Integrand size = 15, antiderivative size = 62 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {8 b \sqrt {a+\frac {b}{x^2}} x}{3 a^3}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]
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Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 198, 197} \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=-\frac {8 b x \sqrt {a+\frac {b}{x^2}}}{3 a^3}+\frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \]
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Rule 197
Rule 198
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}}-\frac {(4 b) \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx}{3 a} \\ & = \frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}}-\frac {(8 b) \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx}{3 a^2} \\ & = \frac {4 b x}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {8 b \sqrt {a+\frac {b}{x^2}} x}{3 a^3}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x^2}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {-8 b^2-4 a b x^2+a^2 x^4}{3 a^3 \sqrt {a+\frac {b}{x^2}} x} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79
method | result | size |
gosper | \(\frac {\left (a \,x^{2}+b \right ) \left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right )}{3 a^{3} x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(49\) |
default | \(\frac {\left (a \,x^{2}+b \right ) \left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right )}{3 a^{3} x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(49\) |
trager | \(\frac {\left (a^{2} x^{4}-4 a b \,x^{2}-8 b^{2}\right ) x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 \left (a \,x^{2}+b \right ) a^{3}}\) | \(53\) |
risch | \(\frac {\left (a \,x^{2}-5 b \right ) \left (a \,x^{2}+b \right )}{3 a^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}-\frac {b^{2}}{a^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\) | \(63\) |
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Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {{\left (a^{2} x^{5} - 4 \, a b x^{3} - 8 \, b^{2} x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{4} x^{2} + a^{3} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (56) = 112\).
Time = 0.64 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.53 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {a^{3} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {3 a^{2} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {12 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} - \frac {8 b^{\frac {15}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a^{5} b^{4} x^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6}} \]
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Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 6 \, \sqrt {a + \frac {b}{x^{2}}} b x}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a + \frac {b}{x^{2}}} a^{3} x} \]
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Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=\frac {8 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a x^{2} + b} a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x^{2} + b} a^{6} b}{3 \, a^{9} \mathrm {sgn}\left (x\right )} \]
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Time = 6.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx=-\frac {-a^2\,x^4+4\,a\,b\,x^2+8\,b^2}{3\,a^3\,x\,\sqrt {a+\frac {b}{x^2}}} \]
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