Integrand size = 19, antiderivative size = 56 \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {680, 674, 213} \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \]
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Rule 213
Rule 674
Rule 680
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{b} \\ & = \frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}+\frac {2 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b} \\ & = \frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {x} \left (\sqrt {b}-\sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{b^{3/2} \sqrt {x (b+c x)}} \]
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Time = 2.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}-\sqrt {b}\right )}{b^{\frac {3}{2}} \sqrt {x}\, \left (c x +b \right )}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.77 \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (c x^{2} + b x\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + b x} b \sqrt {x}}{b^{2} c x^{2} + b^{3} x}, \frac {2 \, {\left ({\left (c x^{2} + b x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + \sqrt {c x^{2} + b x} b \sqrt {x}\right )}}{b^{2} c x^{2} + b^{3} x}\right ] \]
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\[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {x}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )}}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {2}{\sqrt {c x + b} b} \]
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Timed out. \[ \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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