Integrand size = 15, antiderivative size = 173 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=-\frac {3 a^{11}}{2 b^{12} \left (a+\frac {b}{\sqrt [3]{x}}\right )^2}+\frac {33 a^{10}}{b^{12} \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {1}{3 b^3 x^3}+\frac {9 a}{8 b^4 x^{8/3}}-\frac {18 a^2}{7 b^5 x^{7/3}}+\frac {5 a^3}{b^6 x^2}-\frac {9 a^4}{b^7 x^{5/3}}+\frac {63 a^5}{4 b^8 x^{4/3}}-\frac {28 a^6}{b^9 x}+\frac {54 a^7}{b^{10} x^{2/3}}-\frac {135 a^8}{b^{11} \sqrt [3]{x}}+\frac {165 a^9 \log \left (a+\frac {b}{\sqrt [3]{x}}\right )}{b^{12}} \] Output:
-3/2*a^11/b^12/(a+b/x^(1/3))^2+33*a^10/b^12/(a+b/x^(1/3))-1/3/b^3/x^3+9/8* a/b^4/x^(8/3)-18/7*a^2/b^5/x^(7/3)+5*a^3/b^6/x^2-9*a^4/b^7/x^(5/3)+63/4*a^ 5/b^8/x^(4/3)-28*a^6/b^9/x+54*a^7/b^10/x^(2/3)-135*a^8/b^11/x^(1/3)+165*a^ 9*ln(a+b/x^(1/3))/b^12
Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=-\frac {\frac {b \left (56 b^{10}-77 a b^9 \sqrt [3]{x}+110 a^2 b^8 x^{2/3}-165 a^3 b^7 x+264 a^4 b^6 x^{4/3}-462 a^5 b^5 x^{5/3}+924 a^6 b^4 x^2-2310 a^7 b^3 x^{7/3}+9240 a^8 b^2 x^{8/3}+41580 a^9 b x^3+27720 a^{10} x^{10/3}\right )}{\left (b+a \sqrt [3]{x}\right )^2 x^3}-27720 a^9 \log \left (b+a \sqrt [3]{x}\right )+9240 a^9 \log (x)}{168 b^{12}} \] Input:
Integrate[1/((a + b/x^(1/3))^3*x^5),x]
Output:
-1/168*((b*(56*b^10 - 77*a*b^9*x^(1/3) + 110*a^2*b^8*x^(2/3) - 165*a^3*b^7 *x + 264*a^4*b^6*x^(4/3) - 462*a^5*b^5*x^(5/3) + 924*a^6*b^4*x^2 - 2310*a^ 7*b^3*x^(7/3) + 9240*a^8*b^2*x^(8/3) + 41580*a^9*b*x^3 + 27720*a^10*x^(10/ 3)))/((b + a*x^(1/3))^2*x^3) - 27720*a^9*Log[b + a*x^(1/3)] + 9240*a^9*Log [x])/b^12
Time = 0.59 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 798, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {1}{x^4 \left (a \sqrt [3]{x}+b\right )^3}dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \frac {1}{\left (\sqrt [3]{x} a+b\right )^3 x^{10/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \int \left (\frac {55 a^{10}}{b^{12} \left (\sqrt [3]{x} a+b\right )}+\frac {10 a^{10}}{b^{11} \left (\sqrt [3]{x} a+b\right )^2}+\frac {a^{10}}{b^{10} \left (\sqrt [3]{x} a+b\right )^3}-\frac {55 a^9}{b^{12} \sqrt [3]{x}}+\frac {45 a^8}{b^{11} x^{2/3}}-\frac {36 a^7}{b^{10} x}+\frac {28 a^6}{b^9 x^{4/3}}-\frac {21 a^5}{b^8 x^{5/3}}+\frac {15 a^4}{b^7 x^2}-\frac {10 a^3}{b^6 x^{7/3}}+\frac {6 a^2}{b^5 x^{8/3}}-\frac {3 a}{b^4 x^3}+\frac {1}{b^3 x^{10/3}}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {55 a^9 \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac {55 a^9 \log \left (\sqrt [3]{x}\right )}{b^{12}}-\frac {10 a^9}{b^{11} \left (a \sqrt [3]{x}+b\right )}-\frac {a^9}{2 b^{10} \left (a \sqrt [3]{x}+b\right )^2}-\frac {45 a^8}{b^{11} \sqrt [3]{x}}+\frac {18 a^7}{b^{10} x^{2/3}}-\frac {28 a^6}{3 b^9 x}+\frac {21 a^5}{4 b^8 x^{4/3}}-\frac {3 a^4}{b^7 x^{5/3}}+\frac {5 a^3}{3 b^6 x^2}-\frac {6 a^2}{7 b^5 x^{7/3}}+\frac {3 a}{8 b^4 x^{8/3}}-\frac {1}{9 b^3 x^3}\right )\) |
Input:
Int[1/((a + b/x^(1/3))^3*x^5),x]
Output:
3*(-1/2*a^9/(b^10*(b + a*x^(1/3))^2) - (10*a^9)/(b^11*(b + a*x^(1/3))) - 1 /(9*b^3*x^3) + (3*a)/(8*b^4*x^(8/3)) - (6*a^2)/(7*b^5*x^(7/3)) + (5*a^3)/( 3*b^6*x^2) - (3*a^4)/(b^7*x^(5/3)) + (21*a^5)/(4*b^8*x^(4/3)) - (28*a^6)/( 3*b^9*x) + (18*a^7)/(b^10*x^(2/3)) - (45*a^8)/(b^11*x^(1/3)) + (55*a^9*Log [b + a*x^(1/3)])/b^12 - (55*a^9*Log[x^(1/3)])/b^12)
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.50 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(-\frac {1}{3 b^{3} x^{3}}-\frac {55 a^{9} \ln \left (x \right )}{b^{12}}-\frac {135 a^{8}}{b^{11} x^{\frac {1}{3}}}+\frac {54 a^{7}}{b^{10} x^{\frac {2}{3}}}-\frac {28 a^{6}}{b^{9} x}+\frac {63 a^{5}}{4 b^{8} x^{\frac {4}{3}}}-\frac {9 a^{4}}{b^{7} x^{\frac {5}{3}}}+\frac {5 a^{3}}{b^{6} x^{2}}-\frac {18 a^{2}}{7 b^{5} x^{\frac {7}{3}}}+\frac {9 a}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{9}}{2 b^{10} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {165 a^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {30 a^{9}}{b^{11} \left (b +a \,x^{\frac {1}{3}}\right )}\) | \(156\) |
| default | \(-\frac {1}{3 b^{3} x^{3}}-\frac {55 a^{9} \ln \left (x \right )}{b^{12}}-\frac {135 a^{8}}{b^{11} x^{\frac {1}{3}}}+\frac {54 a^{7}}{b^{10} x^{\frac {2}{3}}}-\frac {28 a^{6}}{b^{9} x}+\frac {63 a^{5}}{4 b^{8} x^{\frac {4}{3}}}-\frac {9 a^{4}}{b^{7} x^{\frac {5}{3}}}+\frac {5 a^{3}}{b^{6} x^{2}}-\frac {18 a^{2}}{7 b^{5} x^{\frac {7}{3}}}+\frac {9 a}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{9}}{2 b^{10} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}+\frac {165 a^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {30 a^{9}}{b^{11} \left (b +a \,x^{\frac {1}{3}}\right )}\) | \(156\) |
Input:
int(1/(a+b/x^(1/3))^3/x^5,x,method=_RETURNVERBOSE)
Output:
-1/3/b^3/x^3-55/b^12*a^9*ln(x)-135*a^8/b^11/x^(1/3)+54*a^7/b^10/x^(2/3)-28 *a^6/b^9/x+63/4*a^5/b^8/x^(4/3)-9*a^4/b^7/x^(5/3)+5*a^3/b^6/x^2-18/7*a^2/b ^5/x^(7/3)+9/8*a/b^4/x^(8/3)-3/2*a^9/b^10/(b+a*x^(1/3))^2+165/b^12*a^9*ln( b+a*x^(1/3))-30/b^11*a^9/(b+a*x^(1/3))
Time = 0.09 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=-\frac {9240 \, a^{12} b^{3} x^{4} + 13860 \, a^{9} b^{6} x^{3} + 3080 \, a^{6} b^{9} x^{2} - 728 \, a^{3} b^{12} x + 56 \, b^{15} - 27720 \, {\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) + 27720 \, {\left (a^{15} x^{5} + 2 \, a^{12} b^{3} x^{4} + a^{9} b^{6} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, {\left (1540 \, a^{14} b x^{4} + 2695 \, a^{11} b^{4} x^{3} + 990 \, a^{8} b^{7} x^{2} - 99 \, a^{5} b^{10} x + 24 \, a^{2} b^{13}\right )} x^{\frac {2}{3}} - 63 \, {\left (220 \, a^{13} b^{2} x^{4} + 352 \, a^{10} b^{5} x^{3} + 99 \, a^{7} b^{8} x^{2} - 18 \, a^{4} b^{11} x + 3 \, a b^{14}\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{6} b^{12} x^{5} + 2 \, a^{3} b^{15} x^{4} + b^{18} x^{3}\right )}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^5,x, algorithm="fricas")
Output:
-1/168*(9240*a^12*b^3*x^4 + 13860*a^9*b^6*x^3 + 3080*a^6*b^9*x^2 - 728*a^3 *b^12*x + 56*b^15 - 27720*(a^15*x^5 + 2*a^12*b^3*x^4 + a^9*b^6*x^3)*log(a* x^(1/3) + b) + 27720*(a^15*x^5 + 2*a^12*b^3*x^4 + a^9*b^6*x^3)*log(x^(1/3) ) + 18*(1540*a^14*b*x^4 + 2695*a^11*b^4*x^3 + 990*a^8*b^7*x^2 - 99*a^5*b^1 0*x + 24*a^2*b^13)*x^(2/3) - 63*(220*a^13*b^2*x^4 + 352*a^10*b^5*x^3 + 99* a^7*b^8*x^2 - 18*a^4*b^11*x + 3*a*b^14)*x^(1/3))/(a^6*b^12*x^5 + 2*a^3*b^1 5*x^4 + b^18*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (172) = 344\).
Time = 18.10 (sec) , antiderivative size = 848, normalized size of antiderivative = 4.90 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b/x**(1/3))**3/x**5,x)
Output:
Piecewise((zoo/x**3, Eq(a, 0) & Eq(b, 0)), (-1/(3*b**3*x**3), Eq(a, 0)), ( -1/(4*a**3*x**4), Eq(b, 0)), (-9240*a**11*x**(16/3)*log(x)/(168*a**2*b**12 *x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) + 27720*a**11*x**(16/ 3)*log(x**(1/3) + b/a)/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168* b**14*x**(14/3)) - 18480*a**10*b*x**5*log(x)/(168*a**2*b**12*x**(16/3) + 3 36*a*b**13*x**5 + 168*b**14*x**(14/3)) + 55440*a**10*b*x**5*log(x**(1/3) + b/a)/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) - 27720*a**10*b*x**5/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b* *14*x**(14/3)) - 9240*a**9*b**2*x**(14/3)*log(x)/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) + 27720*a**9*b**2*x**(14/3)*log (x**(1/3) + b/a)/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b**14* x**(14/3)) - 41580*a**9*b**2*x**(14/3)/(168*a**2*b**12*x**(16/3) + 336*a*b **13*x**5 + 168*b**14*x**(14/3)) - 9240*a**8*b**3*x**(13/3)/(168*a**2*b**1 2*x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) + 2310*a**7*b**4*x** 4/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) - 92 4*a**6*b**5*x**(11/3)/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b **14*x**(14/3)) + 462*a**5*b**6*x**(10/3)/(168*a**2*b**12*x**(16/3) + 336* a*b**13*x**5 + 168*b**14*x**(14/3)) - 264*a**4*b**7*x**3/(168*a**2*b**12*x **(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) + 165*a**3*b**8*x**(8/3 )/(168*a**2*b**12*x**(16/3) + 336*a*b**13*x**5 + 168*b**14*x**(14/3)) -...
Time = 0.03 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=\frac {165 \, a^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{12}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{9}}{3 \, b^{12}} + \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8} a}{8 \, b^{12}} - \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a^{2}}{7 \, b^{12}} + \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{3}}{2 \, b^{12}} - \frac {198 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{4}}{b^{12}} + \frac {693 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{5}}{2 \, b^{12}} - \frac {462 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{6}}{b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{7}}{b^{12}} - \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{8}}{b^{12}} + \frac {33 \, a^{10}}{{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} b^{12}} - \frac {3 \, a^{11}}{2 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} b^{12}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^5,x, algorithm="maxima")
Output:
165*a^9*log(a + b/x^(1/3))/b^12 - 1/3*(a + b/x^(1/3))^9/b^12 + 33/8*(a + b /x^(1/3))^8*a/b^12 - 165/7*(a + b/x^(1/3))^7*a^2/b^12 + 165/2*(a + b/x^(1/ 3))^6*a^3/b^12 - 198*(a + b/x^(1/3))^5*a^4/b^12 + 693/2*(a + b/x^(1/3))^4* a^5/b^12 - 462*(a + b/x^(1/3))^3*a^6/b^12 + 495*(a + b/x^(1/3))^2*a^7/b^12 - 495*(a + b/x^(1/3))*a^8/b^12 + 33*a^10/((a + b/x^(1/3))*b^12) - 3/2*a^1 1/((a + b/x^(1/3))^2*b^12)
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=\frac {165 \, a^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{12}} - \frac {55 \, a^{9} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac {27720 \, a^{10} b x^{\frac {10}{3}} + 41580 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac {8}{3}} - 2310 \, a^{7} b^{4} x^{\frac {7}{3}} + 924 \, a^{6} b^{5} x^{2} - 462 \, a^{5} b^{6} x^{\frac {5}{3}} + 264 \, a^{4} b^{7} x^{\frac {4}{3}} - 165 \, a^{3} b^{8} x + 110 \, a^{2} b^{9} x^{\frac {2}{3}} - 77 \, a b^{10} x^{\frac {1}{3}} + 56 \, b^{11}}{168 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} b^{12} x^{3}} \] Input:
integrate(1/(a+b/x^(1/3))^3/x^5,x, algorithm="giac")
Output:
165*a^9*log(abs(a*x^(1/3) + b))/b^12 - 55*a^9*log(abs(x))/b^12 - 1/168*(27 720*a^10*b*x^(10/3) + 41580*a^9*b^2*x^3 + 9240*a^8*b^3*x^(8/3) - 2310*a^7* b^4*x^(7/3) + 924*a^6*b^5*x^2 - 462*a^5*b^6*x^(5/3) + 264*a^4*b^7*x^(4/3) - 165*a^3*b^8*x + 110*a^2*b^9*x^(2/3) - 77*a*b^10*x^(1/3) + 56*b^11)/((a*x ^(1/3) + b)^2*b^12*x^3)
Time = 0.51 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=\frac {330\,a^9\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )}{b^{12}}-\frac {\frac {1}{3\,b}-\frac {11\,a\,x^{1/3}}{24\,b^2}-\frac {55\,a^3\,x}{56\,b^4}+\frac {55\,a^2\,x^{2/3}}{84\,b^3}+\frac {11\,a^6\,x^2}{2\,b^7}+\frac {11\,a^4\,x^{4/3}}{7\,b^5}-\frac {11\,a^5\,x^{5/3}}{4\,b^6}+\frac {495\,a^9\,x^3}{2\,b^{10}}-\frac {55\,a^7\,x^{7/3}}{4\,b^8}+\frac {55\,a^8\,x^{8/3}}{b^9}+\frac {165\,a^{10}\,x^{10/3}}{b^{11}}}{a^2\,x^{11/3}+b^2\,x^3+2\,a\,b\,x^{10/3}} \] Input:
int(1/(x^5*(a + b/x^(1/3))^3),x)
Output:
(330*a^9*atanh((2*a*x^(1/3))/b + 1))/b^12 - (1/(3*b) - (11*a*x^(1/3))/(24* b^2) - (55*a^3*x)/(56*b^4) + (55*a^2*x^(2/3))/(84*b^3) + (11*a^6*x^2)/(2*b ^7) + (11*a^4*x^(4/3))/(7*b^5) - (11*a^5*x^(5/3))/(4*b^6) + (495*a^9*x^3)/ (2*b^10) - (55*a^7*x^(7/3))/(4*b^8) + (55*a^8*x^(8/3))/b^9 + (165*a^10*x^( 10/3))/b^11)/(a^2*x^(11/3) + b^2*x^3 + 2*a*b*x^(10/3))
Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^5} \, dx=\frac {-27720 x^{\frac {11}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{11}+27720 x^{\frac {11}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{11}+13860 x^{\frac {11}{3}} a^{11}-9240 x^{\frac {8}{3}} a^{8} b^{3}+462 x^{\frac {5}{3}} a^{5} b^{6}-110 x^{\frac {2}{3}} a^{2} b^{9}-55440 x^{\frac {10}{3}} \mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{10} b +55440 x^{\frac {10}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{10} b +2310 x^{\frac {7}{3}} a^{7} b^{4}-264 x^{\frac {4}{3}} a^{4} b^{7}+77 x^{\frac {1}{3}} a \,b^{10}-27720 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{9} b^{2} x^{3}+27720 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{9} b^{2} x^{3}-27720 a^{9} b^{2} x^{3}-924 a^{6} b^{5} x^{2}+165 a^{3} b^{8} x -56 b^{11}}{168 b^{12} x^{3} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(1/(a+b/x^(1/3))^3/x^5,x)
Output:
( - 27720*x**(2/3)*log(x**(1/3))*a**11*x**3 + 27720*x**(2/3)*log(x**(1/3)* a + b)*a**11*x**3 + 13860*x**(2/3)*a**11*x**3 - 9240*x**(2/3)*a**8*b**3*x* *2 + 462*x**(2/3)*a**5*b**6*x - 110*x**(2/3)*a**2*b**9 - 55440*x**(1/3)*lo g(x**(1/3))*a**10*b*x**3 + 55440*x**(1/3)*log(x**(1/3)*a + b)*a**10*b*x**3 + 2310*x**(1/3)*a**7*b**4*x**2 - 264*x**(1/3)*a**4*b**7*x + 77*x**(1/3)*a *b**10 - 27720*log(x**(1/3))*a**9*b**2*x**3 + 27720*log(x**(1/3)*a + b)*a* *9*b**2*x**3 - 27720*a**9*b**2*x**3 - 924*a**6*b**5*x**2 + 165*a**3*b**8*x - 56*b**11)/(168*b**12*x**3*(x**(2/3)*a**2 + 2*x**(1/3)*a*b + b**2))