Integrand size = 26, antiderivative size = 163 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {b^3 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \]
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Time = 0.03 (sec) , antiderivative size = 163, 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, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^3 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {a^3 \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
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Rule 45
Rule 272
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{x} \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \left (3 a^2 b^4+\frac {a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {3 a^2 b x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a b^2 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {b^3 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (b x^2 \left (18 a^2+9 a b x^2+2 b^2 x^4\right )+12 a^3 \log (x)\right )}{12 \left (a+b x^2\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.29
method | result | size |
pseudoelliptic | \(\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (2 b^{3} x^{6}+9 b^{2} x^{4} a +6 a^{3} \ln \left (x^{2}\right )+18 a^{2} b \,x^{2}\right )}{12}\) | \(47\) |
default | \(\frac {{\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (2 b^{3} x^{6}+9 b^{2} x^{4} a +18 a^{2} b \,x^{2}+12 a^{3} \ln \left (x \right )\right )}{12 \left (b \,x^{2}+a \right )^{3}}\) | \(57\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \left (\frac {1}{6} b^{2} x^{6}+\frac {3}{4} a b \,x^{4}+\frac {3}{2} a^{2} x^{2}\right )}{b \,x^{2}+a}+\frac {a^{3} \ln \left (x \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\) | \(74\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.20 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\frac {1}{6} \, b^{3} x^{6} + \frac {3}{4} \, a b^{2} x^{4} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} \log \left (x\right ) \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x}\, dx \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\frac {1}{6} \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{4} \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x} \,d x \]
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