Integrand size = 18, antiderivative size = 26 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=-\frac {a A}{3 x^3}-\frac {A b+a B}{x}+b B x \]
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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} \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=-\frac {a B+A b}{x}-\frac {a A}{3 x^3}+b B x \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (b B+\frac {a A}{x^4}+\frac {A b+a B}{x^2}\right ) \, dx \\ & = -\frac {a A}{3 x^3}-\frac {A b+a B}{x}+b B x \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=-\frac {a A}{3 x^3}+\frac {-A b-a B}{x}+b B x \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
default | \(b B x -\frac {a A}{3 x^{3}}-\frac {A b +B a}{x}\) | \(25\) |
risch | \(b B x +\frac {\left (-A b -B a \right ) x^{2}-\frac {A a}{3}}{x^{3}}\) | \(28\) |
norman | \(\frac {b B \,x^{4}+\left (-A b -B a \right ) x^{2}-\frac {A a}{3}}{x^{3}}\) | \(29\) |
gosper | \(-\frac {-3 b B \,x^{4}+3 A b \,x^{2}+3 B a \,x^{2}+A a}{3 x^{3}}\) | \(31\) |
parallelrisch | \(-\frac {-3 b B \,x^{4}+3 A b \,x^{2}+3 B a \,x^{2}+A a}{3 x^{3}}\) | \(31\) |
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none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=\frac {3 \, B b x^{4} - 3 \, {\left (B a + A b\right )} x^{2} - A a}{3 \, x^{3}} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=B b x + \frac {- A a + x^{2} \left (- 3 A b - 3 B a\right )}{3 x^{3}} \]
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none
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=B b x - \frac {3 \, {\left (B a + A b\right )} x^{2} + A a}{3 \, x^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=B b x - \frac {3 \, B a x^{2} + 3 \, A b x^{2} + A a}{3 \, x^{3}} \]
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Time = 4.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx=B\,b\,x-\frac {\left (A\,b+B\,a\right )\,x^2+\frac {A\,a}{3}}{x^3} \]
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