Integrand size = 15, antiderivative size = 28 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=a A x+\frac {1}{3} (A b+a B) x^3+\frac {1}{5} b B x^5 \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {380} \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{3} x^3 (a B+A b)+a A x+\frac {1}{5} b B x^5 \]
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Rule 380
Rubi steps \begin{align*} \text {integral}& = \int \left (a A+(A b+a B) x^2+b B x^4\right ) \, dx \\ & = a A x+\frac {1}{3} (A b+a B) x^3+\frac {1}{5} b B x^5 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=a A x+\frac {1}{3} (A b+a B) x^3+\frac {1}{5} b B x^5 \]
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Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
default | \(a A x +\frac {\left (A b +B a \right ) x^{3}}{3}+\frac {b B \,x^{5}}{5}\) | \(25\) |
norman | \(\frac {b B \,x^{5}}{5}+\left (\frac {A b}{3}+\frac {B a}{3}\right ) x^{3}+a A x\) | \(26\) |
gosper | \(\frac {1}{5} b B \,x^{5}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} B a +a A x\) | \(27\) |
risch | \(\frac {1}{5} b B \,x^{5}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} B a +a A x\) | \(27\) |
parallelrisch | \(\frac {1}{5} b B \,x^{5}+\frac {1}{3} x^{3} A b +\frac {1}{3} x^{3} B a +a A x\) | \(27\) |
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none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{5} \, B b x^{5} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} + A a x \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=A a x + \frac {B b x^{5}}{5} + x^{3} \left (\frac {A b}{3} + \frac {B a}{3}\right ) \]
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none
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{5} \, B b x^{5} + \frac {1}{3} \, {\left (B a + A b\right )} x^{3} + A a x \]
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none
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{5} \, B b x^{5} + \frac {1}{3} \, B a x^{3} + \frac {1}{3} \, A b x^{3} + A a x \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {B\,b\,x^5}{5}+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x^3+A\,a\,x \]
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