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+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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Rubi [A]  time = 0.04, antiderivative size = 64, 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} \[ -\frac {b^4}{2 a^5 (a x+b)^2}+\frac {4 b^3}{a^5 (a x+b)}+\frac {6 b^2 \log (a x+b)}{a^5}-\frac {3 b x}{a^4}+\frac {x^2}{2 a^3} \]

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 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}{\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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 1.06, size = 95, normalized size = 1.48 \[ \frac {a^{4} x^{4} - 4 \, a^{3} b x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + 7 \, b^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{2 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}} \]

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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giac [A]  time = 0.15, size = 61, normalized size = 0.95 \[ \frac {6 \, b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{5}} + \frac {a^{3} x^{2} - 6 \, a^{2} b x}{2 \, a^{6}} + \frac {8 \, a b^{3} x + 7 \, b^{4}}{2 \, {\left (a x + b\right )}^{2} a^{5}} \]

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)

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maple [A]  time = 0.01, size = 61, normalized size = 0.95 \[ \frac {x^{2}}{2 a^{3}}-\frac {b^{4}}{2 \left (a x +b \right )^{2} a^{5}}-\frac {3 b x}{a^{4}}+\frac {4 b^{3}}{\left (a x +b \right ) a^{5}}+\frac {6 b^{2} \ln \left (a x +b \right )}{a^{5}} \]

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 = 1.07, size = 69, normalized size = 1.08 \[ \frac {8 \, a b^{3} x + 7 \, b^{4}}{2 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}} + \frac {6 \, b^{2} \log \left (a x + b\right )}{a^{5}} + \frac {a x^{2} - 6 \, b x}{2 \, a^{4}} \]

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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mupad [B]  time = 1.07, size = 70, normalized size = 1.09 \[ \frac {4\,b^3\,x+\frac {7\,b^4}{2\,a}}{a^6\,x^2+2\,a^5\,b\,x+a^4\,b^2}+\frac {x^2}{2\,a^3}+\frac {6\,b^2\,\ln \left (b+a\,x\right )}{a^5}-\frac {3\,b\,x}{a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x)^3,x)

[Out]

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

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sympy [A]  time = 0.30, size = 70, normalized size = 1.09 \[ \frac {8 a b^{3} x + 7 b^{4}}{2 a^{7} x^{2} + 4 a^{6} b x + 2 a^{5} b^{2}} + \frac {x^{2}}{2 a^{3}} - \frac {3 b x}{a^{4}} + \frac {6 b^{2} \log {\left (a x + b \right )}}{a^{5}} \]

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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