Integrand size = 15, antiderivative size = 18 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]
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Time = 0.00 (sec) , antiderivative size = 18, 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 = {267} \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x^4}}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]
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Time = 4.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
derivativedivides | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
default | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
trager | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
risch | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
elliptic | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
pseudoelliptic | \(\frac {\sqrt {b \,x^{4}+a}}{2 b}\) | \(15\) |
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\begin {cases} \frac {\sqrt {a + b x^{4}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]
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none
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]
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Time = 5.51 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b\,x^4+a}}{2\,b} \]
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