Integrand size = 15, antiderivative size = 47 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} \sqrt {a+\frac {b}{x^3}} x^3+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 43, 65, 214} \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}}+\frac {1}{3} x^3 \sqrt {a+\frac {b}{x^3}} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = \frac {1}{3} \sqrt {a+\frac {b}{x^3}} x^3-\frac {1}{6} b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right ) \\ & = \frac {1}{3} \sqrt {a+\frac {b}{x^3}} x^3-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right ) \\ & = \frac {1}{3} \sqrt {a+\frac {b}{x^3}} x^3+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} \sqrt {a+\frac {b}{x^3}} x^{3/2} \left (x^{3/2}+\frac {b \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )}{\sqrt {a} \sqrt {b+a x^3}}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {\sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x^{2} \left (\sqrt {x \left (a \,x^{3}+b \right )}\, x \sqrt {a}+\operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) b \right )}{3 \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a}}\) | \(71\) |
risch | \(\frac {x^{3} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{3}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) b x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {x \left (a \,x^{3}+b \right )}}{3 \sqrt {a}\, \left (a \,x^{3}+b \right )}\) | \(79\) |
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Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.19 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\left [\frac {4 \, a x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}} + \sqrt {a} b \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}{12 \, a}, \frac {2 \, a x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}} - \sqrt {-a} b \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{6 \, a}\right ] \]
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Time = 1.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {\sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{3} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{3 \sqrt {a}} \]
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Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.13 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} \, \sqrt {a + \frac {b}{x^{3}}} x^{3} - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{6 \, \sqrt {a}} \]
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Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} \, \sqrt {a x^{4} + b x} x - \frac {b \arctan \left (\frac {\sqrt {a + \frac {b}{x^{3}}}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} \]
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Time = 6.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.19 \[ \int \sqrt {a+\frac {b}{x^3}} x^2 \, dx=\frac {x^3\,\sqrt {a+\frac {b}{x^3}}}{3}+\frac {b\,\ln \left (x^6\,\left (\sqrt {a+\frac {b}{x^3}}-\sqrt {a}\right )\,{\left (\sqrt {a+\frac {b}{x^3}}+\sqrt {a}\right )}^3\right )}{6\,\sqrt {a}} \]
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