Optimal. Leaf size=21 \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
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Rubi [A] time = 0.01, antiderivative size = 21, 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} \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int x^m \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{m+2 p} \, dx\\ &=\frac {x^{1+m} \left (b x^2\right )^p}{1+m+2 p}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 24, normalized size = 1.14 \[ \frac {x x^{m} e^{\left (p \log \relax (b) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 24, normalized size = 1.14 \[ \frac {x x^{m} e^{\left (p \log \relax (b) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 1.05 \[ \frac {x^{m +1} \left (b \,x^{2}\right )^{p}}{m +2 p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 24, normalized size = 1.14 \[ \frac {b^{p} x e^{\left (m \log \relax (x) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 21, normalized size = 1.00 \[ \frac {x^{m+1}\,{\left (b\,x^2\right )}^p}{m+2\,p+1} \]
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 \[ \begin {cases} \frac {b^{p} x x^{m} \left (x^{2}\right )^{p}}{m + 2 p + 1} & \text {for}\: m \neq - 2 p - 1 \\\int x^{- 2 p - 1} \left (b x^{2}\right )^{p}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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