Integrand size = 15, antiderivative size = 16 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {\left (a+b x^2\right )^3}{6 b} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 267} \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {\left (a+b x^2\right )^3}{6 b} \]
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Rule 267
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x \left (a+b x^2\right )^2 \, dx \\ & = \frac {\left (a+b x^2\right )^3}{6 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {\left (a+b x^2\right )^3}{6 b} \]
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Time = 1.97 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (b \,x^{2}+a \right )^{3}}{6 b}\) | \(15\) |
norman | \(\frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2}\) | \(25\) |
parallelrisch | \(\frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2}\) | \(25\) |
gosper | \(\frac {x^{2} \left (b^{2} x^{4}+3 a b \,x^{2}+3 a^{2}\right )}{6}\) | \(26\) |
risch | \(\frac {b^{2} x^{6}}{6}+\frac {a b \,x^{4}}{2}+\frac {a^{2} x^{2}}{2}+\frac {a^{3}}{6 b}\) | \(33\) |
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none
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\).
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a x+b x^3\right )^2}{x} \, dx=\frac {a^2\,x^2}{2}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^6}{6} \]
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