Optimal. Leaf size=51 \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
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Rubi [A] time = 0.09835, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c/x]]/x,x]
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Rubi in Sympy [A] time = 9.12758, size = 41, normalized size = 0.8 \[ 4 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )} - 4 \sqrt{a + b \sqrt{\frac{c}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c/x)**(1/2))**(1/2)/x,x)
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Mathematica [A] time = 0.0458404, size = 52, normalized size = 1.02 \[ 4 \left (\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-\sqrt{a+b \sqrt{\frac{c}{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[c/x]]/x,x]
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Maple [B] time = 0.032, size = 150, normalized size = 2.9 \[ 2\,{\frac{1}{bx\sqrt{a}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( \ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{3/2}ab+2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}x-2\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c/x)^(1/2))^(1/2)/x,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(b*sqrt(c/x) + a)/x,x, algorithm="maxima")
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Fricas [A] time = 0.266474, size = 1, normalized size = 0.02 \[ \left [2 \, \sqrt{a} \log \left (\frac{b \sqrt{\frac{c}{x}} + 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a} + 2 \, a}{\sqrt{\frac{c}{x}}}\right ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}, 4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a}}{\sqrt{-a}}\right ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(c/x) + a)/x,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c/x)**(1/2))**(1/2)/x,x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(c/x) + a)/x,x, algorithm="giac")
[Out]