Integrand size = 31, antiderivative size = 83 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {e \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac {(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1261, 654, 622, 31} \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]
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Rule 31
Rule 622
Rule 654
Rule 1261
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {d+e x}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right ) \\ & = \frac {e \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac {(b d-a e) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right )}{2 b} \\ & = \frac {e \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac {\left ((b d-a e) \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^2\right )}{2 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {e \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac {(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \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. \(276\) vs. \(2(83)=166\).
Time = 0.55 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.33 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\left (2 a+b x^2\right ) \left (b e x^2 \left (\sqrt {a^2} b x^2+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right )-(-b d+a e) \left (-a^2-a b x^2+\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right ) \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+(-b d+a e) \left (-a^2-a b x^2+\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}\right ) \log \left (b^2 \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )\right )\right )}{2 b^2 \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right ) \left (\sqrt {a^2} b x^2+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(-\frac {\left (-e \,x^{2} b +\ln \left (b \,x^{2}+a \right ) a e -\ln \left (b \,x^{2}+a \right ) b d \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(45\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-e \,x^{2} b +\ln \left (b \,x^{2}+a \right ) a e -\ln \left (b \,x^{2}+a \right ) b d \right )}{2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{2}}\) | \(55\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, e \,x^{2}}{2 \left (b \,x^{2}+a \right ) b}-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a e -b d \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) b^{2}}\) | \(72\) |
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {b e x^{2} + {\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
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Time = 3.91 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.81 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {\begin {cases} \frac {\left (\frac {a}{b} + x^{2}\right ) \left (- \frac {a e}{b} + d\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{\sqrt {b^{2} \left (\frac {a}{b} + x^{2}\right )^{2}}} + \frac {e \sqrt {a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {2 d \sqrt {a^{2} + 2 a b x^{2}} + \frac {e \left (- a^{2} \sqrt {a^{2} + 2 a b x^{2}} + \frac {\left (a^{2} + 2 a b x^{2}\right )^{\frac {3}{2}}}{3}\right )}{a b}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {d x^{2} + \frac {e x^{4}}{2}}{\sqrt {a^{2}}} & \text {otherwise} \end {cases}}{2} \]
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Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.37 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {e x^{2}}{2 \, b} + \frac {{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.48 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {1}{2} \, {\left (\frac {e x^{2}}{b} + \frac {{\left (b d - a e\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Time = 8.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.24 \[ \int \frac {x \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {e\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,b^2}-\frac {a\,b\,e\,\ln \left (a\,b+\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {b^2}+b^2\,x^2\right )}{2\,{\left (b^2\right )}^{3/2}}+\frac {b^2\,d\,\ln \left (b^2\,x^2+a\,b\right )\,\mathrm {sign}\left (2\,b^2\,x^2+2\,a\,b\right )}{2\,{\left (b^2\right )}^{3/2}} \]
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