Integrand size = 26, antiderivative size = 20 \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d x}{b c \sqrt {c+d x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {21, 197} \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d x}{b c \sqrt {c+d x^2}} \]
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Rule 21
Rule 197
Rubi steps \begin{align*} \text {integral}& = \frac {d \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{b} \\ & = \frac {d x}{b c \sqrt {c+d x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d x}{b c \sqrt {c+d x^2}} \]
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Time = 2.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
default | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
trager | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
pseudoelliptic | \(\frac {d x}{b c \sqrt {d \,x^{2}+c}}\) | \(19\) |
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none
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d x^{2} + c} d x}{b c d x^{2} + b c^{2}} \]
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\[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d \int \frac {1}{c \sqrt {c + d x^{2}} + d x^{2} \sqrt {c + d x^{2}}}\, dx}{b} \]
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\[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + \frac {b c}{d}\right )} \sqrt {d x^{2} + c}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d x}{\sqrt {d x^{2} + c} b c} \]
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Time = 4.81 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\frac {b c}{d}+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {d\,x}{b\,c\,\sqrt {d\,x^2+c}} \]
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