Optimal. Leaf size=24 \[ \frac {2 x^{m+1} \sqrt {b x^n}}{2 m+n+2} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac {2 x^{m+1} \sqrt {b x^n}}{2 m+n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int x^m \sqrt {b x^n} \, dx &=\left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{m+\frac {n}{2}} \, dx\\ &=\frac {2 x^{1+m} \sqrt {b x^n}}{2+2 m+n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.04 \[ \frac {x^{m+1} \sqrt {b x^n}}{m+\frac {n}{2}+1} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {b x^{n}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.96 \[ \frac {2 \sqrt {b \,x^{n}}\, x^{m +1}}{2 m +n +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 22, normalized size = 0.92 \[ \frac {2 \, \sqrt {b} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 22, normalized size = 0.92 \[ \frac {2\,x^{m+1}\,\sqrt {b\,x^n}}{2\,m+n+2} \]
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 \[ \begin {cases} \frac {2 \sqrt {b} x x^{m} \sqrt {x^{n}}}{2 m + n + 2} & \text {for}\: m \neq - \frac {n}{2} - 1 \\\int x^{- \frac {n}{2} - 1} \sqrt {b x^{n}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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