Optimal. Leaf size=14 \[ \frac {\left (b x^n\right )^p}{n p} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, 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}
\begin {gather*} \frac {\left (b x^n\right )^p}{n p} \end {gather*}
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} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-1+n p} \, dx\\ &=\frac {\left (b x^n\right )^p}{n p}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\left (b x^n\right )^p}{n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 15, normalized size = 1.07
method | result | size |
gosper | \(\frac {\left (b \,x^{n}\right )^{p}}{n p}\) | \(15\) |
derivativedivides | \(\frac {\left (b \,x^{n}\right )^{p}}{n p}\) | \(15\) |
default | \(\frac {\left (b \,x^{n}\right )^{p}}{n p}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 15, normalized size = 1.07 \begin {gather*} \frac {b^{p} {\left (x^{n}\right )}^{p}}{n p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 1.29 \begin {gather*} \frac {e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (8) = 16\).
time = 0.10, size = 20, normalized size = 1.43 \begin {gather*} \begin {cases} \log {\left (x \right )} & \text {for}\: n = 0 \wedge p = 0 \\b^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} & \text {for}\: p = 0 \\\frac {\left (b x^{n}\right )^{p}}{n p} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 14, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{n}\right )^{p}}{n p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {{\left (b\,x^n\right )}^p}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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