3.1610 \(\int \frac{x}{a+\frac{b}{x}} \, dx\)

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 = 0.0157775, 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, Rules used = {263, 43} \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{a+\frac{b}{x}} \, dx &=\int \frac{x^2}{b+a x} \, dx\\ &=\int \left (-\frac{b}{a^2}+\frac{x}{a}+\frac{b^2}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac{b x}{a^2}+\frac{x^2}{2 a}+\frac{b^2 \log (b+a x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0031411, 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 = 0.002, size = 30, normalized size = 1. \begin{align*} -{\frac{bx}{{a}^{2}}}+{\frac{{x}^{2}}{2\,a}}+{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{3}}} \end{align*}

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 = 0.991435, size = 39, normalized size = 1.26 \begin{align*} \frac{b^{2} \log \left (a x + b\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \end{align*}

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 = 1.44309, size = 68, normalized size = 2.19 \begin{align*} \frac{a^{2} x^{2} - 2 \, a b x + 2 \, b^{2} \log \left (a x + b\right )}{2 \, a^{3}} \end{align*}

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 = 0.259465, size = 26, normalized size = 0.84 \begin{align*} \frac{x^{2}}{2 a} - \frac{b x}{a^{2}} + \frac{b^{2} \log{\left (a x + b \right )}}{a^{3}} \end{align*}

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 [A]  time = 1.09857, size = 41, normalized size = 1.32 \begin{align*} \frac{b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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