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

Optimal. Leaf size=73 \[ \frac{a x \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}}-\frac{b \log \left (\frac{1}{x}\right ) \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}} \]

[Out]

(a*Sqrt[a^2 + b^2/x^2 + (2*a*b)/x]*x)/(a + b/x) - (b*Sqrt[a^2 + b^2/x^2 + (2*a*b
)/x]*Log[x^(-1)])/(a + b/x)

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Rubi [A]  time = 0.0864296, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a x \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}}-\frac{b \log \left (\frac{1}{x}\right ) \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + b^2/x^2 + (2*a*b)/x],x]

[Out]

(a*Sqrt[a^2 + b^2/x^2 + (2*a*b)/x]*x)/(a + b/x) - (b*Sqrt[a^2 + b^2/x^2 + (2*a*b
)/x]*Log[x^(-1)])/(a + b/x)

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Rubi in Sympy [A]  time = 5.40982, size = 58, normalized size = 0.79 \[ \frac{a x \sqrt{a^{2} + \frac{2 a b}{x} + \frac{b^{2}}{x^{2}}}}{a + \frac{b}{x}} + \frac{b \sqrt{a^{2} + \frac{2 a b}{x} + \frac{b^{2}}{x^{2}}} \log{\left (x \right )}}{a + \frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x*sqrt(a**2 + 2*a*b/x + b**2/x**2)/(a + b/x) + b*sqrt(a**2 + 2*a*b/x + b**2/x*
*2)*log(x)/(a + b/x)

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Mathematica [A]  time = 0.0300928, size = 32, normalized size = 0.44 \[ \frac{x \sqrt{\frac{(a x+b)^2}{x^2}} (a x+b \log (x))}{a x+b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + b^2/x^2 + (2*a*b)/x],x]

[Out]

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

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Maple [A]  time = 0.014, size = 40, normalized size = 0.6 \[{\frac{x \left ( ax+b\ln \left ( x \right ) \right ) }{ax+b}\sqrt{{\frac{{a}^{2}{x}^{2}+2\,abx+{b}^{2}}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((a^2*x^2+2*a*b*x+b^2)/x^2)^(1/2)/(a*x+b)*x*(a*x+b*ln(x))

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Maxima [A]  time = 0.75148, size = 11, normalized size = 0.15 \[ a x + b \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x + b*log(x)

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Fricas [A]  time = 0.267432, size = 11, normalized size = 0.15 \[ a x + b \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x + b*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.293726, size = 39, normalized size = 0.53 \[ a x{\rm sign}\left (a x^{2} + b x\right ) + b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (a x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a^2 + 2*a*b/x + b^2/x^2),x, algorithm="giac")

[Out]

a*x*sign(a*x^2 + b*x) + b*ln(abs(x))*sign(a*x^2 + b*x)