Integrand size = 24, antiderivative size = 38 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{8 b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1121, 621} \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{8 b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
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Rule 621
Rule 1121
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{8 b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {a+b x^2}{8 b \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right )}{8 \left (b \,x^{2}+a \right )^{4} b}\) | \(23\) |
gosper | \(-\frac {b \,x^{2}+a}{8 b {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(24\) |
default | \(-\frac {b \,x^{2}+a}{8 b {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(24\) |
risch | \(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}}{8 \left (b \,x^{2}+a \right )^{5} b}\) | \(26\) |
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none
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{8 \, {\left (b^{5} x^{8} + 4 \, a b^{4} x^{6} + 6 \, a^{2} b^{3} x^{4} + 4 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]
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\[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {x}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{8 \, {\left (b^{5} x^{8} + 4 \, a b^{4} x^{6} + 6 \, a^{2} b^{3} x^{4} + 4 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{8 \, {\left (b x^{2} + a\right )}^{4} b \mathrm {sgn}\left (b x^{2} + a\right )} \]
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Time = 13.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,b\,{\left (b\,x^2+a\right )}^5} \]
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