3.1563 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x} \, dx\)

Optimal. Leaf size=24 \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]

[Out]

-b^2/(2*x^2) - (2*a*b)/x + a^2*Log[x]

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Rubi [A]  time = 0.0350903, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x,x]

[Out]

-b^2/(2*x^2) - (2*a*b)/x + a^2*Log[x]

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Rubi in Sympy [A]  time = 6.38225, size = 20, normalized size = 0.83 \[ a^{2} \log{\left (x \right )} - \frac{2 a b}{x} - \frac{b^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2/x,x)

[Out]

a**2*log(x) - 2*a*b/x - b**2/(2*x**2)

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Mathematica [A]  time = 0.00620191, size = 24, normalized size = 1. \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2/x,x]

[Out]

-b^2/(2*x^2) - (2*a*b)/x + a^2*Log[x]

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Maple [A]  time = 0.008, size = 23, normalized size = 1. \[ -{\frac{{b}^{2}}{2\,{x}^{2}}}-2\,{\frac{ab}{x}}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x,x)

[Out]

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

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Maxima [A]  time = 1.43652, size = 28, normalized size = 1.17 \[ a^{2} \log \left (x\right ) - \frac{4 \, a b x + b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*log(x) - 1/2*(4*a*b*x + b^2)/x^2

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Fricas [A]  time = 0.228346, size = 35, normalized size = 1.46 \[ \frac{2 \, a^{2} x^{2} \log \left (x\right ) - 4 \, a b x - b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a^2*x^2*log(x) - 4*a*b*x - b^2)/x^2

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Sympy [A]  time = 1.19741, size = 20, normalized size = 0.83 \[ a^{2} \log{\left (x \right )} - \frac{4 a b x + b^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2/x,x)

[Out]

a**2*log(x) - (4*a*b*x + b**2)/(2*x**2)

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GIAC/XCAS [A]  time = 0.225109, size = 30, normalized size = 1.25 \[ a^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{4 \, a b x + b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*ln(abs(x)) - 1/2*(4*a*b*x + b^2)/x^2