\(\int \frac {1}{(a+\frac {b}{x})^3 x^5} \, dx\) [1642]

Optimal result

Integrand size = 13, antiderivative size = 57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=-\frac {1}{b^3 x}-\frac {a}{2 b^2 (b+a x)^2}-\frac {2 a}{b^3 (b+a x)}-\frac {3 a \log (x)}{b^4}+\frac {3 a \log (b+a x)}{b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=-\frac {3 a \log (x)}{b^4}+\frac {3 a \log (a x+b)}{b^4}-\frac {2 a}{b^3 (a x+b)}-\frac {a}{2 b^2 (a x+b)^2}-\frac {1}{b^3 x} \]

[In]

Int[1/((a + b/x)^3*x^5),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=-\frac {\frac {b \left (2 b^2+9 a b x+6 a^2 x^2\right )}{x (b+a x)^2}+6 a \log (x)-6 a \log (b+a x)}{2 b^4} \]

[In]

Integrate[1/((a + b/x)^3*x^5),x]

[Out]

-1/2*((b*(2*b^2 + 9*a*b*x + 6*a^2*x^2))/(x*(b + a*x)^2) + 6*a*Log[x] - 6*a*Log[b + a*x])/b^4

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {1}{b^{3} x}-\frac {a}{2 b^{2} \left (a x +b \right )^{2}}-\frac {2 a}{b^{3} \left (a x +b \right )}-\frac {3 a \ln \left (x \right )}{b^{4}}+\frac {3 a \ln \left (a x +b \right )}{b^{4}}\)	\(56\)
risch	\(\frac {-\frac {3 a^{2} x^{2}}{b^{3}}-\frac {9 a x}{2 b^{2}}-\frac {1}{b}}{x \left (a x +b \right )^{2}}-\frac {3 a \ln \left (x \right )}{b^{4}}+\frac {3 a \ln \left (-a x -b \right )}{b^{4}}\)	\(60\)
norman	\(\frac {-\frac {x^{3}}{b}+\frac {6 a^{2} x^{5}}{b^{3}}+\frac {9 a^{3} x^{6}}{2 b^{4}}}{x^{4} \left (a x +b \right )^{2}}-\frac {3 a \ln \left (x \right )}{b^{4}}+\frac {3 a \ln \left (a x +b \right )}{b^{4}}\)	\(64\)
parallelrisch	\(-\frac {6 a^{3} \ln \left (x \right ) x^{3}-6 a^{3} \ln \left (a x +b \right ) x^{3}+12 a^{2} b \ln \left (x \right ) x^{2}-12 \ln \left (a x +b \right ) x^{2} a^{2} b -9 a^{3} x^{3}+6 a \,b^{2} \ln \left (x \right ) x -6 \ln \left (a x +b \right ) x a \,b^{2}-12 a^{2} b \,x^{2}+2 b^{3}}{2 b^{4} x \left (a x +b \right )^{2}}\)	\(111\)

[In]

int(1/(a+b/x)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=-\frac {6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3} - 6 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \left (a x + b\right ) + 6 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \left (x\right )}{2 \, {\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}} \]

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=\frac {3 a \left (- \log {\left (x \right )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {- 6 a^{2} x^{2} - 9 a b x - 2 b^{2}}{2 a^{2} b^{3} x^{3} + 4 a b^{4} x^{2} + 2 b^{5} x} \]

[In]

integrate(1/(a+b/x)**3/x**5,x)

[Out]

3*a*(-log(x) + log(x + b/a))/b**4 + (-6*a**2*x**2 - 9*a*b*x - 2*b**2)/(2*a**2*b**3*x**3 + 4*a*b**4*x**2 + 2*b*
*5*x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=-\frac {6 \, a^{2} x^{2} + 9 \, a b x + 2 \, b^{2}}{2 \, {\left (a^{2} b^{3} x^{3} + 2 \, a b^{4} x^{2} + b^{5} x\right )}} + \frac {3 \, a \log \left (a x + b\right )}{b^{4}} - \frac {3 \, a \log \left (x\right )}{b^{4}} \]

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=\frac {3 \, a \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac {3 \, a \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{2 \, {\left (a x + b\right )}^{2} b^{4} x} \]

[In]

integrate(1/(a+b/x)^3/x^5,x, algorithm="giac")

[Out]

3*a*log(abs(a*x + b))/b^4 - 3*a*log(abs(x))/b^4 - 1/2*(6*a^2*b*x^2 + 9*a*b^2*x + 2*b^3)/((a*x + b)^2*b^4*x)

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^5} \, dx=\frac {6\,a\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4}-\frac {\frac {1}{b}+\frac {3\,a^2\,x^2}{b^3}+\frac {9\,a\,x}{2\,b^2}}{a^2\,x^3+2\,a\,b\,x^2+b^2\,x} \]

[In]

int(1/(x^5*(a + b/x)^3),x)

[Out]

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