Integrand size = 33, antiderivative size = 137 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {d \left (a+b x^2\right )}{2 a x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(b d-a e) \left (a+b x^2\right ) \log (x)}{a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.06 (sec) , antiderivative size = 137, 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} \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right )}{2 a x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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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^3 \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^2 \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^2}+\frac {-b d+a e}{a^2 b x}+\frac {b d-a e}{a^2 (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 )}{2 a x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(b d-a e) \left (a+b x^2\right ) \log (x)}{a^2 \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^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.20 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {a^2 d-\sqrt {a^2} d \sqrt {\left (a+b x^2\right )^2}+2 a (b d-a e) x^2 \log \left (x^2\right )-\left (-a+\sqrt {a^2}\right ) (-b d+a e) x^2 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+\left (a+\sqrt {a^2}\right ) (-b d+a e) x^2 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )}{4 \left (a^2\right )^{3/2} x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.42
method | result | size |
pseudoelliptic | \(-\frac {\left (x^{2} \left (a e -b d \right ) \ln \left (b \,x^{2}+a \right )-x^{2} \left (a e -b d \right ) \ln \left (x^{2}\right )+d a \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{2 a^{2} x^{2}}\) | \(58\) |
default | \(\frac {\left (b \,x^{2}+a \right ) \left (2 \ln \left (x \right ) a e \,x^{2}-2 \ln \left (x \right ) b d \,x^{2}-\ln \left (b \,x^{2}+a \right ) a e \,x^{2}+\ln \left (b \,x^{2}+a \right ) b d \,x^{2}-d a \right )}{2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a^{2} x^{2}}\) | \(79\) |
risch | \(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, d}{2 \left (b \,x^{2}+a \right ) a \,x^{2}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -b d \right ) \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{2}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -b d \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) a^{2}}\) | \(106\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {{\left (b d - a e\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (b d - a e\right )} x^{2} \log \left (x\right ) - a d}{2 \, a^{2} x^{2}} \]
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\[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {d + e x^{2}}{x^{3} \sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac {{\left (b d - a e\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {d}{2 \, a x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {{\left (b d \mathrm {sgn}\left (b x^{2} + a\right ) - a e \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} d \mathrm {sgn}\left (b x^{2} + a\right ) - a b e \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac {b d x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) - a e x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) - a d \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, a^{2} x^{2}} \]
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Time = 8.32 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.91 \[ \int \frac {d+e x^2}{x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {a\,b\,d\,\mathrm {atanh}\left (\frac {a^2+b\,a\,x^2}{\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}\right )}{2\,{\left (a^2\right )}^{3/2}}-\frac {e\,\ln \left (\frac {1}{x^2}\right )}{2\,\sqrt {a^2}}-\frac {d\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,a^2\,x^2}-\frac {e\,\ln \left (\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {a^2}+a^2+a\,b\,x^2\right )}{2\,\sqrt {a^2}} \]
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