3.3048 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \, dx\)

Optimal. Leaf size=113 \[ \frac{\left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{3/2}}+\frac{x \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

[Out]

((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(2*a) + ((4*a*c - b^2*d)*Arc
Tanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(3/2))

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Rubi [A]  time = 0.285122, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{3/2}}+\frac{x \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

((2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(2*a) + ((4*a*c - b^2*d)*Arc
Tanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(4*a^(3/2))

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Rubi in Sympy [A]  time = 23.7626, size = 87, normalized size = 0.77 \[ \frac{x \left (2 a + b \sqrt{\frac{d}{x}}\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{2 a} + \frac{\left (4 a c - b^{2} d\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0939902, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.039, size = 213, normalized size = 1.9 \[{\frac{1}{4}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 2\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b-\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) da{b}^{2}+4\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}+4\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{2}c \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+c/x+b*(d/x)^(1/2))^(1/2),x)

[Out]

1/4*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(2*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)
^(1/2)*(d/x)^(1/2)*x^(1/2)*b-ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*
x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d*a*b^2+4*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+
c)^(1/2)*x^(1/2)+4*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)
*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^2*c)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(d/x) + a + c/x),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x), 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, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.50475, size = 221, normalized size = 1.96 \[ \frac{{\left (2 \, \sqrt{a d x + \sqrt{d x} b d + c d}{\left (\frac{b d}{a} + 2 \, \sqrt{d x}\right )} + \frac{{\left (b^{2} d^{2} - 4 \, a c d\right )}{\rm ln}\left ({\left | -b d - 2 \,{\left (\sqrt{d x} \sqrt{a} - \sqrt{a d x + \sqrt{d x} b d + c d}\right )} \sqrt{a} \right |}\right )}{a^{\frac{3}{2}}}\right )}{\rm sign}\left (x\right )}{4 \, d} - \frac{{\left (b^{2} d{\rm ln}\left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) - 4 \, a c{\rm ln}\left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) + 2 \, \sqrt{c d} \sqrt{a} b\right )}{\rm sign}\left (x\right )}{4 \, a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(d/x) + a + c/x),x, algorithm="giac")

[Out]

1/4*(2*sqrt(a*d*x + sqrt(d*x)*b*d + c*d)*(b*d/a + 2*sqrt(d*x)) + (b^2*d^2 - 4*a*
c*d)*ln(abs(-b*d - 2*(sqrt(d*x)*sqrt(a) - sqrt(a*d*x + sqrt(d*x)*b*d + c*d))*sqr
t(a)))/a^(3/2))*sign(x)/d - 1/4*(b^2*d*ln(abs(-b*d + 2*sqrt(c*d)*sqrt(a))) - 4*a
*c*ln(abs(-b*d + 2*sqrt(c*d)*sqrt(a))) + 2*sqrt(c*d)*sqrt(a)*b)*sign(x)/a^(3/2)