Integrand size = 17, antiderivative size = 19 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=2 b \sqrt {x}+\frac {2}{5} c x^{5/2} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=2 b \sqrt {x}+\frac {2}{5} c x^{5/2} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{\sqrt {x}}+c x^{3/2}\right ) \, dx \\ & = 2 b \sqrt {x}+\frac {2}{5} c x^{5/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=\frac {2}{5} \left (5 b \sqrt {x}+c x^{5/2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {2 c \,x^{\frac {5}{2}}}{5}+2 b \sqrt {x}\) | \(14\) |
default | \(\frac {2 c \,x^{\frac {5}{2}}}{5}+2 b \sqrt {x}\) | \(14\) |
gosper | \(\frac {2 \sqrt {x}\, \left (c \,x^{2}+5 b \right )}{5}\) | \(15\) |
trager | \(\left (\frac {2 c \,x^{2}}{5}+2 b \right ) \sqrt {x}\) | \(15\) |
risch | \(\frac {2 \sqrt {x}\, \left (c \,x^{2}+5 b \right )}{5}\) | \(15\) |
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Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=\frac {2}{5} \, {\left (c x^{2} + 5 \, b\right )} \sqrt {x} \]
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Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=2 b \sqrt {x} + \frac {2 c x^{\frac {5}{2}}}{5} \]
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Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=\frac {2}{5} \, c x^{\frac {5}{2}} + 2 \, b \sqrt {x} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=\frac {2}{5} \, c x^{\frac {5}{2}} + 2 \, b \sqrt {x} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {b x^2+c x^4}{x^{5/2}} \, dx=\frac {2\,\sqrt {x}\,\left (c\,x^2+5\,b\right )}{5} \]
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