3.2926 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x} \, dx\)

Optimal. Leaf size=51 \[ 2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right ) \]

[Out]

2*Sqrt[a + b*Sqrt[c*x^2]] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]]

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Rubi [A]  time = 0.0757166, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ 2 \sqrt{a+b \sqrt{c x^2}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[a + b*Sqrt[c*x^2]] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]]

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Rubi in Sympy [A]  time = 7.76847, size = 44, normalized size = 0.86 \[ - 2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )} + 2 \sqrt{a + b \sqrt{c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*atanh(sqrt(a + b*sqrt(c*x**2))/sqrt(a)) + 2*sqrt(a + b*sqrt(c*x**2))

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Mathematica [A]  time = 0.0325621, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x, x]

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Maple [A]  time = 0.009, size = 40, normalized size = 0.8 \[ -2\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{a+b\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*a^(1/2)+2*(a+b*(c*x^2)^(1/2))^(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(sqrt(c*x^2)*b + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224798, size = 1, normalized size = 0.02 \[ \left [\sqrt{a} \log \left (\frac{\sqrt{c x^{2}} b - 2 \, \sqrt{\sqrt{c x^{2}} b + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{\sqrt{c x^{2}} b + a}, -2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{\sqrt{c x^{2}} b + a}}{\sqrt{-a}}\right ) + 2 \, \sqrt{\sqrt{c x^{2}} b + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[sqrt(a)*log((sqrt(c*x^2)*b - 2*sqrt(sqrt(c*x^2)*b + a)*sqrt(a) + 2*a)/x) + 2*sq
rt(sqrt(c*x^2)*b + a), -2*sqrt(-a)*arctan(sqrt(sqrt(c*x^2)*b + a)/sqrt(-a)) + 2*
sqrt(sqrt(c*x^2)*b + a)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x, x)

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GIAC/XCAS [A]  time = 0.217812, size = 51, normalized size = 1. \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b \sqrt{c} x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/sqrt(-a) + 2*sqrt(b*sqrt(c)*x + a)