Integrand size = 15, antiderivative size = 36 \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\frac {2 x^{1-n} (c x)^m}{a (2+2 m-3 n) \sqrt {a x^n}} \]
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Time = 0.01 (sec) , antiderivative size = 36, 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, 20, 30} \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\frac {2 x^{1-n} (c x)^m}{a (2 m-3 n+2) \sqrt {a x^n}} \]
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Rule 15
Rule 20
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {x^{n/2} \int x^{-3 n/2} (c x)^m \, dx}{a \sqrt {a x^n}} \\ & = \frac {\left (x^{-m+\frac {n}{2}} (c x)^m\right ) \int x^{m-\frac {3 n}{2}} \, dx}{a \sqrt {a x^n}} \\ & = \frac {2 x^{1-n} (c x)^m}{a (2+2 m-3 n) \sqrt {a x^n}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\frac {x (c x)^m}{\left (1+m-\frac {3 n}{2}\right ) \left (a x^n\right )^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {2 x \left (c x \right )^{m}}{\left (2+2 m -3 n \right ) \left (a \,x^{n}\right )^{\frac {3}{2}}}\) | \(26\) |
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Exception generated. \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 2.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\begin {cases} \frac {2 x \left (c x\right )^{m}}{2 m \left (a x^{n}\right )^{\frac {3}{2}} - 3 n \left (a x^{n}\right )^{\frac {3}{2}} + 2 \left (a x^{n}\right )^{\frac {3}{2}}} & \text {for}\: m \neq \frac {3 n}{2} - 1 \\\frac {x \left (c x\right )^{\frac {3 n}{2} - 1} \log {\left (x \right )}}{\left (a x^{n}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\frac {2 \, c^{m} x x^{m}}{a^{\frac {3}{2}} {\left (2 \, m - 3 \, n + 2\right )} {\left (x^{n}\right )}^{\frac {3}{2}}} \]
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\[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{m}}{\left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 5.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {(c x)^m}{\left (a x^n\right )^{3/2}} \, dx=\frac {2\,x^{1-2\,n}\,\sqrt {a\,x^n}\,{\left (c\,x\right )}^m}{a^2\,\left (2\,m-3\,n+2\right )} \]
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