Integrand size = 33, antiderivative size = 26 \[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {2 \sqrt {-b x+a x^3}}{b-a x^2} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2081, 460} \[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {2 x}{\sqrt {a x^3-b x}} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {b+a x^2}{\sqrt {x} \left (-b+a x^2\right )^{3/2}} \, dx}{\sqrt {-b x+a x^3}} \\ & = -\frac {2 x}{\sqrt {-b x+a x^3}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {2 x}{\sqrt {-b x+a x^3}} \]
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Time = 0.99 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {a \,x^{3}-b x}}\) | \(16\) |
default | \(-\frac {2 x}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(17\) |
pseudoelliptic | \(-\frac {2 x}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(17\) |
elliptic | \(-\frac {2 x}{\sqrt {\left (x^{2}-\frac {b}{a}\right ) a x}}\) | \(19\) |
trager | \(-\frac {2 \sqrt {a \,x^{3}-b x}}{a \,x^{2}-b}\) | \(26\) |
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none
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=-\frac {2 \, \sqrt {a x^{3} - b x}}{a x^{2} - b} \]
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\[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int \frac {a x^{2} + b}{\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}\, dx \]
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\[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}} \,d x } \]
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\[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}} \,d x } \]
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Time = 5.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {b+a x^2}{\left (-b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {2\,\sqrt {a\,x^3-b\,x}}{b-a\,x^2} \]
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