3.1562 \(\int \left (a+\frac{b}{x}\right )^2 \, dx\)

Optimal. Leaf size=20 \[ a^2 x+2 a b \log (x)-\frac{b^2}{x} \]

[Out]

-(b^2/x) + a^2*x + 2*a*b*Log[x]

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Rubi [A]  time = 0.0293332, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ a^2 x+2 a b \log (x)-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2,x]

[Out]

-(b^2/x) + a^2*x + 2*a*b*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a b \log{\left (x \right )} - \frac{b^{2}}{x} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2,x)

[Out]

2*a*b*log(x) - b**2/x + Integral(a**2, x)

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Mathematica [A]  time = 0.00563458, size = 20, normalized size = 1. \[ a^2 x+2 a b \log (x)-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2,x]

[Out]

-(b^2/x) + a^2*x + 2*a*b*Log[x]

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Maple [A]  time = 0.008, size = 21, normalized size = 1.1 \[ -{\frac{{b}^{2}}{x}}+x{a}^{2}+2\,ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2,x)

[Out]

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

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Maxima [A]  time = 1.44004, size = 27, normalized size = 1.35 \[ a^{2} x + 2 \, a b \log \left (x\right ) - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x + 2*a*b*log(x) - b^2/x

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Fricas [A]  time = 0.225049, size = 32, normalized size = 1.6 \[ \frac{a^{2} x^{2} + 2 \, a b x \log \left (x\right ) - b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^2*x^2 + 2*a*b*x*log(x) - b^2)/x

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Sympy [A]  time = 1.08675, size = 17, normalized size = 0.85 \[ a^{2} x + 2 a b \log{\left (x \right )} - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2,x)

[Out]

a**2*x + 2*a*b*log(x) - b**2/x

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GIAC/XCAS [A]  time = 0.222449, size = 28, normalized size = 1.4 \[ a^{2} x + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x + 2*a*b*ln(abs(x)) - b^2/x