3.3049 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x} \, dx\)

Optimal. Leaf size=145 \[ -2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}+2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )-\frac{b \sqrt{d} \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{c}} \]

[Out]

-2*Sqrt[a + b*Sqrt[d/x] + c/x] + 2*Sqrt[a]*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a
]*Sqrt[a + b*Sqrt[d/x] + c/x])] - (b*Sqrt[d]*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sq
rt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/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]

[Out]

-2*Sqrt[a + b*Sqrt[d/x] + c/x] + 2*Sqrt[a]*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a
]*Sqrt[a + b*Sqrt[d/x] + c/x])] - (b*Sqrt[d]*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sq
rt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/Sqrt[c]

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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]

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

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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]

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

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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]

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

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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]

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

[Out]

Timed out

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

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]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x, 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(d/x) + a + c/x)/x,x, algorithm="giac")

[Out]

Exception raised: TypeError