Integrand size = 24, antiderivative size = 56 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \left (-b+a x^3\right ) \sqrt {b+a x^3}}{3 x^3}-\frac {4}{3} a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 79, 52, 65, 214} \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=-\frac {4}{3} a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )-\frac {2 \left (a x^3+b\right )^{3/2}}{3 x^3}+\frac {4}{3} a \sqrt {a x^3+b} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {b+a x} (2 b+a x)}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{3} (2 a) \text {Subst}\left (\int \frac {\sqrt {b+a x}}{x} \, dx,x,x^3\right ) \\ & = \frac {4}{3} a \sqrt {b+a x^3}-\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{3} (2 a b) \text {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right ) \\ & = \frac {4}{3} a \sqrt {b+a x^3}-\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{3} (4 b) \text {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right ) \\ & = \frac {4}{3} a \sqrt {b+a x^3}-\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3}-\frac {4}{3} a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \left (-b+a x^3\right ) \sqrt {b+a x^3}}{3 x^3}-\frac {4}{3} a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right ) \]
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Time = 0.89 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88
method | result | size |
elliptic | \(-\frac {2 b \sqrt {a \,x^{3}+b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}+b}}{3}-\frac {4 a \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\) | \(49\) |
risch | \(-\frac {2 b \sqrt {a \,x^{3}+b}}{3 x^{3}}+a \left (\frac {2 \sqrt {a \,x^{3}+b}}{3}-\frac {4 \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\right )\) | \(50\) |
pseudoelliptic | \(\frac {2 a \,x^{3} \sqrt {a \,x^{3}+b}\, \sqrt {b}-4 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right ) a b \,x^{3}-2 b^{\frac {3}{2}} \sqrt {a \,x^{3}+b}}{3 x^{3} \sqrt {b}}\) | \(63\) |
default | \(a \left (\frac {2 \sqrt {a \,x^{3}+b}}{3}-\frac {2 \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\right )+2 b \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )\) | \(73\) |
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Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\left [\frac {2 \, {\left (a \sqrt {b} x^{3} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) + \sqrt {a x^{3} + b} {\left (a x^{3} - b\right )}\right )}}{3 \, x^{3}}, \frac {2 \, {\left (2 \, a \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right ) + \sqrt {a x^{3} + b} {\left (a x^{3} - b\right )}\right )}}{3 \, x^{3}}\right ] \]
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Time = 11.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 a^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} + \frac {2 \sqrt {a} b}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {4 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (44) = 88\).
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {1}{3} \, {\left (\sqrt {b} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right ) + 2 \, \sqrt {a x^{3} + b}\right )} a + \frac {1}{3} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} b \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \, {\left (\frac {2 \, a^{2} b \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + \sqrt {a x^{3} + b} a^{2} - \frac {\sqrt {a x^{3} + b} a b}{x^{3}}\right )}}{3 \, a} \]
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Time = 5.99 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2\,a\,\sqrt {a\,x^3+b}}{3}-\frac {2\,b\,\sqrt {a\,x^3+b}}{3\,x^3}+\frac {2\,a\,\sqrt {b}\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{3} \]
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