Integrand size = 26, antiderivative size = 127 \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^4 \left (a+b x^2\right )}{4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.07 (sec) , antiderivative size = 127, 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 = {1125, 660, 45} \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {x^4 \left (a+b x^2\right )}{4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 45
Rule 660
Rule 1125
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right ) \\ & = \frac {\left (a b+b^2 x^2\right ) \text {Subst}\left (\int \frac {x^2}{a b+b^2 x} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {\left (a b+b^2 x^2\right ) \text {Subst}\left (\int \left (-\frac {a}{b^3}+\frac {x}{b^2}+\frac {a^2}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {a x^2 \left (a+b x^2\right )}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^4 \left (a+b x^2\right )}{4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^2 \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.31 \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=-\frac {\frac {b x^2 \left (-2 a+b x^2\right ) \left (\sqrt {a^2} b x^2+a \left (\sqrt {a^2}-\sqrt {\left (a+b x^2\right )^2}\right )\right )}{a^2+a b x^2-\sqrt {a^2} \sqrt {\left (a+b x^2\right )^2}}-2 a^2 \log \left (\sqrt {a^2}-b x^2-\sqrt {\left (a+b x^2\right )^2}\right )+2 a^2 \log \left (b^3 \left (\sqrt {a^2}+b x^2-\sqrt {\left (a+b x^2\right )^2}\right )\right )}{4 b^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.33
method | result | size |
pseudoelliptic | \(\frac {\left (b^{2} x^{4}-2 a b \,x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2}\right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{4 b^{3}}\) | \(42\) |
default | \(\frac {\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}-2 a b \,x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2}\right )}{4 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{3}}\) | \(52\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-b \,x^{2}+a \right )^{2}}{4 \left (b \,x^{2}+a \right ) b^{3}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, a^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) b^{3}}\) | \(73\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.26 \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {b^{2} x^{4} - 2 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
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\[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {x^{5}}{\sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{4 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.46 \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\frac {a^{2} \log \left ({\left | b x^{2} + a \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {b x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) - 2 \, a x^{2} \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, b^{2}} \]
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Timed out. \[ \int \frac {x^5}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx=\int \frac {x^5}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
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