3.1623 \(\int \frac{x}{(a+\frac{b}{x})^2} \, dx\)

Optimal. Leaf size=46 \[ \frac{b^3}{a^4 (a x+b)}+\frac{3 b^2 \log (a x+b)}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2} \]

[Out]

(-2*b*x)/a^3 + x^2/(2*a^2) + b^3/(a^4*(b + a*x)) + (3*b^2*Log[b + a*x])/a^4

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Rubi [A]  time = 0.0281744, antiderivative size = 46, 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} \[ \frac{b^3}{a^4 (a x+b)}+\frac{3 b^2 \log (a x+b)}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x)^2,x]

[Out]

(-2*b*x)/a^3 + x^2/(2*a^2) + b^3/(a^4*(b + a*x)) + (3*b^2*Log[b + a*x])/a^4

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}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^3}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{2 b}{a^3}+\frac{x}{a^2}-\frac{b^3}{a^3 (b+a x)^2}+\frac{3 b^2}{a^3 (b+a x)}\right ) \, dx\\ &=-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2}+\frac{b^3}{a^4 (b+a x)}+\frac{3 b^2 \log (b+a x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0128846, size = 43, normalized size = 0.93 \[ \frac{a^2 x^2+\frac{2 b^3}{a x+b}+6 b^2 \log (a x+b)-4 a b x}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x)^2,x]

[Out]

(-4*a*b*x + a^2*x^2 + (2*b^3)/(b + a*x) + 6*b^2*Log[b + a*x])/(2*a^4)

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Maple [A]  time = 0.005, size = 45, normalized size = 1. \begin{align*} -2\,{\frac{bx}{{a}^{3}}}+{\frac{{x}^{2}}{2\,{a}^{2}}}+{\frac{{b}^{3}}{{a}^{4} \left ( ax+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b/x)^2,x)

[Out]

-2*b*x/a^3+1/2*x^2/a^2+b^3/a^4/(a*x+b)+3*b^2*ln(a*x+b)/a^4

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Maxima [A]  time = 0.973541, size = 63, normalized size = 1.37 \begin{align*} \frac{b^{3}}{a^{5} x + a^{4} b} + \frac{3 \, b^{2} \log \left (a x + b\right )}{a^{4}} + \frac{a x^{2} - 4 \, b x}{2 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3/(a^5*x + a^4*b) + 3*b^2*log(a*x + b)/a^4 + 1/2*(a*x^2 - 4*b*x)/a^3

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Fricas [A]  time = 1.47135, size = 132, normalized size = 2.87 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 2 \, b^{3} + 6 \,{\left (a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{5} x + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^3*x^3 - 3*a^2*b*x^2 - 4*a*b^2*x + 2*b^3 + 6*(a*b^2*x + b^3)*log(a*x + b))/(a^5*x + a^4*b)

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Sympy [A]  time = 0.323778, size = 44, normalized size = 0.96 \begin{align*} \frac{b^{3}}{a^{5} x + a^{4} b} + \frac{x^{2}}{2 a^{2}} - \frac{2 b x}{a^{3}} + \frac{3 b^{2} \log{\left (a x + b \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)**2,x)

[Out]

b**3/(a**5*x + a**4*b) + x**2/(2*a**2) - 2*b*x/a**3 + 3*b**2*log(a*x + b)/a**4

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Giac [A]  time = 1.09019, size = 65, normalized size = 1.41 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac{b^{3}}{{\left (a x + b\right )} a^{4}} + \frac{a^{2} x^{2} - 4 \, a b x}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b^2*log(abs(a*x + b))/a^4 + b^3/((a*x + b)*a^4) + 1/2*(a^2*x^2 - 4*a*b*x)/a^4