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 = 0.0125165, antiderivative size = 37, normalized size of antiderivative = 1., 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 80
Rule 53
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0242423, 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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Maple [C] time = 0.015, size = 70, normalized size = 1.9 \begin{align*} -{\frac{x{\it csgn} \left ( a \right ) }{2}\sqrt{-ax+1} \left ( 2\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) -3\,\arctan \left ( 1/2\,{\frac{{\it csgn} \left ( a \right ) \left ( 2\,ax-1 \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30267, size = 100, normalized size = 2.7 \begin{align*} -\frac{\sqrt{a x} \sqrt{-a x + 1} + 3 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.76718, size = 133, normalized size = 3.59 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.54551, size = 38, normalized size = 1.03 \begin{align*} -\frac{\sqrt{a x} \sqrt{-a x + 1} - 3 \, \arcsin \left (\sqrt{a x}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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