Integrand size = 15, antiderivative size = 25 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=a^2 x+\frac {2}{3} a b x^3+\frac {b^2 x^5}{5} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 200} \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=a^2 x+\frac {2}{3} a b x^3+\frac {b^2 x^5}{5} \]
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Rule 200
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \left (a+b x^2\right )^2 \, dx \\ & = \int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx \\ & = a^2 x+\frac {2}{3} a b x^3+\frac {b^2 x^5}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=a^2 x+\frac {2}{3} a b x^3+\frac {b^2 x^5}{5} \]
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Time = 1.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
default | \(a^{2} x +\frac {2}{3} a b \,x^{3}+\frac {1}{5} b^{2} x^{5}\) | \(22\) |
risch | \(a^{2} x +\frac {2}{3} a b \,x^{3}+\frac {1}{5} b^{2} x^{5}\) | \(22\) |
parallelrisch | \(a^{2} x +\frac {2}{3} a b \,x^{3}+\frac {1}{5} b^{2} x^{5}\) | \(22\) |
gosper | \(\frac {x \left (3 b^{2} x^{4}+10 a b \,x^{2}+15 a^{2}\right )}{15}\) | \(25\) |
norman | \(\frac {a^{2} x^{2}+\frac {1}{5} b^{2} x^{6}+\frac {2}{3} a b \,x^{4}}{x}\) | \(28\) |
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none
Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=\frac {1}{5} \, b^{2} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=a^{2} x + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{5}}{5} \]
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none
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=\frac {1}{5} \, b^{2} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
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none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=\frac {1}{5} \, b^{2} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a x+b x^3\right )^2}{x^2} \, dx=a^2\,x+\frac {2\,a\,b\,x^3}{3}+\frac {b^2\,x^5}{5} \]
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