Optimal. Leaf size=20 \[ -\frac {\left (b x^n\right )^p}{x^2 (2-n p)} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \[ -\frac {\left (b x^n\right )^p}{x^2 (2-n p)} \]
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^3} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-3+n p} \, dx\\ &=-\frac {\left (b x^n\right )^p}{(2-n p) x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.90 \[ \frac {\left (b x^n\right )^p}{x^2 (n p-2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 22, normalized size = 1.10 \[ \frac {e^{\left (n p \log \relax (x) + p \log \relax (b)\right )}}{{\left (n p - 2\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x^{n}\right )^{p}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.95 \[ \frac {\left (b \,x^{n}\right )^{p}}{\left (n p -2\right ) x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 19, normalized size = 0.95 \[ \frac {b^{p} {\left (x^{n}\right )}^{p}}{{\left (n p - 2\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (b\,x^n\right )}^p}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b^{p} \left (x^{n}\right )^{p}}{n p x^{2} - 2 x^{2}} & \text {for}\: n \neq \frac {2}{p} \\\int \frac {\left (b x^{\frac {2}{p}}\right )^{p}}{x^{3}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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