{(1-n p) x}\\ \end{align*}
Mathematica [A] time = 0.003191, size = 18, normalized size = 0.9 \[ \frac{\left (b x^n\right )^p}{x (n p-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( b{x}^{n} \right ) ^{p}}{x \left ( np-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973192, size = 26, normalized size = 1.3 \begin{align*} \frac{b^{p}{\left (x^{n}\right )}^{p}}{{\left (n p - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36715, size = 55, normalized size = 2.75 \begin{align*} \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{{\left (n p - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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