Integrand size = 26, antiderivative size = 189 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \left (a+b x^2\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1126, 272, 46} \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \log (x) \left (a+b x^2\right )}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 46
Rule 272
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^3 \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 {1}{x^2 \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 {1}{a^3 b^3 x^2}-\frac {3}{a^4 b^2 x}+\frac {1}{a^2 b (a+b x)^3}+\frac {2}{a^3 b (a+b x)^2}+\frac {3}{a^4 b (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {b}{a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b \left (a+b x^2\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(901\) vs. \(2(189)=378\).
Time = 1.41 (sec) , antiderivative size = 901, normalized size of antiderivative = 4.77 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {-2 a^6-5 a^5 b x^2+2 a^4 b^2 x^4+4 a^3 b^3 x^6-a b^5 x^{10}+2 a^4 \sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}+3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2}-5 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2}+a \sqrt {a^2} b^3 x^6 \sqrt {\left (a+b x^2\right )^2}-\sqrt {a^2} b^4 x^8 \sqrt {\left (a+b x^2\right )^2}+6 b x^2 \left (\left (a^2\right )^{3/2} b^2 x^4+a^4 \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )+a^3 b x^2 \left (2 \sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right ) \text {arctanh}\left (\frac {b x^2}{\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}}\right )-6 b x^2 \left (a^5+2 a^4 b x^2-\left (a^2\right )^{3/2} b x^2 \sqrt {\left (a+b x^2\right )^2}+a^3 \left (b^2 x^4-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right )\right ) \log \left (x^2\right )+3 a^5 b x^2 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+6 a^4 b^2 x^4 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^3 b^3 x^6 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^5 b x^2 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+6 a^4 b^2 x^4 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+3 a^3 b^3 x^6 \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 a^3 \sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )-3 \left (a^2\right )^{3/2} b^2 x^4 \sqrt {\left (a+b x^2\right )^2} \log \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )}{2 a^4 \sqrt {a^2} x^2 \left (a^2+a b x^2-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right )^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.48
method | result | size |
pseudoelliptic | \(-\frac {\left (-3 b \,x^{2} \left (b \,x^{2}+a \right )^{2} \ln \left (b \,x^{2}+a \right )+3 b \,x^{2} \left (b \,x^{2}+a \right )^{2} \ln \left (x^{2}\right )+a \left (3 b^{2} x^{4}+\frac {9}{2} a b \,x^{2}+a^{2}\right )\right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right )^{2} x^{2} a^{4}}\) | \(90\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {3 b^{2} x^{4}}{2 a^{3}}-\frac {9 b \,x^{2}}{4 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{2}+a \right )^{3} x^{2}}-\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{4}}+\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \ln \left (-b \,x^{2}-a \right )}{2 \left (b \,x^{2}+a \right ) a^{4}}\) | \(117\) |
default | \(-\frac {\left (12 \ln \left (x \right ) x^{6} b^{3}-6 \ln \left (b \,x^{2}+a \right ) x^{6} b^{3}+24 \ln \left (x \right ) x^{4} a \,b^{2}-12 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2}+6 b^{2} x^{4} a +12 \ln \left (x \right ) x^{2} a^{2} b -6 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b +9 a^{2} b \,x^{2}+2 a^{3}\right ) \left (b \,x^{2}+a \right )}{4 x^{2} a^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(133\) |
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Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]
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\[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {6 \, b^{2} x^{4} + 9 \, a b x^{2} + 2 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} + \frac {3 \, b \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \left (x\right )}{a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9 \, b^{3} x^{4} + 22 \, a b^{2} x^{2} + 14 \, a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b x^{2} - a}{2 \, a^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
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Timed out. \[ \int \frac {1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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