Integrand size = 22, antiderivative size = 47 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A}{a x \sqrt {a+b x^2}}-\frac {(2 A b-a B) x}{a^2 \sqrt {a+b x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {464, 197} \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {x (2 A b-a B)}{a^2 \sqrt {a+b x^2}}-\frac {A}{a x \sqrt {a+b x^2}} \]
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Rule 197
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{a x \sqrt {a+b x^2}}-\frac {(2 A b-a B) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{a} \\ & = -\frac {A}{a x \sqrt {a+b x^2}}-\frac {(2 A b-a B) x}{a^2 \sqrt {a+b x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a A-2 A b x^2+a B x^2}{a^2 x \sqrt {a+b x^2}} \]
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Time = 2.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(-\frac {2 A b \,x^{2}-B a \,x^{2}+A a}{x \sqrt {b \,x^{2}+a}\, a^{2}}\) | \(36\) |
trager | \(-\frac {2 A b \,x^{2}-B a \,x^{2}+A a}{x \sqrt {b \,x^{2}+a}\, a^{2}}\) | \(36\) |
pseudoelliptic | \(-\frac {\left (-x^{2} B +A \right ) a +2 A b \,x^{2}}{\sqrt {b \,x^{2}+a}\, x \,a^{2}}\) | \(36\) |
risch | \(-\frac {A \sqrt {b \,x^{2}+a}}{a^{2} x}-\frac {x \left (A b -B a \right )}{\sqrt {b \,x^{2}+a}\, a^{2}}\) | \(43\) |
default | \(\frac {B x}{a \sqrt {b \,x^{2}+a}}+A \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )\) | \(53\) |
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (B a - 2 \, A b\right )} x^{2} - A a\right )} \sqrt {b x^{2} + a}}{a^{2} b x^{3} + a^{3} x} \]
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Time = 2.49 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) + \frac {B x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B x}{\sqrt {b x^{2} + a} a} - \frac {2 \, A b x}{\sqrt {b x^{2} + a} a^{2}} - \frac {A}{\sqrt {b x^{2} + a} a x} \]
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a} + \frac {{\left (B a - A b\right )} x}{\sqrt {b x^{2} + a} a^{2}} \]
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Time = 5.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {b\,x^2+a}\,\left (\frac {A}{a}-x^2\,\left (\frac {B}{a}-\frac {2\,A\,b}{a^2}\right )\right )}{b\,x^3+a\,x} \]
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