Integrand size = 25, antiderivative size = 19 \[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\frac {2 \sqrt {-b x+a x^3}}{x} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2061} \[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\frac {2 \sqrt {a x^3-b x}}{x} \]
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Rule 2061
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {-b x+a x^3}}{x} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\frac {2 \sqrt {-b x+a x^3}}{x} \]
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Time = 0.85 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
trager | \(\frac {2 \sqrt {a \,x^{3}-b x}}{x}\) | \(18\) |
gosper | \(\frac {2 a \,x^{2}-2 b}{\sqrt {a \,x^{3}-b x}}\) | \(24\) |
default | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
risch | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
elliptic | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
pseudoelliptic | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} - b x}}{x} \]
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\[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\int \frac {a x^{2} + b}{x \sqrt {x \left (a x^{2} - b\right )}}\, dx \]
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\[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} x} \,d x } \]
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\[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} x} \,d x } \]
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Time = 5.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx=\frac {2\,\sqrt {a\,x^3-b\,x}}{x} \]
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