Integrand size = 26, antiderivative size = 39 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^4} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 660, 37} \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^4} \]
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Rule 37
Rule 660
Rule 1125
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {a b+b^2 x}{x^3} \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )} \\ & = -\frac {\left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 a x^4} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (a+2 b x^2\right )}{4 x^4 \left (a+b x^2\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56
method | result | size |
pseudoelliptic | \(-\frac {\left (2 b \,x^{2}+a \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{4 x^{4}}\) | \(22\) |
gosper | \(-\frac {\left (2 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{4 x^{4} \left (b \,x^{2}+a \right )}\) | \(34\) |
default | \(-\frac {\left (2 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{4 x^{4} \left (b \,x^{2}+a \right )}\) | \(34\) |
risch | \(\frac {\left (-\frac {b \,x^{2}}{2}-\frac {a}{4}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{x^{4} \left (b \,x^{2}+a \right )}\) | \(35\) |
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {2 \, b x^{2} + a}{4 \, x^{4}} \]
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\[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=\int \frac {\sqrt {\left (a + b x^{2}\right )^{2}}}{x^{5}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {2 \, b x^{2} + a}{4 \, x^{4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {2 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, x^{4}} \]
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Time = 13.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^5} \, dx=-\frac {\left (2\,b\,x^2+a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{4\,x^4\,\left (b\,x^2+a\right )} \]
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