Integrand size = 19, antiderivative size = 23 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {662} \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]
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Rule 662
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {x (b+c x)}} \]
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Time = 2.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) x^{\frac {3}{2}}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(25\) |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}}{\sqrt {x}\, \left (c x +b \right ) c}\) | \(25\) |
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none
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x} \sqrt {x}}{c^{2} x^{2} + b c x} \]
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\[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\sqrt {c x + b} c} + \frac {2}{\sqrt {b} c} \]
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Timed out. \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{3/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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