Integrand size = 26, antiderivative size = 29 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \sqrt {1-a x}}{\sqrt {a x}}-\arcsin (1-2 a x) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {16, 79, 55, 633, 222} \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=-\arcsin (1-2 a x)-\frac {2 \sqrt {1-a x}}{\sqrt {a x}} \]
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Rule 16
Rule 55
Rule 79
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = a \int \frac {1+a x}{(a x)^{3/2} \sqrt {1-a x}} \, dx \\ & = -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}+a \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx \\ & = -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}+a \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx \\ & = -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{a} \\ & = -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}-\sin ^{-1}(1-2 a x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(29)=58\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=\frac {2 \left (-1+a x+2 \sqrt {a} \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )\right )}{\sqrt {-a x (-1+a x)}} \]
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Time = 1.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
method | result | size |
meijerg | \(\frac {2 \sqrt {a}\, \sqrt {x}\, \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{\sqrt {a x}}-\frac {2 \sqrt {-a x +1}}{\sqrt {a x}}\) | \(38\) |
default | \(\frac {\sqrt {-a x +1}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 a x -1\right )}{2 \sqrt {-x \left (a x -1\right ) a}}\right ) a x -2 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}\right ) \operatorname {csgn}\left (a \right )}{\sqrt {a x}\, \sqrt {-x \left (a x -1\right ) a}}\) | \(69\) |
risch | \(\frac {2 \left (a x -1\right ) \sqrt {a x \left (-a x +1\right )}}{\sqrt {-x \left (a x -1\right ) a}\, \sqrt {a x}\, \sqrt {-a x +1}}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+a x}}\right ) \sqrt {a x \left (-a x +1\right )}}{\sqrt {a^{2}}\, \sqrt {a x}\, \sqrt {-a x +1}}\) | \(103\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, {\left (a x \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right ) + \sqrt {a x} \sqrt {-a x + 1}\right )}}{a x} \]
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Result contains complex when optimal does not.
Time = 3.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {2 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {for}\: \left |{a x}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {otherwise} \end {cases}\right ) + \begin {cases} - 2 \sqrt {-1 + \frac {1}{a x}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- 2 i \sqrt {1 - \frac {1}{a x}} & \text {otherwise} \end {cases} \]
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none
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{a x} - \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=\frac {2 \, a \arcsin \left (\sqrt {a x}\right ) - \frac {a {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} + \frac {\sqrt {a x} a}{\sqrt {-a x + 1} - 1}}{a} \]
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Time = 3.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x}}-\frac {4\,a\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {1-a\,x}-1\right )}{\sqrt {a\,x}\,\sqrt {a^2}}\right )}{\sqrt {a^2}} \]
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