3.1608 \(\int \frac{x}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0483696, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int x\, dx}{a} - \frac{\int b\, dx}{a^{2}} + \frac{b^{2} \log{\left (a x + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x),x)

[Out]

Integral(x, x)/a - Integral(b, x)/a**2 + b**2*log(a*x + b)/a**3

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Mathematica [A]  time = 0.00557378, size = 31, normalized size = 1. \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 1. \[ -{\frac{bx}{{a}^{2}}}+{\frac{{x}^{2}}{2\,a}}+{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x),x)

[Out]

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

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Maxima [A]  time = 1.43764, size = 39, normalized size = 1.26 \[ \frac{b^{2} \log \left (a x + b\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*log(a*x + b)/a^3 + 1/2*(a*x^2 - 2*b*x)/a^2

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Fricas [A]  time = 0.217784, size = 39, normalized size = 1.26 \[ \frac{a^{2} x^{2} - 2 \, a b x + 2 \, b^{2} \log \left (a x + b\right )}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a^2*x^2 - 2*a*b*x + 2*b^2*log(a*x + b))/a^3

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Sympy [A]  time = 1.09741, size = 26, normalized size = 0.84 \[ \frac{x^{2}}{2 a} - \frac{b x}{a^{2}} + \frac{b^{2} \log{\left (a x + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x),x)

[Out]

x**2/(2*a) - b*x/a**2 + b**2*log(a*x + b)/a**3

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GIAC/XCAS [A]  time = 0.223661, size = 41, normalized size = 1.32 \[ \frac{b^{2}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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