Integrand size = 24, antiderivative size = 18 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 75} \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 \left (a x^3+b\right )^{3/2}}{3 x^3} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-2 b+a x) \sqrt {b+a x}}{x^2} \, dx,x,x^3\right ) \\ & = \frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3} \]
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Time = 3.43 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(15\) |
trager | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(15\) |
risch | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(15\) |
pseudoelliptic | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(15\) |
elliptic | \(\frac {2 b \sqrt {a \,x^{3}+b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}+b}}{3}\) | \(29\) |
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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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (15) = 30\).
Time = 10.80 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.33 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 a^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {b}{a x^{3}}}} + \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} + \frac {2 \sqrt {a} b}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 5.94 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {1}{3} \, {\left (\sqrt {b} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right ) + 2 \, \sqrt {a x^{3} + b}\right )} a - \frac {1}{3} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2 \, {\left (\sqrt {a x^{3} + b} a^{2} + \frac {\sqrt {a x^{3} + b} a b}{x^{3}}\right )}}{3 \, a} \]
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Time = 5.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx=\frac {2\,{\left (a\,x^3+b\right )}^{3/2}}{3\,x^3} \]
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