Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.108697, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - a^2*x^2]/(Sqrt[x]*Sqrt[1 + a*x]),x]
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Rubi in Sympy [A] time = 9.75256, size = 29, normalized size = 0.83 \[ \sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-a**2*x**2+1)**(1/2)/x**(1/2)/(a*x+1)**(1/2),x)
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Mathematica [C] time = 0.107416, size = 80, normalized size = 2.29 \[ \frac{\sqrt{x} \sqrt{1-a^2 x^2}}{\sqrt{a x+1}}+\frac{i \log \left (\frac{2 \sqrt{1-a^2 x^2}}{\sqrt{a x+1}}-2 i \sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - a^2*x^2]/(Sqrt[x]*Sqrt[1 + a*x]),x]
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Maple [B] time = 0.019, size = 76, normalized size = 2.2 \[{\frac{1}{2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{x} \left ( 2\,\sqrt{a}\sqrt{-x \left ( ax-1 \right ) }+\arctan \left ({\frac{2\,ax-1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}} \right ) \right ){\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-a^2*x^2+1)^(1/2)/x^(1/2)/(a*x+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)/(sqrt(a*x + 1)*sqrt(x)),x, algorithm="maxima")
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Fricas [A] time = 0.311637, size = 1, normalized size = 0.03 \[ \left [\frac{4 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} \sqrt{-a} \sqrt{x} +{\left (a x + 1\right )} \log \left (-\frac{4 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a^{2} x - a\right )} \sqrt{a x + 1} \sqrt{x} +{\left (8 \, a^{3} x^{3} - 7 \, a x + 1\right )} \sqrt{-a}}{a x + 1}\right )}{4 \,{\left (a x + 1\right )} \sqrt{-a}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} \sqrt{a} \sqrt{x} -{\left (a x + 1\right )} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} \sqrt{a} \sqrt{x}}{2 \, a^{2} x^{2} + a x - 1}\right )}{2 \,{\left (a x + 1\right )} \sqrt{a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)/(sqrt(a*x + 1)*sqrt(x)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt{x} \sqrt{a x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-a**2*x**2+1)**(1/2)/x**(1/2)/(a*x+1)**(1/2),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)/(sqrt(a*x + 1)*sqrt(x)),x, algorithm="giac")
[Out]