Integrand size = 21, antiderivative size = 36 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2054, 212} \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \]
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Rule 212
Rule 2054
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a x^2+b x^5}}\right ) \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {b}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 x \sqrt {a+b x^3} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{3 \sqrt {b} \sqrt {x^2 \left (a+b x^3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).
Time = 1.78 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64
method | result | size |
default | \(\frac {2 x^{\frac {3}{2}} \left (b \,x^{3}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, \sqrt {b}}\) | \(59\) |
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none
Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.81 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} + a\right )} \sqrt {b} \sqrt {x} - a^{2}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {-b} \sqrt {x}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]
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\[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{\frac {3}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
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\[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {3}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{3 \, \sqrt {b}} - \frac {2 \, \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{3 \, \sqrt {b} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{3/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \]
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