Integrand size = 15, antiderivative size = 28 \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {(c x)^{2+m} \sqrt {b x^2}}{c^2 (2+m) x} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 16, 32} \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {\sqrt {b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]
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Rule 15
Rule 16
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^2} \int x (c x)^m \, dx}{x} \\ & = \frac {\sqrt {b x^2} \int (c x)^{1+m} \, dx}{c x} \\ & = \frac {(c x)^{2+m} \sqrt {b x^2}}{c^2 (2+m) x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {x (c x)^m \sqrt {b x^2}}{2+m} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {x \left (c x \right )^{m} \sqrt {b \,x^{2}}}{2+m}\) | \(20\) |
risch | \(\frac {x \left (c x \right )^{m} \sqrt {b \,x^{2}}}{2+m}\) | \(20\) |
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {\sqrt {b x^{2}} \left (c x\right )^{m} x}{m + 2} \]
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Time = 0.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int (c x)^m \sqrt {b x^2} \, dx=\begin {cases} \frac {x \sqrt {b x^{2}} \left (c x\right )^{m}}{m + 2} & \text {for}\: m \neq -2 \\\frac {\sqrt {b x^{2}} \log {\left (x \right )}}{c^{2} x} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {\sqrt {b} c^{m} x^{2} x^{m}}{m + 2} \]
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Exception generated. \[ \int (c x)^m \sqrt {b x^2} \, dx=\text {Exception raised: TypeError} \]
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Time = 5.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int (c x)^m \sqrt {b x^2} \, dx=\frac {\sqrt {b}\,x\,{\left (c\,x\right )}^m\,\sqrt {x^2}}{m+2} \]
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