Integrand size = 33, antiderivative size = 223 \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {2 b d-a e}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(3 b d-a e) \left (a+b x^2\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(3 b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.12 (sec) , antiderivative size = 223, 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} \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\log (x) \left (a+b x^2\right ) (3 b d-a e)}{a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) (3 b d-a e) \log \left (a+b x^2\right )}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {2 b d-a e}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 78
Rule 457
Rule 1264
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {d+e x^2}{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 {d+e x}{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 {d}{a^3 b^3 x^2}+\frac {-3 b d+a e}{a^4 b^3 x}+\frac {b d-a e}{a^2 b^2 (a+b x)^3}+\frac {2 b d-a e}{a^3 b^2 (a+b x)^2}+\frac {3 b d-a e}{a^4 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 {2 b d-a e}{2 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b d-a e}{4 a^2 \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 )}{2 a^3 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {(3 b d-a e) \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 d-a e) \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*}
Time = 1.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.58 \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {a \left (-6 b^2 d x^4+a^2 \left (-2 d+3 e x^2\right )+a b \left (-9 d x^2+2 e x^4\right )\right )+4 (-3 b d+a e) x^2 \left (a+b x^2\right )^2 \log (x)+2 (3 b d-a e) x^2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^4 x^2 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (x^{2} \left (b \,x^{2}+a \right )^{2} \left (a e -3 b d \right ) \ln \left (b \,x^{2}+a \right )-x^{2} \left (b \,x^{2}+a \right )^{2} \left (a e -3 b d \right ) \ln \left (x^{2}\right )+a \left (\left (-a b e +3 b^{2} d \right ) x^{4}-\frac {3 a \left (a e -3 b d \right ) x^{2}}{2}+d \,a^{2}\right )\right )}{2 \left (b \,x^{2}+a \right )^{2} x^{2} a^{4}}\) | \(120\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {b \left (a e -3 b d \right ) x^{4}}{2 a^{3}}+\frac {3 \left (a e -3 b d \right ) x^{2}}{4 a^{2}}-\frac {d}{2 a}\right )}{\left (b \,x^{2}+a \right )^{3} x^{2}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -3 b d \right ) \ln \left (x \right )}{\left (b \,x^{2}+a \right ) a^{4}}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -3 b d \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) a^{4}}\) | \(141\) |
default | \(\frac {\left (4 \ln \left (x \right ) a \,b^{2} e \,x^{6}-12 \ln \left (x \right ) b^{3} d \,x^{6}-2 \ln \left (b \,x^{2}+a \right ) a \,b^{2} e \,x^{6}+6 \ln \left (b \,x^{2}+a \right ) b^{3} d \,x^{6}+8 \ln \left (x \right ) a^{2} b e \,x^{4}-24 \ln \left (x \right ) a \,b^{2} d \,x^{4}-4 \ln \left (b \,x^{2}+a \right ) a^{2} b e \,x^{4}+12 \ln \left (b \,x^{2}+a \right ) a \,b^{2} d \,x^{4}+2 a^{2} b e \,x^{4}-6 a \,b^{2} d \,x^{4}+4 \ln \left (x \right ) a^{3} e \,x^{2}-12 \ln \left (x \right ) a^{2} b d \,x^{2}-2 \ln \left (b \,x^{2}+a \right ) a^{3} e \,x^{2}+6 \ln \left (b \,x^{2}+a \right ) a^{2} b d \,x^{2}+3 a^{3} e \,x^{2}-9 a^{2} b d \,x^{2}-2 a^{3} d \right ) \left (b \,x^{2}+a \right )}{4 x^{2} a^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(249\) |
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Time = 0.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} + 2 \, a^{3} d + 3 \, {\left (3 \, a^{2} b d - a^{3} e\right )} x^{2} - 2 \, {\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \, {\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} + {\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \, {\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} + {\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]
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\[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {d + e x^{2}}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.62 \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {1}{4} \, d {\left (\frac {6 \, b^{2} x^{4} + 9 \, a b x^{2} + 2 \, a^{2}}{a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}} - \frac {6 \, b \log \left (b x^{2} + a\right )}{a^{4}} + \frac {12 \, b \log \left (x\right )}{a^{4}}\right )} + \frac {1}{4} \, e {\left (\frac {2 \, b x^{2} + 3 \, a}{a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}} - \frac {2 \, \log \left (b x^{2} + a\right )}{a^{3}} + \frac {4 \, \log \left (x\right )}{a^{3}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {{\left (3 \, b d - a e\right )} \log \left (x^{2}\right )}{2 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {{\left (3 \, b^{2} d - a b e\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9 \, b^{3} d x^{4} - 3 \, a b^{2} e x^{4} + 22 \, a b^{2} d x^{2} - 8 \, a^{2} b e x^{2} + 14 \, a^{2} b d - 6 \, a^{3} e}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b d x^{2} - a e x^{2} - a d}{2 \, a^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
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Timed out. \[ \int \frac {d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {e\,x^2+d}{x^3\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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