Integrand size = 19, antiderivative size = 25 \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2039} \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {x^{21/2}}{7 a \left (a x+b x^3\right )^{7/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{21/2}}{7 a \left (x \left (a+b x^2\right )\right )^{7/2}} \]
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Time = 1.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08
method | result | size |
gosper | \(\frac {\left (b \,x^{2}+a \right ) x^{\frac {23}{2}}}{7 a \left (b \,x^{3}+a x \right )^{\frac {9}{2}}}\) | \(27\) |
default | \(\frac {x^{\frac {13}{2}} \sqrt {x \left (b \,x^{2}+a \right )}}{7 a \left (b \,x^{2}+a \right )^{4}}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.48 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {\sqrt {b x^{3} + a x} x^{\frac {13}{2}}}{7 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )}} \]
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Timed out. \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {x^{\frac {21}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{7}}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} \]
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Timed out. \[ \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{21/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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