Integrand size = 19, antiderivative size = 25 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 25, 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} \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Rule 662
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 (x (b+c x))^{3/2}}{3 c x^{3/2}} \]
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Time = 1.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {2 \left (c x +b \right ) \sqrt {x \left (c x +b \right )}}{3 c \sqrt {x}}\) | \(23\) |
gosper | \(\frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 c \sqrt {x}}\) | \(25\) |
risch | \(\frac {2 \left (c x +b \right )^{2} \sqrt {x}}{3 \sqrt {x \left (c x +b \right )}\, c}\) | \(25\) |
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none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + b\right )}}{3 \, c \sqrt {x}} \]
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\[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\sqrt {x}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c} \]
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none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c} - \frac {2 \, b^{\frac {3}{2}}}{3 \, c} \]
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Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{\sqrt {x}} \,d x \]
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