Integrand size = 22, antiderivative size = 294 \[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}} \]
1/2*x*hypergeom([1/3, 2/3],[4/3],x^3)-1/12*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3 ))*2^(1/3)+1/12*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1) ^(1/3))*2^(1/3)-1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/24*ln(2*2 ^(1/3)+(1-x)^2/(-x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*ar ctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)-1/12* arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 10.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{4} x^4 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},x^3,-x^3\right ) \]
Time = 0.45 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {983, 778, 928, 778, 927, 982, 821, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 983 |
| \(\displaystyle \int \frac {1}{\left (1-x^3\right )^{2/3}}dx-\int \frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )-\int \frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx\) |
| \(\Big \downarrow \) 928 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{\left (1-x^3\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{1-x^3}}{x^3+1}dx+x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )-\frac {1}{2} \int \frac {\sqrt [3]{1-x^3}}{x^3+1}dx\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle \frac {9}{2} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right ) \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle \frac {9}{2} \left (\frac {1}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {2}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {\int -\frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\int \frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {-3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )+\frac {1}{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )\) |
(x*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/2 + (9*((2*((-((Sqrt[3]*ArcTan[( 1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/2^(1/3)) + Log[1 + (2^( 2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(2*2^ (1/3)))/(3*2^(1/3)) - Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(2/3 ))))/9 + (-1/3*Log[2^(2/3) - (1 - x)/(1 - x^3)^(1/3)]/2^(2/3) - (Sqrt[3]*A rcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]] - Log[2*2^(1/3) + ( 1 - x)^2/(1 - x^3)^(2/3) + (2^(2/3)*(1 - x))/(1 - x^3)^(1/3)]/2)/(3*2^(2/3 )))/9))/2
3.7.35.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[9*(a/(c*q)) Subst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[ b*c - a*d, 0] && EqQ[b*c + a*d, 0]
Int[1/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Sim p[b/(b*c - a*d) Int[1/(a + b*x^3)^(2/3), x], x] - Simp[d/(b*c - a*d) In t[(a + b*x^3)^(1/3)/(c + d*x^3), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b* c - a*d, 0] && EqQ[b*c + a*d, 0]
Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[(e*x)^m/(a + b*x^n), x], x] - Simp[d /(b*c - a*d) Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^( n_)), x_Symbol] :> Simp[e^n/b Int[(e*x)^(m - n)*(c + d*x^n)^q, x], x] - S imp[a*(e^n/b) Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /; Fr eeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
\[\int \frac {x^{3}}{\left (-x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+1\right )}d x\]
\[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{3}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^3}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \]