Integrand size = 31, antiderivative size = 77 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1261, 654, 621} \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 621
Rule 654
Rule 1261
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) \text {Subst}\left (\int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 b} \\ & = -\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \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. \(181\) vs. \(2(77)=154\).
Time = 0.65 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.35 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {x^2 \left (a^3 b d x^2+a^2 b^2 e x^6+a^4 \left (2 d+e x^2\right )+a \left (-b^3 d x^6+\sqrt {a^2} b x^2 \sqrt {\left (a+b x^2\right )^2} \left (d+e x^2\right )\right )-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2} \left (b^2 d x^4+a^2 \left (2 d+e x^2\right )\right )\right )}{4 a^4 \left (a+b x^2\right ) \left (\sqrt {a^2} b x^2+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.48
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (2 e \,x^{2}+d \right ) b +a e \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{4 \left (b \,x^{2}+a \right )^{2} b^{2}}\) | \(37\) |
gosper | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 e \,x^{2} b +a e +b d \right )}{4 b^{2} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(38\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 e \,x^{2} b +a e +b d \right )}{4 b^{2} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(38\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {e \,x^{2}}{2 b}-\frac {a e +b d}{4 b^{2}}\right )}{\left (b \,x^{2}+a \right )^{3}}\) | \(44\) |
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.55 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {2 \, b e x^{2} + b d + a e}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
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\[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {x \left (d + e x^{2}\right )}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {{\left (2 \, b x^{2} + a\right )} e}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} - \frac {d}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.49 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {2 \, b e x^{2} + b d + a e}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
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Time = 7.73 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (2\,b\,e\,x^2+a\,e+b\,d\right )}{4\,b^2\,{\left (b\,x^2+a\right )}^3} \]
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