Optimal. Leaf size=29 \[ \frac {2 \sqrt {b x^n} (c x)^{m+1}}{c (2 m+n+2)} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 30} \[ \frac {2 x \sqrt {b x^n} (c x)^m}{2 m+n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin {align*} \int (c x)^m \sqrt {b x^n} \, dx &=\left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{n/2} (c x)^m \, dx\\ &=\left (x^{-m-\frac {n}{2}} (c x)^m \sqrt {b x^n}\right ) \int x^{m+\frac {n}{2}} \, dx\\ &=\frac {2 x (c x)^m \sqrt {b x^n}}{2+2 m+n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 0.90 \[ \frac {x \sqrt {b x^n} (c x)^m}{m+\frac {n}{2}+1} \]
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 \[ \int \sqrt {b x^{n}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.83 \[ \frac {2 \sqrt {b \,x^{n}}\, x \left (c x \right )^{m}}{2 m +n +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 25, normalized size = 0.86 \[ \frac {2 \, \sqrt {b} c^{m} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 23, normalized size = 0.79 \[ \frac {2\,x\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+n+2} \]
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 \[ \begin {cases} \frac {2 \sqrt {b} c^{m} x x^{m} \sqrt {x^{n}}}{2 m + n + 2} & \text {for}\: m \neq - \frac {n}{2} - 1 \\\int \sqrt {b x^{n}} \left (c x\right )^{- \frac {n}{2} - 1}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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