Integrand size = 15, antiderivative size = 19 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 x}{b \sqrt {b x+c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {650} \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 x}{b \sqrt {b x+c x^2}} \]
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Rule 650
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{b \sqrt {b x+c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 x}{b \sqrt {x (b+c x)}} \]
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Time = 2.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {2 x}{b \sqrt {x \left (c x +b \right )}}\) | \(16\) |
trager | \(\frac {2 \sqrt {c \,x^{2}+b x}}{\left (c x +b \right ) b}\) | \(24\) |
gosper | \(\frac {2 x^{2} \left (c x +b \right )}{b \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(25\) |
default | \(-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\) | \(42\) |
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none
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x}}{b c x + b^{2}} \]
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\[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, x}{\sqrt {c x^{2} + b x} b} \]
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none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )} \sqrt {c}} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2\,x}{b\,\sqrt {x\,\left (b+c\,x\right )}} \]
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