Integrand size = 13, antiderivative size = 26 \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=-\frac {2 b x^{-3+n} \sqrt {b x^n}}{3 (2-n)} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=-\frac {2 b x^{n-3} \sqrt {b x^n}}{3 (2-n)} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{-4+\frac {3 n}{2}} \, dx \\ & = -\frac {2 b x^{-3+n} \sqrt {b x^n}}{3 (2-n)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\frac {\left (b x^n\right )^{3/2}}{\left (-3+\frac {3 n}{2}\right ) x^3} \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 x^{3} \left (-2+n \right )}\) | \(18\) |
risch | \(\frac {2 b^{2} x^{2 n}}{3 \left (-2+n \right ) x^{3} \sqrt {b \,x^{n}}}\) | \(26\) |
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Exception generated. \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Time = 1.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\begin {cases} \frac {2 \left (b x^{n}\right )^{\frac {3}{2}}}{3 n x^{3} - 6 x^{3}} & \text {for}\: n \neq 2 \\\frac {\left (b x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\int { \frac {\left (b x^{n}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Time = 5.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x^n\right )^{3/2}}{x^4} \, dx=\frac {2\,b\,x^{n-3}\,\sqrt {b\,x^n}}{3\,n-6} \]
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