Integrand size = 18, antiderivative size = 26 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=-\frac {a A}{x}+(A b+a B) x+\frac {1}{3} b B x^3 \]
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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 = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {459} \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=x (a B+A b)-\frac {a A}{x}+\frac {1}{3} b B x^3 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (A b \left (1+\frac {a B}{A b}\right )+\frac {a A}{x^2}+b B x^2\right ) \, dx \\ & = -\frac {a A}{x}+(A b+a B) x+\frac {1}{3} b B x^3 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=-\frac {a A}{x}+(A b+a B) x+\frac {1}{3} b B x^3 \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {b B \,x^{3}}{3}+A b x +B a x -\frac {a A}{x}\) | \(24\) |
risch | \(\frac {b B \,x^{3}}{3}+A b x +B a x -\frac {a A}{x}\) | \(24\) |
norman | \(\frac {\frac {b B \,x^{4}}{3}+\left (A b +B a \right ) x^{2}-A a}{x}\) | \(28\) |
parallelrisch | \(\frac {b B \,x^{4}+3 A b \,x^{2}+3 B a \,x^{2}-3 A a}{3 x}\) | \(31\) |
gosper | \(-\frac {-b B \,x^{4}-3 A b \,x^{2}-3 B a \,x^{2}+3 A a}{3 x}\) | \(32\) |
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none
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=\frac {B b x^{4} + 3 \, {\left (B a + A b\right )} x^{2} - 3 \, A a}{3 \, x} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=- \frac {A a}{x} + \frac {B b x^{3}}{3} + x \left (A b + B a\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{3} \, B b x^{3} + {\left (B a + A b\right )} x - \frac {A a}{x} \]
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{3} \, B b x^{3} + B a x + A b x - \frac {A a}{x} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^2} \, dx=x\,\left (A\,b+B\,a\right )-\frac {A\,a}{x}+\frac {B\,b\,x^3}{3} \]
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