Optimal. Leaf size=37 \[ -\frac {\sqrt {a x} \sqrt {1-a x}}{a}-\frac {3 \sin ^{-1}(1-2 a x)}{2 a} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {80, 53, 619, 216} \[ -\frac {\sqrt {a x} \sqrt {1-a x}}{a}-\frac {3 \sin ^{-1}(1-2 a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 53
Rule 80
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx &=-\frac {\sqrt {a x} \sqrt {1-a x}}{a}+\frac {3}{2} \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx\\ &=-\frac {\sqrt {a x} \sqrt {1-a x}}{a}+\frac {3}{2} \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {a x} \sqrt {1-a x}}{a}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{2 a^2}\\ &=-\frac {\sqrt {a x} \sqrt {1-a x}}{a}-\frac {3 \sin ^{-1}(1-2 a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 1.65 \[ \frac {\sqrt {a} x (a x-1)+3 \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {-a x (a x-1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 43, normalized size = 1.16 \[ -\frac {\sqrt {a x} \sqrt {-a x + 1} + 3 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 28, normalized size = 0.76 \[ -\frac {\sqrt {a x} \sqrt {-a x + 1} - 3 \, \arcsin \left (\sqrt {a x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 70, normalized size = 1.89 \[ -\frac {\sqrt {-a x +1}\, \left (-3 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+2 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{2 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 41, normalized size = 1.11 \[ -\frac {3 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + a x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 118, normalized size = 3.19 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{a}-\frac {4\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {1-a\,x}-1\right )}{\sqrt {a\,x}\,\sqrt {a^2}}\right )}{\sqrt {a^2}}-\frac {\frac {2\,\sqrt {a\,x}}{\sqrt {1-a\,x}-1}-\frac {2\,{\left (a\,x\right )}^{3/2}}{{\left (\sqrt {1-a\,x}-1\right )}^3}}{a\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.71, size = 133, normalized size = 3.59 \[ a \left (\begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} - \frac {i \sqrt {x} \sqrt {a x - 1}}{a^{\frac {3}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} \sqrt {- a x + 1}} - \frac {\sqrt {x}}{a^{\frac {3}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {for}\: \left |{a x}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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