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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.00, size = 31, normalized size = 1.00 \[ \frac {b^2 \log (a x+b)}{a^3}-\frac {b x}{a^2}+\frac {x^2}{2 a} \]

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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fricas [A]  time = 0.68, size = 29, normalized size = 0.94 \[ \frac {a^{2} x^{2} - 2 \, a b x + 2 \, b^{2} \log \left (a x + b\right )}{2 \, a^{3}} \]

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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giac [A]  time = 0.15, size = 30, normalized size = 0.97 \[ \frac {b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac {a x^{2} - 2 \, b x}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 30, normalized size = 0.97 \[ \frac {x^{2}}{2 a}-\frac {b x}{a^{2}}+\frac {b^{2} \ln \left (a x +b \right )}{a^{3}} \]

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 = 1.10, size = 29, normalized size = 0.94 \[ \frac {b^{2} \log \left (a x + b\right )}{a^{3}} + \frac {a x^{2} - 2 \, b x}{2 \, a^{2}} \]

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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mupad [B]  time = 0.04, size = 29, normalized size = 0.94 \[ \frac {2\,b^2\,\ln \left (b+a\,x\right )+a^2\,x^2-2\,a\,b\,x}{2\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x),x)

[Out]

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

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sympy [A]  time = 0.12, 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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