Integrand size = 13, antiderivative size = 85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=-\frac {a^5}{2 b^6 \left (a+\frac {b}{x}\right )^2}+\frac {5 a^4}{b^6 \left (a+\frac {b}{x}\right )}-\frac {1}{3 b^3 x^3}+\frac {3 a}{2 b^4 x^2}-\frac {6 a^2}{b^5 x}+\frac {10 a^3 \log \left (a+\frac {b}{x}\right )}{b^6} \] Output:
-1/2*a^5/b^6/(a+b/x)^2+5*a^4/b^6/(a+b/x)-1/3/b^3/x^3+3/2*a/b^4/x^2-6*a^2/b ^5/x+10*a^3*ln(a+b/x)/b^6
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=-\frac {\frac {b \left (2 b^4-5 a b^3 x+20 a^2 b^2 x^2+90 a^3 b x^3+60 a^4 x^4\right )}{x^3 (b+a x)^2}+60 a^3 \log (x)-60 a^3 \log (b+a x)}{6 b^6} \] Input:
Integrate[1/((a + b/x)^3*x^7),x]
Output:
-1/6*((b*(2*b^4 - 5*a*b^3*x + 20*a^2*b^2*x^2 + 90*a^3*b*x^3 + 60*a^4*x^4)) /(x^3*(b + a*x)^2) + 60*a^3*Log[x] - 60*a^3*Log[b + a*x])/b^6
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {795, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^7 \left (a+\frac {b}{x}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^4 (a x+b)^3}dx\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \int \left (\frac {10 a^4}{b^6 (a x+b)}+\frac {4 a^4}{b^5 (a x+b)^2}+\frac {a^4}{b^4 (a x+b)^3}-\frac {10 a^3}{b^6 x}+\frac {6 a^2}{b^5 x^2}-\frac {3 a}{b^4 x^3}+\frac {1}{b^3 x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {10 a^3 \log (x)}{b^6}+\frac {10 a^3 \log (a x+b)}{b^6}-\frac {4 a^3}{b^5 (a x+b)}-\frac {a^3}{2 b^4 (a x+b)^2}-\frac {6 a^2}{b^5 x}+\frac {3 a}{2 b^4 x^2}-\frac {1}{3 b^3 x^3}\) |
Input:
Int[1/((a + b/x)^3*x^7),x]
Output:
-1/3*1/(b^3*x^3) + (3*a)/(2*b^4*x^2) - (6*a^2)/(b^5*x) - a^3/(2*b^4*(b + a *x)^2) - (4*a^3)/(b^5*(b + a*x)) - (10*a^3*Log[x])/b^6 + (10*a^3*Log[b + a *x])/b^6
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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]
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {1}{3 b^{3} x^{3}}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}-\frac {6 a^{2}}{b^{5} x}+\frac {3 a}{2 b^{4} x^{2}}-\frac {a^{3}}{2 b^{4} \left (a x +b \right )^{2}}+\frac {10 a^{3} \ln \left (a x +b \right )}{b^{6}}-\frac {4 a^{3}}{b^{5} \left (a x +b \right )}\) | \(84\) |
risch | \(\frac {-\frac {10 a^{4} x^{4}}{b^{5}}-\frac {15 a^{3} x^{3}}{b^{4}}-\frac {10 a^{2} x^{2}}{3 b^{3}}+\frac {5 a x}{6 b^{2}}-\frac {1}{3 b}}{x^{3} \left (a x +b \right )^{2}}+\frac {10 a^{3} \ln \left (-a x -b \right )}{b^{6}}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}\) | \(86\) |
norman | \(\frac {\frac {15 a^{5} x^{8}}{b^{6}}-\frac {x^{3}}{3 b}+\frac {5 a \,x^{4}}{6 b^{2}}-\frac {10 a^{2} x^{5}}{3 b^{3}}+\frac {20 a^{4} x^{7}}{b^{5}}}{\left (a x +b \right )^{2} x^{6}}-\frac {10 a^{3} \ln \left (x \right )}{b^{6}}+\frac {10 a^{3} \ln \left (a x +b \right )}{b^{6}}\) | \(88\) |
parallelrisch | \(-\frac {60 a^{5} \ln \left (x \right ) x^{5}-60 a^{5} \ln \left (a x +b \right ) x^{5}+120 \ln \left (x \right ) x^{4} a^{4} b -120 \ln \left (a x +b \right ) x^{4} a^{4} b -90 a^{5} x^{5}+60 \ln \left (x \right ) x^{3} a^{3} b^{2}-60 \ln \left (a x +b \right ) x^{3} a^{3} b^{2}-120 a^{4} b \,x^{4}+20 a^{2} b^{3} x^{2}-5 b^{4} x a +2 b^{5}}{6 b^{6} \left (a x +b \right )^{2} x^{3}}\) | \(137\) |
Input:
int(1/(a+b/x)^3/x^7,x,method=_RETURNVERBOSE)
Output:
-1/3/b^3/x^3-10/b^6*a^3*ln(x)-6*a^2/b^5/x+3/2*a/b^4/x^2-1/2*a^3/b^4/(a*x+b )^2+10/b^6*a^3*ln(a*x+b)-4*a^3/b^5/(a*x+b)
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.66 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=-\frac {60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5} - 60 \, {\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{5} x^{5} + 2 \, a^{4} b x^{4} + a^{3} b^{2} x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + b^{8} x^{3}\right )}} \] Input:
integrate(1/(a+b/x)^3/x^7,x, algorithm="fricas")
Output:
-1/6*(60*a^4*b*x^4 + 90*a^3*b^2*x^3 + 20*a^2*b^3*x^2 - 5*a*b^4*x + 2*b^5 - 60*(a^5*x^5 + 2*a^4*b*x^4 + a^3*b^2*x^3)*log(a*x + b) + 60*(a^5*x^5 + 2*a ^4*b*x^4 + a^3*b^2*x^3)*log(x))/(a^2*b^6*x^5 + 2*a*b^7*x^4 + b^8*x^3)
Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=\frac {10 a^{3} \left (- \log {\left (x \right )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{6}} + \frac {- 60 a^{4} x^{4} - 90 a^{3} b x^{3} - 20 a^{2} b^{2} x^{2} + 5 a b^{3} x - 2 b^{4}}{6 a^{2} b^{5} x^{5} + 12 a b^{6} x^{4} + 6 b^{7} x^{3}} \] Input:
integrate(1/(a+b/x)**3/x**7,x)
Output:
10*a**3*(-log(x) + log(x + b/a))/b**6 + (-60*a**4*x**4 - 90*a**3*b*x**3 - 20*a**2*b**2*x**2 + 5*a*b**3*x - 2*b**4)/(6*a**2*b**5*x**5 + 12*a*b**6*x** 4 + 6*b**7*x**3)
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=-\frac {60 \, a^{4} x^{4} + 90 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a b^{3} x + 2 \, b^{4}}{6 \, {\left (a^{2} b^{5} x^{5} + 2 \, a b^{6} x^{4} + b^{7} x^{3}\right )}} + \frac {10 \, a^{3} \log \left (a x + b\right )}{b^{6}} - \frac {10 \, a^{3} \log \left (x\right )}{b^{6}} \] Input:
integrate(1/(a+b/x)^3/x^7,x, algorithm="maxima")
Output:
-1/6*(60*a^4*x^4 + 90*a^3*b*x^3 + 20*a^2*b^2*x^2 - 5*a*b^3*x + 2*b^4)/(a^2 *b^5*x^5 + 2*a*b^6*x^4 + b^7*x^3) + 10*a^3*log(a*x + b)/b^6 - 10*a^3*log(x )/b^6
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=\frac {10 \, a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{6}} - \frac {10 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{4} + 90 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 5 \, a b^{4} x + 2 \, b^{5}}{6 \, {\left (a x + b\right )}^{2} b^{6} x^{3}} \] Input:
integrate(1/(a+b/x)^3/x^7,x, algorithm="giac")
Output:
10*a^3*log(abs(a*x + b))/b^6 - 10*a^3*log(abs(x))/b^6 - 1/6*(60*a^4*b*x^4 + 90*a^3*b^2*x^3 + 20*a^2*b^3*x^2 - 5*a*b^4*x + 2*b^5)/((a*x + b)^2*b^6*x^ 3)
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=\frac {20\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^6}-\frac {\frac {1}{3\,b}+\frac {10\,a^2\,x^2}{3\,b^3}+\frac {15\,a^3\,x^3}{b^4}+\frac {10\,a^4\,x^4}{b^5}-\frac {5\,a\,x}{6\,b^2}}{a^2\,x^5+2\,a\,b\,x^4+b^2\,x^3} \] Input:
int(1/(x^7*(a + b/x)^3),x)
Output:
(20*a^3*atanh((2*a*x)/b + 1))/b^6 - (1/(3*b) + (10*a^2*x^2)/(3*b^3) + (15* a^3*x^3)/b^4 + (10*a^4*x^4)/b^5 - (5*a*x)/(6*b^2))/(a^2*x^5 + b^2*x^3 + 2* a*b*x^4)
Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^7} \, dx=\frac {60 \,\mathrm {log}\left (a x +b \right ) a^{5} x^{5}+120 \,\mathrm {log}\left (a x +b \right ) a^{4} b \,x^{4}+60 \,\mathrm {log}\left (a x +b \right ) a^{3} b^{2} x^{3}-60 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-120 \,\mathrm {log}\left (x \right ) a^{4} b \,x^{4}-60 \,\mathrm {log}\left (x \right ) a^{3} b^{2} x^{3}+30 a^{5} x^{5}-60 a^{3} b^{2} x^{3}-20 a^{2} b^{3} x^{2}+5 a \,b^{4} x -2 b^{5}}{6 b^{6} x^{3} \left (a^{2} x^{2}+2 a b x +b^{2}\right )} \] Input:
int(1/(a+b/x)^3/x^7,x)
Output:
(60*log(a*x + b)*a**5*x**5 + 120*log(a*x + b)*a**4*b*x**4 + 60*log(a*x + b )*a**3*b**2*x**3 - 60*log(x)*a**5*x**5 - 120*log(x)*a**4*b*x**4 - 60*log(x )*a**3*b**2*x**3 + 30*a**5*x**5 - 60*a**3*b**2*x**3 - 20*a**2*b**3*x**2 + 5*a*b**4*x - 2*b**5)/(6*b**6*x**3*(a**2*x**2 + 2*a*b*x + b**2))