\(\int \frac {1}{(a+\frac {b}{x})^3 x^6} \, dx\) [1643]

Optimal result

Integrand size = 13, antiderivative size = 76 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=-\frac {1}{2 b^3 x^2}+\frac {3 a}{b^4 x}+\frac {a^2}{2 b^3 (b+a x)^2}+\frac {3 a^2}{b^4 (b+a x)}+\frac {6 a^2 \log (x)}{b^5}-\frac {6 a^2 \log (b+a x)}{b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 46} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {6 a^2 \log (x)}{b^5}-\frac {6 a^2 \log (a x+b)}{b^5}+\frac {3 a^2}{b^4 (a x+b)}+\frac {a^2}{2 b^3 (a x+b)^2}+\frac {3 a}{b^4 x}-\frac {1}{2 b^3 x^2} \]

[In]

Int[1/((a + b/x)^3*x^6),x]

[Out]

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 269

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*} \text {integral}& = \int \frac {1}{x^3 (b+a x)^3} \, dx \\ & = \int \left (\frac {1}{b^3 x^3}-\frac {3 a}{b^4 x^2}+\frac {6 a^2}{b^5 x}-\frac {a^3}{b^3 (b+a x)^3}-\frac {3 a^3}{b^4 (b+a x)^2}-\frac {6 a^3}{b^5 (b+a x)}\right ) \, dx \\ & = -\frac {1}{2 b^3 x^2}+\frac {3 a}{b^4 x}+\frac {a^2}{2 b^3 (b+a x)^2}+\frac {3 a^2}{b^4 (b+a x)}+\frac {6 a^2 \log (x)}{b^5}-\frac {6 a^2 \log (b+a x)}{b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {\frac {b \left (-b^3+4 a b^2 x+18 a^2 b x^2+12 a^3 x^3\right )}{x^2 (b+a x)^2}+12 a^2 \log (x)-12 a^2 \log (b+a x)}{2 b^5} \]

[In]

Integrate[1/((a + b/x)^3*x^6),x]

[Out]

((b*(-b^3 + 4*a*b^2*x + 18*a^2*b*x^2 + 12*a^3*x^3))/(x^2*(b + a*x)^2) + 12*a^2*Log[x] - 12*a^2*Log[b + a*x])/(
2*b^5)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {1}{2 b^{3} x^{2}}+\frac {3 a}{b^{4} x}+\frac {a^{2}}{2 b^{3} \left (a x +b \right )^{2}}+\frac {3 a^{2}}{b^{4} \left (a x +b \right )}+\frac {6 a^{2} \ln \left (x \right )}{b^{5}}-\frac {6 a^{2} \ln \left (a x +b \right )}{b^{5}}\)	\(73\)
risch	\(\frac {\frac {6 a^{3} x^{3}}{b^{4}}+\frac {9 a^{2} x^{2}}{b^{3}}+\frac {2 a x}{b^{2}}-\frac {1}{2 b}}{x^{2} \left (a x +b \right )^{2}}+\frac {6 a^{2} \ln \left (-x \right )}{b^{5}}-\frac {6 a^{2} \ln \left (a x +b \right )}{b^{5}}\)	\(74\)
norman	\(\frac {-\frac {x^{3}}{2 b}+\frac {2 a \,x^{4}}{b^{2}}-\frac {12 a^{3} x^{6}}{b^{4}}-\frac {9 a^{4} x^{7}}{b^{5}}}{x^{5} \left (a x +b \right )^{2}}+\frac {6 a^{2} \ln \left (x \right )}{b^{5}}-\frac {6 a^{2} \ln \left (a x +b \right )}{b^{5}}\)	\(77\)
parallelrisch	\(\frac {12 \ln \left (x \right ) x^{4} a^{6}-12 \ln \left (a x +b \right ) x^{4} a^{6}+24 \ln \left (x \right ) x^{3} a^{5} b -24 \ln \left (a x +b \right ) x^{3} a^{5} b +12 \ln \left (x \right ) x^{2} a^{4} b^{2}-12 \ln \left (a x +b \right ) x^{2} a^{4} b^{2}+12 a^{5} b \,x^{3}+18 a^{4} x^{2} b^{2}+4 x \,a^{3} b^{3}-a^{2} b^{4}}{2 a^{2} b^{5} x^{2} \left (a x +b \right )^{2}}\)	\(137\)

[In]

int(1/(a+b/x)^3/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.71 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {12 \, a^{3} b x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - b^{4} - 12 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (a x + b\right ) + 12 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}} \]

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="fricas")

[Out]

1/2*(12*a^3*b*x^3 + 18*a^2*b^2*x^2 + 4*a*b^3*x - b^4 - 12*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*log(a*x + b) +
12*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*log(x))/(a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {6 a^{2} \left (\log {\left (x \right )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{5}} + \frac {12 a^{3} x^{3} + 18 a^{2} b x^{2} + 4 a b^{2} x - b^{3}}{2 a^{2} b^{4} x^{4} + 4 a b^{5} x^{3} + 2 b^{6} x^{2}} \]

[In]

integrate(1/(a+b/x)**3/x**6,x)

[Out]

6*a**2*(log(x) - log(x + b/a))/b**5 + (12*a**3*x**3 + 18*a**2*b*x**2 + 4*a*b**2*x - b**3)/(2*a**2*b**4*x**4 +
4*a*b**5*x**3 + 2*b**6*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \, {\left (a^{2} b^{4} x^{4} + 2 \, a b^{5} x^{3} + b^{6} x^{2}\right )}} - \frac {6 \, a^{2} \log \left (a x + b\right )}{b^{5}} + \frac {6 \, a^{2} \log \left (x\right )}{b^{5}} \]

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="maxima")

[Out]

1/2*(12*a^3*x^3 + 18*a^2*b*x^2 + 4*a*b^2*x - b^3)/(a^2*b^4*x^4 + 2*a*b^5*x^3 + b^6*x^2) - 6*a^2*log(a*x + b)/b
^5 + 6*a^2*log(x)/b^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=-\frac {6 \, a^{2} \log \left ({\left | a x + b \right |}\right )}{b^{5}} + \frac {6 \, a^{2} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \, {\left (a x^{2} + b x\right )}^{2} b^{4}} \]

[In]

integrate(1/(a+b/x)^3/x^6,x, algorithm="giac")

[Out]

-6*a^2*log(abs(a*x + b))/b^5 + 6*a^2*log(abs(x))/b^5 + 1/2*(12*a^3*x^3 + 18*a^2*b*x^2 + 4*a*b^2*x - b^3)/((a*x
^2 + b*x)^2*b^4)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^6} \, dx=\frac {\frac {9\,a^2\,x^2}{b^3}-\frac {1}{2\,b}+\frac {6\,a^3\,x^3}{b^4}+\frac {2\,a\,x}{b^2}}{a^2\,x^4+2\,a\,b\,x^3+b^2\,x^2}-\frac {12\,a^2\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^5} \]

[In]

int(1/(x^6*(a + b/x)^3),x)

[Out]

((9*a^2*x^2)/b^3 - 1/(2*b) + (6*a^3*x^3)/b^4 + (2*a*x)/b^2)/(a^2*x^4 + b^2*x^2 + 2*a*b*x^3) - (12*a^2*atanh((2
*a*x)/b + 1))/b^5