Optimal. Leaf size=29 \[ -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}-\sin ^{-1}(1-2 a x) \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {16, 78, 53, 619, 216} \[ -\frac {2 \sqrt {1-a x}}{\sqrt {a x}}-\sin ^{-1}(1-2 a x) \]
Antiderivative was successfully verified.
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Rule 16
Rule 53
Rule 78
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1+a x}{x \sqrt {a x} \sqrt {1-a x}} \, dx &=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 {\operatorname {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*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.83 \[ \frac {2 \left (a x+\sqrt {a} \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )-1\right )}{\sqrt {-a x (a x-1)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 47, normalized size = 1.62 \[ -\frac {2 \, {\left (a x \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right ) + \sqrt {a x} \sqrt {-a x + 1}\right )}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 44, normalized size = 1.52 \[ -\frac {\sqrt {-a x + 1} - 1}{\sqrt {a x}} + \frac {\sqrt {a x}}{\sqrt {-a x + 1} - 1} + 2 \, \arcsin \left (\sqrt {a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 69, normalized size = 2.38 \[ \frac {\left (a x \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )-2 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) \sqrt {-a x +1}\, \mathrm {csgn}\relax (a )}{\sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 41, normalized size = 1.41 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{a x} - \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 47, normalized size = 1.62 \[ -\frac {2\,\sqrt {1-a\,x}}{\sqrt {a\,x}}-\frac {4\,a\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {1-a\,x}-1\right )}{\sqrt {a\,x}\,\sqrt {a^2}}\right )}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.62, size = 71, normalized size = 2.45 \[ a \left (\begin {cases} - \frac {2 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {for}\: \left |{a x}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {otherwise} \end {cases}\right ) + \begin {cases} - 2 \sqrt {-1 + \frac {1}{a x}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- 2 i \sqrt {1 - \frac {1}{a x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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