Optimal. Leaf size=16 \[ \frac {x \left (b x^2\right )^p}{1+2 p} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30}
\begin {gather*} \frac {x \left (b x^2\right )^p}{2 p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{2 p} \, dx\\ &=\frac {x \left (b x^2\right )^p}{1+2 p}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x \left (b x^2\right )^p}{1+2 p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 1.06
method | result | size |
gosper | \(\frac {x \left (b \,x^{2}\right )^{p}}{1+2 p}\) | \(17\) |
risch | \(\frac {x \left (b \,x^{2}\right )^{p}}{1+2 p}\) | \(17\) |
norman | \(\frac {x \,{\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{1+2 p}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 17, normalized size = 1.06 \begin {gather*} \frac {b^{p} x x^{2 \, p}}{2 \, p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x}{2 \, p + 1} \end {gather*}
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 {gather*} \begin {cases} \frac {x \left (b x^{2}\right )^{p}}{2 p + 1} & \text {for}\: p \neq - \frac {1}{2} \\\int \frac {1}{\sqrt {b x^{2}}}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.27, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x}{2 \, p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 16, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (b\,x^2\right )}^p}{2\,p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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