Optimal. Leaf size=92 \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]
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Rubi [A] time = 0.112581, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c/x]],x]
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Rubi in Sympy [A] time = 11.6816, size = 73, normalized size = 0.79 \[ x \sqrt{a + b \sqrt{\frac{c}{x}}} + \frac{b c \sqrt{a + b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}} - \frac{b^{2} c \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c/x)**(1/2))**(1/2),x)
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Mathematica [A] time = 0.105072, size = 79, normalized size = 0.86 \[ \frac{\sqrt{a} x \sqrt{a+b \sqrt{\frac{c}{x}}} \left (2 a+b \sqrt{\frac{c}{x}}\right )-b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[c/x]],x]
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Maple [B] time = 0.031, size = 147, normalized size = 1.6 \[{\frac{1}{4}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{{\frac{c}{x}}}\sqrt{x}b-{b}^{2}c\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) a+4\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{x} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c/x)^(1/2))^(1/2),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, algorithm="maxima")
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Fricas [A] time = 0.291769, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} c \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right ) + 2 \,{\left (b x \sqrt{\frac{c}{x}} + 2 \, a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a}}{4 \, a^{\frac{3}{2}}}, \frac{b^{2} c \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right ) +{\left (\sqrt{-a} b x \sqrt{\frac{c}{x}} + 2 \, \sqrt{-a} a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{2 \, \sqrt{-a} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(c/x) + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{\frac{c}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c/x)**(1/2))**(1/2),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, algorithm="giac")
[Out]