Integrand size = 15, antiderivative size = 55 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=-2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+2 a^{3/2} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 65, 211} \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=2 a^{3/2} \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-2 a \sqrt {b x-a}+\frac {2}{3} (b x-a)^{3/2} \]
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Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} (-a+b x)^{3/2}-a \int \frac {\sqrt {-a+b x}}{x} \, dx \\ & = -2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+a^2 \int \frac {1}{x \sqrt {-a+b x}} \, dx \\ & = -2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{b} \\ & = -2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=\frac {2}{3} (-4 a+b x) \sqrt {-a+b x}+2 a^{3/2} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-\frac {2 \sqrt {b x -a}\, \left (-b x +4 a \right )}{3}\) | \(40\) |
derivativedivides | \(\frac {2 \left (b x -a \right )^{\frac {3}{2}}}{3}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-2 a \sqrt {b x -a}\) | \(44\) |
default | \(\frac {2 \left (b x -a \right )^{\frac {3}{2}}}{3}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-2 a \sqrt {b x -a}\) | \(44\) |
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none
Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=\left [\sqrt {-a} a \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}, 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}\right ] \]
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Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.40 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=\begin {cases} - \frac {8 a^{\frac {3}{2}} \sqrt {-1 + \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - 2 a^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 \sqrt {a} b x \sqrt {-1 + \frac {b x}{a}}}{3} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {8 i a^{\frac {3}{2}} \sqrt {1 - \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )} + \frac {2 i \sqrt {a} b x \sqrt {1 - \frac {b x}{a}}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \]
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Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {(-a+b x)^{3/2}}{x} \, dx=2\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )-2\,a\,\sqrt {b\,x-a}+\frac {2\,{\left (b\,x-a\right )}^{3/2}}{3} \]
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