Integrand size = 29, antiderivative size = 21 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {a+c x^4}}{2 a x^2} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4, 270} \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {a+c x^4}}{2 a x^2} \]
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Rule 4
Rule 270
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \sqrt {a+c x^4}} \, dx \\ & = -\frac {\sqrt {a+c x^4}}{2 a x^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {a+c x^4}}{2 a x^2} \]
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Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
default | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
trager | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
risch | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
pseudoelliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \]
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Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=- \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{2 a} \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=\frac {\sqrt {c}}{{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a} \]
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Time = 12.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx=-\frac {\sqrt {c\,x^4+a}}{2\,a\,x^2} \]
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