Integrand size = 38, antiderivative size = 66 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {587, 162, 65, 211, 209} \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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Rule 65
Rule 162
Rule 209
Rule 211
Rule 587
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {-b+4 a x}{x \sqrt {-b+a x} (2 b+a x)} \, dx,x,x^3\right ) \\ & = -\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {-b+a x}} \, dx,x,x^3\right )\right )+\frac {1}{2} (3 a) \text {Subst}\left (\int \frac {1}{\sqrt {-b+a x} (2 b+a x)} \, dx,x,x^3\right ) \\ & = 3 \text {Subst}\left (\int \frac {1}{3 b+x^2} \, dx,x,\sqrt {-b+a x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {-b+a x^3}\right )}{3 a} \\ & = -\frac {\arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )-3 \sqrt {3} \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{3 \sqrt {b}} \]
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Time = 1.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {3 \arctan \left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {3}}{3 \sqrt {b}}\right ) \sqrt {3}-\arctan \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\) | \(50\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}+\frac {\arctan \left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {3}}{3 \sqrt {b}}\right ) \sqrt {3}}{\sqrt {b}}\) | \(54\) |
elliptic | \(\text {Expression too large to display}\) | \(1430\) |
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Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.39 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\left [\frac {3 \, \sqrt {3} b \sqrt {-\frac {1}{b}} \log \left (\frac {a x^{3} + 2 \, \sqrt {3} \sqrt {a x^{3} - b} b \sqrt {-\frac {1}{b}} - 4 \, b}{a x^{3} + 2 \, b}\right ) - \sqrt {-b} \log \left (\frac {a x^{3} + 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{6 \, b}, -\frac {3 \, \sqrt {3} \sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt {b}}{\sqrt {a x^{3} - b}}\right ) + \sqrt {b} \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, b}\right ] \]
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Time = 6.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\begin {cases} \frac {2 \left (- \frac {a \operatorname {atan}{\left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}} \right )}}{6 \sqrt {b}} + \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \sqrt {b}} \right )}}{2 \sqrt {b}}\right )}{a} & \text {for}\: a \neq 0 \\- \frac {\log {\left (x^{3} \right )}}{6 \sqrt {- b}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\int { \frac {4 \, a x^{3} - b}{{\left (a x^{3} + 2 \, b\right )} \sqrt {a x^{3} - b} x} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right )}{\sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \]
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Time = 7.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.45 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {\ln \left (\frac {2\,b-a\,x^3+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,b\,4{}\mathrm {i}+6\,\sqrt {b}\,\sqrt {a\,x^3-b}-\sqrt {3}\,a\,x^3\,1{}\mathrm {i}}{6\,a\,x^3+12\,b}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}} \]
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