Integrand size = 19, antiderivative size = 22 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {b x^2+c x^4}}{c x} \]
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Time = 0.01 (sec) , antiderivative size = 22, 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 = {1602} \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {b x^2+c x^4}}{c x} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^2+c x^4}}{c x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {x^2 \left (b+c x^2\right )}}{c x} \]
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
trager | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{c x}\) | \(21\) |
gosper | \(\frac {\left (c \,x^{2}+b \right ) x}{c \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
default | \(\frac {\left (c \,x^{2}+b \right ) x}{c \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
risch | \(\frac {x \left (c \,x^{2}+b \right )}{\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, c}\) | \(26\) |
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {c x^{4} + b x^{2}}}{c x} \]
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\[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {x^{2}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {c x^{2} + b}}{c} \]
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none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=-\frac {\sqrt {b} \mathrm {sgn}\left (x\right )}{c} + \frac {\sqrt {c x^{2} + b}}{c \mathrm {sgn}\left (x\right )} \]
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Time = 12.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx=\frac {\sqrt {c\,x^4+b\,x^2}}{c\,x} \]
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