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

Optimal. Leaf size=22 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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Rubi [A]  time = 0.00918767, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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Rubi in Sympy [A]  time = 7.45317, size = 17, normalized size = 0.77 \[ \frac{\sqrt{a x^{2}} \sqrt{x^{2} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**2)**(1/2)/(x**2+1)**(1/2),x)

[Out]

sqrt(a*x**2)*sqrt(x**2 + 1)/x

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Mathematica [A]  time = 0.0079487, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^2]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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Maple [A]  time = 0.003, size = 19, normalized size = 0.9 \[{\frac{1}{x}\sqrt{a{x}^{2}}\sqrt{{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^2)^(1/2)/(x^2+1)^(1/2),x)

[Out]

(a*x^2)^(1/2)*(x^2+1)^(1/2)/x

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Maxima [A]  time = 0.768608, size = 26, normalized size = 1.18 \[ \frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

(sqrt(a)*x^2 + sqrt(a))/sqrt(x^2 + 1)

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Fricas [A]  time = 0.290682, size = 74, normalized size = 3.36 \[ -\frac{\sqrt{a x^{2}}{\left (2 \, x^{3} -{\left (2 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 2 \, x\right )}}{2 \, x^{3} - 2 \, \sqrt{x^{2} + 1} x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

-sqrt(a*x^2)*(2*x^3 - (2*x^2 + 1)*sqrt(x^2 + 1) + 2*x)/(2*x^3 - 2*sqrt(x^2 + 1)*
x^2 + x)

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Sympy [A]  time = 1.11581, size = 20, normalized size = 0.91 \[ \frac{\sqrt{a} \sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**2)**(1/2)/(x**2+1)**(1/2),x)

[Out]

sqrt(a)*sqrt(x**2 + 1)*sqrt(x**2)/x

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GIAC/XCAS [A]  time = 0.264582, size = 26, normalized size = 1.18 \[{\left (\sqrt{x^{2} + 1}{\rm sign}\left (x\right ) -{\rm sign}\left (x\right )\right )} \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^2)/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

(sqrt(x^2 + 1)*sign(x) - sign(x))*sqrt(a)