Integrand size = 33, antiderivative size = 161 \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {d}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b d-a e}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \left (a+b x^2\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 161, 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} \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {b d-a e}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \log (x) \left (a+b x^2\right )}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 78
Rule 457
Rule 1264
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {d+e x^2}{x \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {d+e x}{x \left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \left (\frac {d}{a^3 b^3 x}+\frac {-b d+a e}{a b^3 (a+b x)^3}-\frac {d}{a^2 b^2 (a+b x)^2}-\frac {d}{a^3 b^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}{2 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {b d-a e}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \left (a+b x^2\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.57 \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {a \left (3 a b d-a^2 e+2 b^2 d x^2\right )+4 b d \left (a+b x^2\right )^2 \log (x)-2 b d \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^3 b \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (2 d b \left (b \,x^{2}+a \right )^{2} \ln \left (b \,x^{2}+a \right )-2 d b \left (b \,x^{2}+a \right )^{2} \ln \left (x^{2}\right )+a \left (-2 b^{2} d \,x^{2}+e \,a^{2}-3 d a b \right )\right )}{4 a^{3} b \left (b \,x^{2}+a \right )^{2}}\) | \(87\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {b d \,x^{2}}{2 a^{2}}-\frac {a e -3 b d}{4 a b}\right )}{\left (b \,x^{2}+a \right )^{3}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, d \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{3}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, d \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) a^{3}}\) | \(111\) |
default | \(\frac {\left (4 \ln \left (x \right ) b^{3} d \,x^{4}-2 \ln \left (b \,x^{2}+a \right ) b^{3} d \,x^{4}+8 \ln \left (x \right ) a \,b^{2} d \,x^{2}-4 \ln \left (b \,x^{2}+a \right ) a \,b^{2} d \,x^{2}+2 b^{2} d \,x^{2} a +4 \ln \left (x \right ) a^{2} b d -2 \ln \left (b \,x^{2}+a \right ) a^{2} b d -a^{3} e +3 a^{2} b d \right ) \left (b \,x^{2}+a \right )}{4 b \,a^{3} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(133\) |
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Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {2 \, a b^{2} d x^{2} + 3 \, a^{2} b d - a^{3} e - 2 \, {\left (b^{3} d x^{4} + 2 \, a b^{2} d x^{2} + a^{2} b d\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{3} d x^{4} + 2 \, a b^{2} d x^{2} + a^{2} b d\right )} \log \left (x\right )}{4 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]
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\[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {d + e x^{2}}{x \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55 \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {1}{4} \, d {\left (\frac {2 \, b x^{2} + 3 \, a}{a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}} - \frac {2 \, \log \left (b x^{2} + a\right )}{a^{3}} + \frac {4 \, \log \left (x\right )}{a^{3}}\right )} - \frac {e}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {d \log \left (x^{2}\right )}{2 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {d \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b^{3} d x^{4} + 8 \, a b^{2} d x^{2} + 6 \, a^{2} b d - a^{3} e}{4 \, {\left (b x^{2} + a\right )}^{2} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )} \]
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Timed out. \[ \int \frac {d+e x^2}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {e\,x^2+d}{x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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