3.1636 \(\int \frac{x}{(a+\frac{b}{x})^3} \, dx\)

Optimal. Leaf size=64 \[ -\frac{b^4}{2 a^5 (a x+b)^2}+\frac{4 b^3}{a^5 (a x+b)}+\frac{6 b^2 \log (a x+b)}{a^5}-\frac{3 b x}{a^4}+\frac{x^2}{2 a^3} \]

[Out]

(-3*b*x)/a^4 + x^2/(2*a^3) - b^4/(2*a^5*(b + a*x)^2) + (4*b^3)/(a^5*(b + a*x)) + (6*b^2*Log[b + a*x])/a^5

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Rubi [A]  time = 0.0367898, antiderivative size = 64, 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} \[ -\frac{b^4}{2 a^5 (a x+b)^2}+\frac{4 b^3}{a^5 (a x+b)}+\frac{6 b^2 \log (a x+b)}{a^5}-\frac{3 b x}{a^4}+\frac{x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x)^3,x]

[Out]

(-3*b*x)/a^4 + x^2/(2*a^3) - b^4/(2*a^5*(b + a*x)^2) + (4*b^3)/(a^5*(b + a*x)) + (6*b^2*Log[b + a*x])/a^5

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 )^3} \, dx &=\int \frac{x^4}{(b+a x)^3} \, dx\\ &=\int \left (-\frac{3 b}{a^4}+\frac{x}{a^3}+\frac{b^4}{a^4 (b+a x)^3}-\frac{4 b^3}{a^4 (b+a x)^2}+\frac{6 b^2}{a^4 (b+a x)}\right ) \, dx\\ &=-\frac{3 b x}{a^4}+\frac{x^2}{2 a^3}-\frac{b^4}{2 a^5 (b+a x)^2}+\frac{4 b^3}{a^5 (b+a x)}+\frac{6 b^2 \log (b+a x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0332175, size = 50, normalized size = 0.78 \[ \frac{a^2 x^2+\frac{b^3 (8 a x+7 b)}{(a x+b)^2}+12 b^2 \log (a x+b)-6 a b x}{2 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x)^3,x]

[Out]

(-6*a*b*x + a^2*x^2 + (b^3*(7*b + 8*a*x))/(b + a*x)^2 + 12*b^2*Log[b + a*x])/(2*a^5)

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Maple [A]  time = 0.007, size = 61, normalized size = 1. \begin{align*} -3\,{\frac{bx}{{a}^{4}}}+{\frac{{x}^{2}}{2\,{a}^{3}}}-{\frac{{b}^{4}}{2\,{a}^{5} \left ( ax+b \right ) ^{2}}}+4\,{\frac{{b}^{3}}{{a}^{5} \left ( ax+b \right ) }}+6\,{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b/x)^3,x)

[Out]

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

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Maxima [A]  time = 1.05226, size = 93, normalized size = 1.45 \begin{align*} \frac{8 \, a b^{3} x + 7 \, b^{4}}{2 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}} + \frac{6 \, b^{2} \log \left (a x + b\right )}{a^{5}} + \frac{a x^{2} - 6 \, b x}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)^3,x, algorithm="maxima")

[Out]

1/2*(8*a*b^3*x + 7*b^4)/(a^7*x^2 + 2*a^6*b*x + a^5*b^2) + 6*b^2*log(a*x + b)/a^5 + 1/2*(a*x^2 - 6*b*x)/a^4

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Fricas [A]  time = 1.47112, size = 200, normalized size = 3.12 \begin{align*} \frac{a^{4} x^{4} - 4 \, a^{3} b x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + 7 \, b^{4} + 12 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)^3,x, algorithm="fricas")

[Out]

1/2*(a^4*x^4 - 4*a^3*b*x^3 - 11*a^2*b^2*x^2 + 2*a*b^3*x + 7*b^4 + 12*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*log(a*x +
b))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2)

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Sympy [A]  time = 0.419766, size = 70, normalized size = 1.09 \begin{align*} \frac{8 a b^{3} x + 7 b^{4}}{2 a^{7} x^{2} + 4 a^{6} b x + 2 a^{5} b^{2}} + \frac{x^{2}}{2 a^{3}} - \frac{3 b x}{a^{4}} + \frac{6 b^{2} \log{\left (a x + b \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)**3,x)

[Out]

(8*a*b**3*x + 7*b**4)/(2*a**7*x**2 + 4*a**6*b*x + 2*a**5*b**2) + x**2/(2*a**3) - 3*b*x/a**4 + 6*b**2*log(a*x +
b)/a**5

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Giac [A]  time = 1.09574, size = 82, normalized size = 1.28 \begin{align*} \frac{6 \, b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{5}} + \frac{a^{3} x^{2} - 6 \, a^{2} b x}{2 \, a^{6}} + \frac{8 \, a b^{3} x + 7 \, b^{4}}{2 \,{\left (a x + b\right )}^{2} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)^3,x, algorithm="giac")

[Out]

6*b^2*log(abs(a*x + b))/a^5 + 1/2*(a^3*x^2 - 6*a^2*b*x)/a^6 + 1/2*(8*a*b^3*x + 7*b^4)/((a*x + b)^2*a^5)