3.221 \(\int \frac{\sqrt{1-a^2 x^2}}{\sqrt{x} \sqrt{1+a x}} \, dx\)

Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

[Out]

Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/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]

[Out]

Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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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)

[Out]

sqrt(x)*sqrt(-a*x + 1) + asin(sqrt(a)*sqrt(x))/sqrt(a)

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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]

[Out]

(Sqrt[x]*Sqrt[1 - a^2*x^2])/Sqrt[1 + a*x] + (I*Log[(-2*I)*Sqrt[a]*Sqrt[x] + (2*S
qrt[1 - a^2*x^2])/Sqrt[1 + a*x]])/Sqrt[a]

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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)

[Out]

1/2*(-a^2*x^2+1)^(1/2)*x^(1/2)/(a*x+1)^(1/2)*(2*a^(1/2)*(-x*(a*x-1))^(1/2)+arcta
n(1/2/a^(1/2)*(2*a*x-1)/(-x*(a*x-1))^(1/2)))/(-x*(a*x-1))^(1/2)/a^(1/2)

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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")

[Out]

Exception raised: ValueError

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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")

[Out]

[1/4*(4*sqrt(-a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(-a)*sqrt(x) + (a*x + 1)*log(-(4*sq
rt(-a^2*x^2 + 1)*(2*a^2*x - a)*sqrt(a*x + 1)*sqrt(x) + (8*a^3*x^3 - 7*a*x + 1)*s
qrt(-a))/(a*x + 1)))/((a*x + 1)*sqrt(-a)), 1/2*(2*sqrt(-a^2*x^2 + 1)*sqrt(a*x +
1)*sqrt(a)*sqrt(x) - (a*x + 1)*arctan(2*sqrt(-a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(a)
*sqrt(x)/(2*a^2*x^2 + a*x - 1)))/((a*x + 1)*sqrt(a))]

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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)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(x)*sqrt(a*x + 1)), 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]

Exception raised: NotImplementedError