3.3050 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^2} \, dx\)

Optimal. Leaf size=155 \[ \frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{5/2}}+\frac{b \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{4 c^2}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 c} \]

[Out]

(b*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(4*c^2) - (2*(a + b*Sqrt[d
/x] + c/x)^(3/2))/(3*c) + (b*Sqrt[d]*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x
])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(8*c^(5/2))

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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]

(b*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(4*c^2) - (2*(a + b*Sqrt[d
/x] + c/x)^(3/2))/(3*c) + (b*Sqrt[d]*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x
])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(8*c^(5/2))

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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]

b*(b*d + 2*c*sqrt(d/x))*sqrt(a + b*sqrt(d/x) + c/x)/(4*c**2) + b*sqrt(d)*(4*a*c
- b**2*d)*atanh((b*d + 2*c*sqrt(d/x))/(2*sqrt(c)*sqrt(d)*sqrt(a + b*sqrt(d/x) +
c/x)))/(8*c**(5/2)) - 2*(a + b*sqrt(d/x) + c/x)**(3/2)/(3*c)

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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]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^2, x]

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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]

-1/24*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(3*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*
(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(1/2)*(d/x)^(3/2)*x^3*b^3-6*(b*(d/x)^(1/2
)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^3*b^3-12*a*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(
d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(3/2)*(d/x)^(1/2)*x^2*b-6*a*(b*(d/x)^(1/2)
*x+a*x+c)^(1/2)*d*x^2*b^2+6*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2+12*a*(b*(d/x)^
(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^2*b*c-12*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^
(1/2)*x*b*c+16*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^2)/x/(b*(d/x)^(1/2)*x+a*x+c)^(1/2
)/c^3

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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]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^2, x)

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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]

Timed 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]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**2, x)

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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]

Timed out