Integrand size = 33, antiderivative size = 92 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {d \left (a+b x^2\right ) \log (x)}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1264, 457, 78} \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {d \log (x) \left (a+b x^2\right )}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 78
Rule 457
Rule 1264
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^2\right ) \int \frac {d+e x^2}{x \left (a b+b^2 x^2\right )} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\left (a b+b^2 x^2\right ) \text {Subst}\left (\int \frac {d+e x}{x \left (a b+b^2 x\right )} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\left (a b+b^2 x^2\right ) \text {Subst}\left (\int \left (\frac {d}{a b x}+\frac {-b d+a e}{a b (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {d \left (a+b x^2\right ) \log (x)}{a \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.36 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {1}{4} \left (\left (\frac {2 d}{a}-\frac {4 e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}}{b x^2}\right )+\frac {d \left (-2 \log \left (x^2\right )+\log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+\log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )\right )}{\sqrt {a^2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(\frac {\left (d \ln \left (x^{2}\right ) b +\ln \left (b \,x^{2}+a \right ) a e -\ln \left (b \,x^{2}+a \right ) b d \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{2 a b}\) | \(48\) |
default | \(\frac {\left (b \,x^{2}+a \right ) \left (2 d \ln \left (x \right ) b +\ln \left (b \,x^{2}+a \right ) a e -\ln \left (b \,x^{2}+a \right ) b d \right )}{2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a b}\) | \(57\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, d \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -b d \right ) \ln \left (-b \,x^{2}-a \right )}{2 \left (b \,x^{2}+a \right ) a b}\) | \(76\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.36 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {2 \, b d \log \left (x\right ) - {\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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\[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {d + e x^{2}}{x \sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.38 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {d \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {d \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, a} - \frac {{\left (b d \mathrm {sgn}\left (b x^{2} + a\right ) - a e \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \]
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Time = 8.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x^2}{x \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {e\,\ln \left (b^2\,x^2+a\,b\right )\,\mathrm {sign}\left (2\,b^2\,x^2+2\,a\,b\right )}{2\,\sqrt {b^2}}-\frac {d\,\ln \left (\frac {1}{x^2}\right )}{2\,\sqrt {a^2}}-\frac {d\,\ln \left (\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {a^2}+a^2+a\,b\,x^2\right )}{2\,\sqrt {a^2}} \]
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