Optimal. Leaf size=14 \[ \frac {\left (b x^2\right )^p}{2 p} \]
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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^2\right )^p}{2 p} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {\left (b x^2\right )^p}{x} \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{-1+2 p} \, dx\\ &=\frac {\left (b x^2\right )^p}{2 p}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\left (b x^2\right )^p}{2 p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 13, normalized size = 0.93
method | result | size |
gosper | \(\frac {\left (b \,x^{2}\right )^{p}}{2 p}\) | \(13\) |
derivativedivides | \(\frac {\left (b \,x^{2}\right )^{p}}{2 p}\) | \(13\) |
default | \(\frac {\left (b \,x^{2}\right )^{p}}{2 p}\) | \(13\) |
risch | \(\frac {\left (b \,x^{2}\right )^{p}}{2 p}\) | \(13\) |
norman | \(\frac {{\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{2 p}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.93 \begin {gather*} \frac {b^{p} {\left (x^{2}\right )}^{p}}{2 \, p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (b x^{2}\right )^{p}}{2 \, p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 12, normalized size = 0.86 \begin {gather*} \begin {cases} \frac {\left (b x^{2}\right )^{p}}{2 p} & \text {for}\: p \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.90, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (b x^{2}\right )^{p}}{2 \, p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 12, normalized size = 0.86 \begin {gather*} \frac {{\left (b\,x^2\right )}^p}{2\,p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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