\(\int \frac {1}{(a+\frac {b}{x})^3 x^8} \, dx\) [1645]

Optimal result

Integrand size = 13, antiderivative size = 97 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=-\frac {1}{4 b^3 x^4}+\frac {a}{b^4 x^3}-\frac {3 a^2}{b^5 x^2}+\frac {10 a^3}{b^6 x}+\frac {a^4}{2 b^5 (b+a x)^2}+\frac {5 a^4}{b^6 (b+a x)}+\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (b+a x)}{b^7} \]

[Out]

-1/4/b^3/x^4+a/b^4/x^3-3*a^2/b^5/x^2+10*a^3/b^6/x+1/2*a^4/b^5/(a*x+b)^2+5*a^4/b^6/(a*x+b)+15*a^4*ln(x)/b^7-15*
a^4*ln(a*x+b)/b^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (a x+b)}{b^7}+\frac {5 a^4}{b^6 (a x+b)}+\frac {a^4}{2 b^5 (a x+b)^2}+\frac {10 a^3}{b^6 x}-\frac {3 a^2}{b^5 x^2}+\frac {a}{b^4 x^3}-\frac {1}{4 b^3 x^4} \]

[In]

Int[1/((a + b/x)^3*x^8),x]

[Out]

-1/4*1/(b^3*x^4) + a/(b^4*x^3) - (3*a^2)/(b^5*x^2) + (10*a^3)/(b^6*x) + a^4/(2*b^5*(b + a*x)^2) + (5*a^4)/(b^6
*(b + a*x)) + (15*a^4*Log[x])/b^7 - (15*a^4*Log[b + a*x])/b^7

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^5 (b+a x)^3} \, dx \\ & = \int \left (\frac {1}{b^3 x^5}-\frac {3 a}{b^4 x^4}+\frac {6 a^2}{b^5 x^3}-\frac {10 a^3}{b^6 x^2}+\frac {15 a^4}{b^7 x}-\frac {a^5}{b^5 (b+a x)^3}-\frac {5 a^5}{b^6 (b+a x)^2}-\frac {15 a^5}{b^7 (b+a x)}\right ) \, dx \\ & = -\frac {1}{4 b^3 x^4}+\frac {a}{b^4 x^3}-\frac {3 a^2}{b^5 x^2}+\frac {10 a^3}{b^6 x}+\frac {a^4}{2 b^5 (b+a x)^2}+\frac {5 a^4}{b^6 (b+a x)}+\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (b+a x)}{b^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {\frac {b \left (-b^5+2 a b^4 x-5 a^2 b^3 x^2+20 a^3 b^2 x^3+90 a^4 b x^4+60 a^5 x^5\right )}{x^4 (b+a x)^2}+60 a^4 \log (x)-60 a^4 \log (b+a x)}{4 b^7} \]

[In]

Integrate[1/((a + b/x)^3*x^8),x]

[Out]

((b*(-b^5 + 2*a*b^4*x - 5*a^2*b^3*x^2 + 20*a^3*b^2*x^3 + 90*a^4*b*x^4 + 60*a^5*x^5))/(x^4*(b + a*x)^2) + 60*a^
4*Log[x] - 60*a^4*Log[b + a*x])/(4*b^7)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {1}{4 b^{3} x^{4}}+\frac {a}{b^{4} x^{3}}-\frac {3 a^{2}}{b^{5} x^{2}}+\frac {10 a^{3}}{b^{6} x}+\frac {a^{4}}{2 b^{5} \left (a x +b \right )^{2}}+\frac {5 a^{4}}{b^{6} \left (a x +b \right )}+\frac {15 a^{4} \ln \left (x \right )}{b^{7}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}\)	\(94\)
risch	\(\frac {\frac {15 a^{5} x^{5}}{b^{6}}+\frac {45 a^{4} x^{4}}{2 b^{5}}+\frac {5 a^{3} x^{3}}{b^{4}}-\frac {5 a^{2} x^{2}}{4 b^{3}}+\frac {a x}{2 b^{2}}-\frac {1}{4 b}}{x^{4} \left (a x +b \right )^{2}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}+\frac {15 a^{4} \ln \left (-x \right )}{b^{7}}\)	\(96\)
norman	\(\frac {-\frac {x^{3}}{4 b}+\frac {a \,x^{4}}{2 b^{2}}-\frac {5 a^{2} x^{5}}{4 b^{3}}+\frac {5 a^{3} x^{6}}{b^{4}}-\frac {30 a^{5} x^{8}}{b^{6}}-\frac {45 a^{6} x^{9}}{2 b^{7}}}{\left (a x +b \right )^{2} x^{7}}+\frac {15 a^{4} \ln \left (x \right )}{b^{7}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}\)	\(99\)
parallelrisch	\(\frac {60 \ln \left (x \right ) x^{6} a^{6}-60 \ln \left (a x +b \right ) x^{6} a^{6}+120 \ln \left (x \right ) x^{5} a^{5} b -120 \ln \left (a x +b \right ) x^{5} a^{5} b -90 a^{6} x^{6}+60 \ln \left (x \right ) x^{4} a^{4} b^{2}-60 \ln \left (a x +b \right ) x^{4} a^{4} b^{2}-120 a^{5} b \,x^{5}+20 a^{3} x^{3} b^{3}-5 b^{4} x^{2} a^{2}+2 a x \,b^{5}-b^{6}}{4 b^{7} x^{4} \left (a x +b \right )^{2}}\)	\(148\)

[In]

int(1/(a+b/x)^3/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/4/b^3/x^4+a/b^4/x^3-3*a^2/b^5/x^2+10*a^3/b^6/x+1/2*a^4/b^5/(a*x+b)^2+5*a^4/b^6/(a*x+b)+15*a^4*ln(x)/b^7-15*
a^4*ln(a*x+b)/b^7

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6} - 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{2} b^{7} x^{6} + 2 \, a b^{8} x^{5} + b^{9} x^{4}\right )}} \]

[In]

integrate(1/(a+b/x)^3/x^8,x, algorithm="fricas")

[Out]

1/4*(60*a^5*b*x^5 + 90*a^4*b^2*x^4 + 20*a^3*b^3*x^3 - 5*a^2*b^4*x^2 + 2*a*b^5*x - b^6 - 60*(a^6*x^6 + 2*a^5*b*
x^5 + a^4*b^2*x^4)*log(a*x + b) + 60*(a^6*x^6 + 2*a^5*b*x^5 + a^4*b^2*x^4)*log(x))/(a^2*b^7*x^6 + 2*a*b^8*x^5
+ b^9*x^4)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {15 a^{4} \left (\log {\left (x \right )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{7}} + \frac {60 a^{5} x^{5} + 90 a^{4} b x^{4} + 20 a^{3} b^{2} x^{3} - 5 a^{2} b^{3} x^{2} + 2 a b^{4} x - b^{5}}{4 a^{2} b^{6} x^{6} + 8 a b^{7} x^{5} + 4 b^{8} x^{4}} \]

[In]

integrate(1/(a+b/x)**3/x**8,x)

[Out]

15*a**4*(log(x) - log(x + b/a))/b**7 + (60*a**5*x**5 + 90*a**4*b*x**4 + 20*a**3*b**2*x**3 - 5*a**2*b**3*x**2 +
2*a*b**4*x - b**5)/(4*a**2*b**6*x**6 + 8*a*b**7*x**5 + 4*b**8*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {60 \, a^{5} x^{5} + 90 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} + 2 \, a b^{4} x - b^{5}}{4 \, {\left (a^{2} b^{6} x^{6} + 2 \, a b^{7} x^{5} + b^{8} x^{4}\right )}} - \frac {15 \, a^{4} \log \left (a x + b\right )}{b^{7}} + \frac {15 \, a^{4} \log \left (x\right )}{b^{7}} \]

[In]

integrate(1/(a+b/x)^3/x^8,x, algorithm="maxima")

[Out]

1/4*(60*a^5*x^5 + 90*a^4*b*x^4 + 20*a^3*b^2*x^3 - 5*a^2*b^3*x^2 + 2*a*b^4*x - b^5)/(a^2*b^6*x^6 + 2*a*b^7*x^5
+ b^8*x^4) - 15*a^4*log(a*x + b)/b^7 + 15*a^4*log(x)/b^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=-\frac {15 \, a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{7}} + \frac {15 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{7}} + \frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6}}{4 \, {\left (a x + b\right )}^{2} b^{7} x^{4}} \]

[In]

integrate(1/(a+b/x)^3/x^8,x, algorithm="giac")

[Out]

-15*a^4*log(abs(a*x + b))/b^7 + 15*a^4*log(abs(x))/b^7 + 1/4*(60*a^5*b*x^5 + 90*a^4*b^2*x^4 + 20*a^3*b^3*x^3 -
5*a^2*b^4*x^2 + 2*a*b^5*x - b^6)/((a*x + b)^2*b^7*x^4)

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {\frac {5\,a^3\,x^3}{b^4}-\frac {5\,a^2\,x^2}{4\,b^3}-\frac {1}{4\,b}+\frac {45\,a^4\,x^4}{2\,b^5}+\frac {15\,a^5\,x^5}{b^6}+\frac {a\,x}{2\,b^2}}{a^2\,x^6+2\,a\,b\,x^5+b^2\,x^4}-\frac {30\,a^4\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^7} \]

[In]

int(1/(x^8*(a + b/x)^3),x)

[Out]

((5*a^3*x^3)/b^4 - (5*a^2*x^2)/(4*b^3) - 1/(4*b) + (45*a^4*x^4)/(2*b^5) + (15*a^5*x^5)/b^6 + (a*x)/(2*b^2))/(a
^2*x^6 + b^2*x^4 + 2*a*b*x^5) - (30*a^4*atanh((2*a*x)/b + 1))/b^7