Integrand size = 15, antiderivative size = 25 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2025} \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
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Rule 2025
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 c x^3} \]
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Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c x}\) | \(29\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c x}\) | \(29\) |
trager | \(\frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c x}\) | \(29\) |
risch | \(\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (c \,x^{2}+b \right )}{3 x c}\) | \(29\) |
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none
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{3 \, c x} \]
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\[ \int \sqrt {b x^2+c x^4} \, dx=\int \sqrt {b x^{2} + c x^{4}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}}}{3 \, c} \]
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none
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, c} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, c} \]
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Time = 12.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \sqrt {b x^2+c x^4} \, dx=\frac {\left (\frac {b}{3\,c}+\frac {x^2}{3}\right )\,\sqrt {c\,x^4+b\,x^2}}{x} \]
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