Optimal. Leaf size=145 \[ -2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}+2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.528261, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}+2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x,x]
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Rubi in Sympy [A] time = 39.4534, size = 117, normalized size = 0.81 \[ 2 \sqrt{a} \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )} - \frac{b \sqrt{d} \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{\sqrt{c}} - 2 \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0779955, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x,x]
[Out]
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Maple [B] time = 0.038, size = 237, normalized size = 1.6 \[ -{\frac{1}{c}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ({a}^{{\frac{3}{2}}}\ln \left ({1 \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ){\frac{1}{\sqrt{x}}}} \right ) \sqrt{c}\sqrt{{\frac{d}{x}}}xb-2\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}x-2\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}xb+2\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{a}^{3/2}-2\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{x}{a}^{2}c \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+c/x+b*(d/x)^(1/2))^(1/2)/x,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(d/x) + a + c/x)/x,x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(d/x) + a + c/x)/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x,x)
[Out]
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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(d/x) + a + c/x)/x,x, algorithm="giac")
[Out]