Integrand size = 26, antiderivative size = 167 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {a^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {3 a b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]
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Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1126, 276} \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {3 a b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {a^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
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Rule 276
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^2 \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 x^2+3 a^2 b^4 x^4+3 a b^5 x^6+b^6 x^8\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {a^3 x^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b x^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac {3 a b^2 x^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac {b^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right )}{315 \left (a+b x^2\right )} \]
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Time = 0.76 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {x^{3} \left (35 b^{3} x^{6}+135 b^{2} x^{4} a +189 a^{2} b \,x^{2}+105 a^{3}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{315 \left (b \,x^{2}+a \right )^{3}}\) | \(58\) |
default | \(\frac {x^{3} \left (35 b^{3} x^{6}+135 b^{2} x^{4} a +189 a^{2} b \,x^{2}+105 a^{3}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{315 \left (b \,x^{2}+a \right )^{3}}\) | \(58\) |
risch | \(\frac {a^{3} x^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}+\frac {3 a^{2} b \,x^{5} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 \left (b \,x^{2}+a \right )}+\frac {3 a \,b^{2} x^{7} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{7 \left (b \,x^{2}+a \right )}+\frac {b^{3} x^{9} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{9 b \,x^{2}+9 a}\) | \(116\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {1}{9} \, b^{3} x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \]
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\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.21 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {1}{9} \, b^{3} x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.40 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {1}{9} \, b^{3} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{7} \, a b^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{5} \, a^{2} b x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{3} \, a^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \]
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Timed out. \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int x^2\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \]
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