Integrand size = 15, antiderivative size = 171 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {3 b^{11}}{2 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {33 b^{10}}{a^{12} \left (b+a \sqrt [3]{x}\right )}+\frac {135 b^8 \sqrt [3]{x}}{a^{11}}-\frac {54 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{a^9}-\frac {63 b^5 x^{4/3}}{4 a^8}+\frac {9 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{a^6}+\frac {18 b^2 x^{7/3}}{7 a^5}-\frac {9 b x^{8/3}}{8 a^4}+\frac {x^3}{3 a^3}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}} \] Output:
3/2*b^11/a^12/(b+a*x^(1/3))^2-33*b^10/a^12/(b+a*x^(1/3))+135*b^8*x^(1/3)/a ^11-54*b^7*x^(2/3)/a^10+28*b^6*x/a^9-63/4*b^5*x^(4/3)/a^8+9*b^4*x^(5/3)/a^ 7-5*b^3*x^2/a^6+18/7*b^2*x^(7/3)/a^5-9/8*b*x^(8/3)/a^4+1/3*x^3/a^3-165*b^9 *ln(b+a*x^(1/3))/a^12
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {-5292 b^{11}+17136 a b^{10} \sqrt [3]{x}+36288 a^2 b^9 x^{2/3}+9240 a^3 b^8 x-2310 a^4 b^7 x^{4/3}+924 a^5 b^6 x^{5/3}-462 a^6 b^5 x^2+264 a^7 b^4 x^{7/3}-165 a^8 b^3 x^{8/3}+110 a^9 b^2 x^3-77 a^{10} b x^{10/3}+56 a^{11} x^{11/3}}{168 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac {165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}} \] Input:
Integrate[x^2/(a + b/x^(1/3))^3,x]
Output:
(-5292*b^11 + 17136*a*b^10*x^(1/3) + 36288*a^2*b^9*x^(2/3) + 9240*a^3*b^8* x - 2310*a^4*b^7*x^(4/3) + 924*a^5*b^6*x^(5/3) - 462*a^6*b^5*x^2 + 264*a^7 *b^4*x^(7/3) - 165*a^8*b^3*x^(8/3) + 110*a^9*b^2*x^3 - 77*a^10*b*x^(10/3) + 56*a^11*x^(11/3))/(168*a^12*(b + a*x^(1/3))^2) - (165*b^9*Log[b + a*x^(1 /3)])/a^12
Time = 0.56 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {795, 798, 49, 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.
| \(\displaystyle \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \int \frac {x^3}{\left (a \sqrt [3]{x}+b\right )^3}dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 3 \int \frac {x^{11/3}}{\left (\sqrt [3]{x} a+b\right )^3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 3 \int \left (-\frac {b^{11}}{a^{11} \left (\sqrt [3]{x} a+b\right )^3}+\frac {11 b^{10}}{a^{11} \left (\sqrt [3]{x} a+b\right )^2}-\frac {55 b^9}{a^{11} \left (\sqrt [3]{x} a+b\right )}+\frac {45 b^8}{a^{11}}-\frac {36 \sqrt [3]{x} b^7}{a^{10}}+\frac {28 x^{2/3} b^6}{a^9}-\frac {21 x b^5}{a^8}+\frac {15 x^{4/3} b^4}{a^7}-\frac {10 x^{5/3} b^3}{a^6}+\frac {6 x^2 b^2}{a^5}-\frac {3 x^{7/3} b}{a^4}+\frac {x^{8/3}}{a^3}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac {11 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}-\frac {55 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}+\frac {45 b^8 \sqrt [3]{x}}{a^{11}}-\frac {18 b^7 x^{2/3}}{a^{10}}+\frac {28 b^6 x}{3 a^9}-\frac {21 b^5 x^{4/3}}{4 a^8}+\frac {3 b^4 x^{5/3}}{a^7}-\frac {5 b^3 x^2}{3 a^6}+\frac {6 b^2 x^{7/3}}{7 a^5}-\frac {3 b x^{8/3}}{8 a^4}+\frac {x^3}{9 a^3}\right )\) |
Input:
Int[x^2/(a + b/x^(1/3))^3,x]
Output:
3*(b^11/(2*a^12*(b + a*x^(1/3))^2) - (11*b^10)/(a^12*(b + a*x^(1/3))) + (4 5*b^8*x^(1/3))/a^11 - (18*b^7*x^(2/3))/a^10 + (28*b^6*x)/(3*a^9) - (21*b^5 *x^(4/3))/(4*a^8) + (3*b^4*x^(5/3))/a^7 - (5*b^3*x^2)/(3*a^6) + (6*b^2*x^( 7/3))/(7*a^5) - (3*b*x^(8/3))/(8*a^4) + x^3/(9*a^3) - (55*b^9*Log[b + a*x^ (1/3)])/a^12)
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] && IGtQ[m, 0] && IGtQ[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 = 0.45 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {9 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {18 b^{2} x^{\frac {7}{3}} a^{6}}{7}-5 a^{5} b^{3} x^{2}+9 x^{\frac {5}{3}} a^{4} b^{4}-\frac {63 b^{5} x^{\frac {4}{3}} a^{3}}{4}+28 a^{2} b^{6} x -54 a \,b^{7} x^{\frac {2}{3}}+135 b^{8} x^{\frac {1}{3}}}{a^{11}}+\frac {3 b^{11}}{2 a^{12} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {33 b^{10}}{a^{12} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {165 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{12}}\) | \(145\) |
| default | \(\frac {\frac {a^{8} x^{3}}{3}-\frac {9 b \,x^{\frac {8}{3}} a^{7}}{8}+\frac {18 b^{2} x^{\frac {7}{3}} a^{6}}{7}-5 a^{5} b^{3} x^{2}+9 x^{\frac {5}{3}} a^{4} b^{4}-\frac {63 b^{5} x^{\frac {4}{3}} a^{3}}{4}+28 a^{2} b^{6} x -54 a \,b^{7} x^{\frac {2}{3}}+135 b^{8} x^{\frac {1}{3}}}{a^{11}}+\frac {3 b^{11}}{2 a^{12} \left (b +a \,x^{\frac {1}{3}}\right )^{2}}-\frac {33 b^{10}}{a^{12} \left (b +a \,x^{\frac {1}{3}}\right )}-\frac {165 b^{9} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{12}}\) | \(145\) |
Input:
int(x^2/(a+b/x^(1/3))^3,x,method=_RETURNVERBOSE)
Output:
3/a^11*(1/9*a^8*x^3-3/8*b*x^(8/3)*a^7+6/7*b^2*x^(7/3)*a^6-5/3*a^5*b^3*x^2+ 3*x^(5/3)*a^4*b^4-21/4*b^5*x^(4/3)*a^3+28/3*a^2*b^6*x-18*a*b^7*x^(2/3)+45* b^8*x^(1/3))+3/2*b^11/a^12/(b+a*x^(1/3))^2-33*b^10/a^12/(b+a*x^(1/3))-165* b^9*ln(b+a*x^(1/3))/a^12
Time = 0.08 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.32 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {56 \, a^{15} x^{5} - 728 \, a^{12} b^{3} x^{4} + 3080 \, a^{9} b^{6} x^{3} + 8568 \, a^{6} b^{9} x^{2} - 1344 \, a^{3} b^{12} x - 5292 \, b^{15} - 27720 \, {\left (a^{6} b^{9} x^{2} + 2 \, a^{3} b^{12} x + b^{15}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 63 \, {\left (3 \, a^{14} b x^{4} - 18 \, a^{11} b^{4} x^{3} + 99 \, a^{8} b^{7} x^{2} + 352 \, a^{5} b^{10} x + 220 \, a^{2} b^{13}\right )} x^{\frac {2}{3}} + 18 \, {\left (24 \, a^{13} b^{2} x^{4} - 99 \, a^{10} b^{5} x^{3} + 990 \, a^{7} b^{8} x^{2} + 2695 \, a^{4} b^{11} x + 1540 \, a b^{14}\right )} x^{\frac {1}{3}}}{168 \, {\left (a^{18} x^{2} + 2 \, a^{15} b^{3} x + a^{12} b^{6}\right )}} \] Input:
integrate(x^2/(a+b/x^(1/3))^3,x, algorithm="fricas")
Output:
1/168*(56*a^15*x^5 - 728*a^12*b^3*x^4 + 3080*a^9*b^6*x^3 + 8568*a^6*b^9*x^ 2 - 1344*a^3*b^12*x - 5292*b^15 - 27720*(a^6*b^9*x^2 + 2*a^3*b^12*x + b^15 )*log(a*x^(1/3) + b) - 63*(3*a^14*b*x^4 - 18*a^11*b^4*x^3 + 99*a^8*b^7*x^2 + 352*a^5*b^10*x + 220*a^2*b^13)*x^(2/3) + 18*(24*a^13*b^2*x^4 - 99*a^10* b^5*x^3 + 990*a^7*b^8*x^2 + 2695*a^4*b^11*x + 1540*a*b^14)*x^(1/3))/(a^18* x^2 + 2*a^15*b^3*x + a^12*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (170) = 340\).
Time = 1.21 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.65 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\begin {cases} \frac {56 a^{11} x^{\frac {11}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {77 a^{10} b x^{\frac {10}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {110 a^{9} b^{2} x^{3}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {165 a^{8} b^{3} x^{\frac {8}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {264 a^{7} b^{4} x^{\frac {7}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {462 a^{6} b^{5} x^{2}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {924 a^{5} b^{6} x^{\frac {5}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {2310 a^{4} b^{7} x^{\frac {4}{3}}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac {9240 a^{3} b^{8} x}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 a^{2} b^{9} x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {55440 a b^{10} \sqrt [3]{x}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {27720 b^{11} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac {41580 b^{11}}{168 a^{14} x^{\frac {2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4 b^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2/(a+b/x**(1/3))**3,x)
Output:
Piecewise((56*a**11*x**(11/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 77*a**10*b*x**(10/3)/(168*a**14*x**(2/3) + 336*a**13*b* x**(1/3) + 168*a**12*b**2) + 110*a**9*b**2*x**3/(168*a**14*x**(2/3) + 336* a**13*b*x**(1/3) + 168*a**12*b**2) - 165*a**8*b**3*x**(8/3)/(168*a**14*x** (2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 264*a**7*b**4*x**(7/3)/(1 68*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 462*a**6*b**5 *x**2/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 924*a **5*b**6*x**(5/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b **2) - 2310*a**4*b**7*x**(4/3)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) + 9240*a**3*b**8*x/(168*a**14*x**(2/3) + 336*a**13*b*x** (1/3) + 168*a**12*b**2) - 27720*a**2*b**9*x**(2/3)*log(x**(1/3) + b/a)/(16 8*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 55440*a*b**10* x**(1/3)*log(x**(1/3) + b/a)/(168*a**14*x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 55440*a*b**10*x**(1/3)/(168*a**14*x**(2/3) + 336*a**13*b *x**(1/3) + 168*a**12*b**2) - 27720*b**11*log(x**(1/3) + b/a)/(168*a**14*x **(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2) - 41580*b**11/(168*a**14* x**(2/3) + 336*a**13*b*x**(1/3) + 168*a**12*b**2), Ne(a, 0)), (x**4/(4*b** 3), True))
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {56 \, a^{10} - \frac {77 \, a^{9} b}{x^{\frac {1}{3}}} + \frac {110 \, a^{8} b^{2}}{x^{\frac {2}{3}}} - \frac {165 \, a^{7} b^{3}}{x} + \frac {264 \, a^{6} b^{4}}{x^{\frac {4}{3}}} - \frac {462 \, a^{5} b^{5}}{x^{\frac {5}{3}}} + \frac {924 \, a^{4} b^{6}}{x^{2}} - \frac {2310 \, a^{3} b^{7}}{x^{\frac {7}{3}}} + \frac {9240 \, a^{2} b^{8}}{x^{\frac {8}{3}}} + \frac {41580 \, a b^{9}}{x^{3}} + \frac {27720 \, b^{10}}{x^{\frac {10}{3}}}}{168 \, {\left (\frac {a^{13}}{x^{3}} + \frac {2 \, a^{12} b}{x^{\frac {10}{3}}} + \frac {a^{11} b^{2}}{x^{\frac {11}{3}}}\right )}} - \frac {165 \, b^{9} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{12}} - \frac {55 \, b^{9} \log \left (x\right )}{a^{12}} \] Input:
integrate(x^2/(a+b/x^(1/3))^3,x, algorithm="maxima")
Output:
1/168*(56*a^10 - 77*a^9*b/x^(1/3) + 110*a^8*b^2/x^(2/3) - 165*a^7*b^3/x + 264*a^6*b^4/x^(4/3) - 462*a^5*b^5/x^(5/3) + 924*a^4*b^6/x^2 - 2310*a^3*b^7 /x^(7/3) + 9240*a^2*b^8/x^(8/3) + 41580*a*b^9/x^3 + 27720*b^10/x^(10/3))/( a^13/x^3 + 2*a^12*b/x^(10/3) + a^11*b^2/x^(11/3)) - 165*b^9*log(a + b/x^(1 /3))/a^12 - 55*b^9*log(x)/a^12
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=-\frac {165 \, b^{9} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{12}} - \frac {3 \, {\left (22 \, a b^{10} x^{\frac {1}{3}} + 21 \, b^{11}\right )}}{2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{2} a^{12}} + \frac {56 \, a^{24} x^{3} - 189 \, a^{23} b x^{\frac {8}{3}} + 432 \, a^{22} b^{2} x^{\frac {7}{3}} - 840 \, a^{21} b^{3} x^{2} + 1512 \, a^{20} b^{4} x^{\frac {5}{3}} - 2646 \, a^{19} b^{5} x^{\frac {4}{3}} + 4704 \, a^{18} b^{6} x - 9072 \, a^{17} b^{7} x^{\frac {2}{3}} + 22680 \, a^{16} b^{8} x^{\frac {1}{3}}}{168 \, a^{27}} \] Input:
integrate(x^2/(a+b/x^(1/3))^3,x, algorithm="giac")
Output:
-165*b^9*log(abs(a*x^(1/3) + b))/a^12 - 3/2*(22*a*b^10*x^(1/3) + 21*b^11)/ ((a*x^(1/3) + b)^2*a^12) + 1/168*(56*a^24*x^3 - 189*a^23*b*x^(8/3) + 432*a ^22*b^2*x^(7/3) - 840*a^21*b^3*x^2 + 1512*a^20*b^4*x^(5/3) - 2646*a^19*b^5 *x^(4/3) + 4704*a^18*b^6*x - 9072*a^17*b^7*x^(2/3) + 22680*a^16*b^8*x^(1/3 ))/a^27
Time = 0.37 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {x^3}{3\,a^3}-\frac {\frac {63\,b^{11}}{2\,a}+33\,b^{10}\,x^{1/3}}{a^{11}\,b^2+a^{13}\,x^{2/3}+2\,a^{12}\,b\,x^{1/3}}-\frac {9\,b\,x^{8/3}}{8\,a^4}+\frac {28\,b^6\,x}{a^9}-\frac {165\,b^9\,\ln \left (b+a\,x^{1/3}\right )}{a^{12}}-\frac {5\,b^3\,x^2}{a^6}+\frac {18\,b^2\,x^{7/3}}{7\,a^5}+\frac {9\,b^4\,x^{5/3}}{a^7}-\frac {63\,b^5\,x^{4/3}}{4\,a^8}-\frac {54\,b^7\,x^{2/3}}{a^{10}}+\frac {135\,b^8\,x^{1/3}}{a^{11}} \] Input:
int(x^2/(a + b/x^(1/3))^3,x)
Output:
x^3/(3*a^3) - ((63*b^11)/(2*a) + 33*b^10*x^(1/3))/(a^11*b^2 + a^13*x^(2/3) + 2*a^12*b*x^(1/3)) - (9*b*x^(8/3))/(8*a^4) + (28*b^6*x)/a^9 - (165*b^9*l og(b + a*x^(1/3)))/a^12 - (5*b^3*x^2)/a^6 + (18*b^2*x^(7/3))/(7*a^5) + (9* b^4*x^(5/3))/a^7 - (63*b^5*x^(4/3))/(4*a^8) - (54*b^7*x^(2/3))/a^10 + (135 *b^8*x^(1/3))/a^11
Time = 0.21 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^3} \, dx=\frac {-27720 x^{\frac {2}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a^{2} b^{9}+56 x^{\frac {11}{3}} a^{11}-165 x^{\frac {8}{3}} a^{8} b^{3}+924 x^{\frac {5}{3}} a^{5} b^{6}+27720 x^{\frac {2}{3}} a^{2} b^{9}-55440 x^{\frac {1}{3}} \mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) a \,b^{10}-77 x^{\frac {10}{3}} a^{10} b +264 x^{\frac {7}{3}} a^{7} b^{4}-2310 x^{\frac {4}{3}} a^{4} b^{7}-27720 \,\mathrm {log}\left (x^{\frac {1}{3}} a +b \right ) b^{11}+110 a^{9} b^{2} x^{3}-462 a^{6} b^{5} x^{2}+9240 a^{3} b^{8} x -13860 b^{11}}{168 a^{12} \left (x^{\frac {2}{3}} a^{2}+2 x^{\frac {1}{3}} a b +b^{2}\right )} \] Input:
int(x^2/(a+b/x^(1/3))^3,x)
Output:
( - 27720*x**(2/3)*log(x**(1/3)*a + b)*a**2*b**9 + 56*x**(2/3)*a**11*x**3 - 165*x**(2/3)*a**8*b**3*x**2 + 924*x**(2/3)*a**5*b**6*x + 27720*x**(2/3)* a**2*b**9 - 55440*x**(1/3)*log(x**(1/3)*a + b)*a*b**10 - 77*x**(1/3)*a**10 *b*x**3 + 264*x**(1/3)*a**7*b**4*x**2 - 2310*x**(1/3)*a**4*b**7*x - 27720* log(x**(1/3)*a + b)*b**11 + 110*a**9*b**2*x**3 - 462*a**6*b**5*x**2 + 9240 *a**3*b**8*x - 13860*b**11)/(168*a**12*(x**(2/3)*a**2 + 2*x**(1/3)*a*b + b **2))