Integrand size = 17, antiderivative size = 48 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}+\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x}-\frac {2 \sqrt {b x+c x^2}}{3 b x^2} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {b x+c x^2}}{3 b x^2}-\frac {(2 c) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{3 b} \\ & = -\frac {2 \sqrt {b x+c x^2}}{3 b x^2}+\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x (b+c x)} (-b+2 c x)}{3 b^2 x^2} \]
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Time = 1.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.50
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-2 c x +b \right ) \sqrt {x \left (c x +b \right )}}{3 b^{2} x^{2}}\) | \(24\) |
trager | \(-\frac {2 \left (-2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 b^{2} x^{2}}\) | \(26\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 b^{2} x \sqrt {x \left (c x +b \right )}}\) | \(29\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 x \,b^{2} \sqrt {c \,x^{2}+b x}}\) | \(31\) |
default | \(-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \]
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\[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {x \left (b + c x\right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {4 \, \sqrt {c x^{2} + b x} c}{3 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, b x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \]
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Time = 9.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=-\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b-2\,c\,x\right )}{3\,b^2\,x^2} \]
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