Optimal. Leaf size=155 \[ \frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{5/2}}+\frac{b \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{4 c^2}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.33331, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{5/2}}+\frac{b \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{4 c^2}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^2,x]
[Out]
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Rubi in Sympy [A] time = 26.9829, size = 126, normalized size = 0.81 \[ \frac{b \left (b d + 2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{4 c^{2}} + \frac{b \sqrt{d} \left (4 a c - b^{2} d\right ) \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{8 c^{\frac{5}{2}}} - \frac{2 \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0706302, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^2,x]
[Out]
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Maple [B] time = 0.039, size = 331, normalized size = 2.1 \[ -{\frac{1}{24\,x{c}^{3}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 3\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}-6\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}-12\,a\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ){c}^{3/2}\sqrt{{\frac{d}{x}}}{x}^{2}b-6\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{2}{b}^{2}+6\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}dx{b}^{2}+12\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{2}bc-12\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}xbc+16\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{c}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+c/x+b*(d/x)^(1/2))^(1/2)/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^2,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^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [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^2,x, algorithm="giac")
[Out]