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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]