3.452 \(\int \frac{1}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=67 \[ \frac{x \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a}-\frac{b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0956167, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a}-\frac{b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + c/x^2 + b/x],x]

[Out]

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

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Rubi in Sympy [A]  time = 11.689, size = 53, normalized size = 0.79 \[ \frac{x \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}}{a} - \frac{b \operatorname{atanh}{\left (\frac{2 a + \frac{b}{x}}{2 \sqrt{a} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0970812, size = 87, normalized size = 1.3 \[ \frac{2 \sqrt{a} (x (a x+b)+c)-b \sqrt{x (a x+b)+c} \log \left (2 \sqrt{a} \sqrt{x (a x+b)+c}+2 a x+b\right )}{2 a^{3/2} x \sqrt{a+\frac{b x+c}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + c/x^2 + b/x],x]

[Out]

(2*Sqrt[a]*(c + x*(b + a*x)) - b*Sqrt[c + x*(b + a*x)]*Log[b + 2*a*x + 2*Sqrt[a]
*Sqrt[c + x*(b + a*x)]])/(2*a^(3/2)*x*Sqrt[a + (c + b*x)/x^2])

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Maple [A]  time = 0.01, size = 88, normalized size = 1.3 \[{\frac{1}{2\,x}\sqrt{a{x}^{2}+bx+c} \left ( 2\,\sqrt{a{x}^{2}+bx+c}{a}^{3/2}-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+bx+c}{{x}^{2}}}}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(a + b/x + c/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276129, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, a x \sqrt{\frac{a x^{2} + b x + c}{x^{2}}} + \sqrt{a} b \log \left (-{\left (8 \, a^{2} x^{2} + 8 \, a b x + b^{2} + 4 \, a c\right )} \sqrt{a} + 4 \,{\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}\right )}{4 \, a^{2}}, \frac{2 \, a x \sqrt{\frac{a x^{2} + b x + c}{x^{2}}} + \sqrt{-a} b \arctan \left (\frac{{\left (2 \, a x + b\right )} \sqrt{-a}}{2 \, a x \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}\right )}{2 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(a + b/x + c/x^2),x, algorithm="fricas")

[Out]

[1/4*(4*a*x*sqrt((a*x^2 + b*x + c)/x^2) + sqrt(a)*b*log(-(8*a^2*x^2 + 8*a*b*x +
b^2 + 4*a*c)*sqrt(a) + 4*(2*a^2*x^2 + a*b*x)*sqrt((a*x^2 + b*x + c)/x^2)))/a^2,
1/2*(2*a*x*sqrt((a*x^2 + b*x + c)/x^2) + sqrt(-a)*b*arctan(1/2*(2*a*x + b)*sqrt(
-a)/(a*x*sqrt((a*x^2 + b*x + c)/x^2))))/a^2]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a + b/x + c/x**2), 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(1/sqrt(a + b/x + c/x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError