Integrand size = 15, antiderivative size = 29 \[ \int (c x)^m \sqrt {b x^n} \, dx=\frac {2 (c x)^{1+m} \sqrt {b x^n}}{c (2+2 m+n)} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 20, 30} \[ \int (c x)^m \sqrt {b x^n} \, dx=\frac {2 x \sqrt {b x^n} (c x)^m}{2 m+n+2} \]
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Rule 15
Rule 20
Rule 30
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int (c x)^m \sqrt {b x^n} \, dx=\frac {x (c x)^m \sqrt {b x^n}}{1+m+\frac {n}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {2 x \left (c x \right )^{m} \sqrt {b \,x^{n}}}{2+2 m +n}\) | \(24\) |
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Exception generated. \[ \int (c x)^m \sqrt {b x^n} \, dx=\text {Exception raised: TypeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.93 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int (c x)^m \sqrt {b x^n} \, dx=\begin {cases} \frac {2 x \sqrt {b x^{n}} \left (c x\right )^{m}}{2 m + n + 2} & \text {for}\: m \neq - \frac {n}{2} - 1 \\x \sqrt {b x^{n}} \left (c x\right )^{- \frac {n}{2} - 1} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int (c x)^m \sqrt {b x^n} \, dx=\frac {2 \, \sqrt {b} c^{m} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \]
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\[ \int (c x)^m \sqrt {b x^n} \, dx=\int { \sqrt {b x^{n}} \left (c x\right )^{m} \,d x } \]
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Time = 5.86 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int (c x)^m \sqrt {b x^n} \, dx=\frac {2\,x\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+n+2} \]
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