Integrand size = 22, antiderivative size = 473 \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3} a^{7/3} \sqrt [3]{b}}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{12\ 2^{2/3} a^{7/3} \sqrt [3]{b}}-\frac {\log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{7/3} \sqrt [3]{b}} \] Output:
1/4*x/a^2/(b*x^3+a)^(2/3)-1/12*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x) /(b*x^3+a)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)/a^(7/3)/b^(1/3)-1/24*arctan(1/3 *(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)/ a^(7/3)/b^(1/3)+1/2*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3, 2/3],[4/3],-b*x^3/ a)/a^2/(b*x^3+a)^(2/3)-1/24*ln(2^(2/3)-(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3) )*2^(1/3)/a^(7/3)/b^(1/3)+1/24*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a )^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/b^(1/ 3)-1/12*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/ b^(1/3)+1/48*ln(2*2^(1/3)+(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)+2^(2/3)*(a ^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/b^(1/3)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 10.10 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\frac {x \left (\frac {4}{a^2}-\frac {b x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{a^3}+\frac {48 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (4 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+b x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )\right )}\right )}{16 \left (a+b x^3\right )^{2/3}} \] Input:
Integrate[1/((a - b*x^3)*(a + b*x^3)^(5/3)),x]
Output:
(x*(4/a^2 - (b*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x ^3)/a), (b*x^3)/a])/a^3 + (48*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b* x^3)/a])/((a - b*x^3)*(4*a*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3 )/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*Ap pellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))))/(16*(a + b*x^3)^(2/ 3))
Time = 1.17 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.14, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {931, 25, 27, 1026, 779, 778, 928, 779, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}-\frac {\int -\frac {b \left (3 a-b x^3\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{4 a^2 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b \left (3 a-b x^3\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{4 a^2 b}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a-b x^3}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {\int \frac {1}{\left (b x^3+a\right )^{2/3}}dx+2 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {2 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {2 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 928 |
| \(\displaystyle \frac {2 a \left (\frac {\int \frac {1}{\left (b x^3+a\right )^{2/3}}dx}{2 a}+\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {2 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {2 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle \frac {2 a \left (\frac {9 \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right ) \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {1}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {2}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{4 a^2 \left (b x^3+a\right )^{2/3}}+\frac {\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (b x^3+a\right )^{2/3}}+2 a \left (\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (b x^3+a\right )^{2/3}}+\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}\right )}{4 a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{4 a^2 \left (b x^3+a\right )^{2/3}}+\frac {\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (b x^3+a\right )^{2/3}}+2 a \left (\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (b x^3+a\right )^{2/3}}+\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}\right )}{4 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {-\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}+\frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{4 a^2 \left (a+b x^3\right )^{2/3}}+\frac {2 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}}{4 a^2}\) |
Input:
Int[1/((a - b*x^3)*(a + b*x^3)^(5/3)),x]
Output:
x/(4*a^2*(a + b*x^3)^(2/3)) + ((x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[ 1/3, 2/3, 4/3, -((b*x^3)/a)])/(a + b*x^3)^(2/3) + 2*a*((x*(1 + (b*x^3)/a)^ (2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(2*a*(a + b*x^3)^(2/ 3)) + (9*((2*((-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*a^(1/3))) + Log[1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b *x^3)^(1/3)]/(2*2^(1/3)*a^(1/3)))/(3*2^(1/3)*a^(1/3)) - Log[1 + (2^(1/3)*( a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(3*2^(2/3)*a^(2/3))))/9 + (-1/3*L og[2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3)]/(2^(2/3)*a^(2/3)) - ((Sqrt[3]*ArcTan[(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/S qrt[3]])/a^(1/3) - Log[2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/ 3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(2*a^(1/3)))/(3*2^ (2/3)*a^(1/3)))/9))/(2*a^(2/3)*b^(1/3))))/(4*a^2)
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x ^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p, x], x ] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Si mplify[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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
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[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, 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]
\[\int \frac {1}{\left (-b \,x^{3}+a \right ) \left (b \,x^{3}+a \right )^{\frac {5}{3}}}d x\]
Input:
int(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x)
Output:
int(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x)
Timed out. \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=- \int \frac {1}{- a^{2} \left (a + b x^{3}\right )^{\frac {2}{3}} + b^{2} x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(-b*x**3+a)/(b*x**3+a)**(5/3),x)
Output:
-Integral(1/(-a**2*(a + b*x**3)**(2/3) + b**2*x**6*(a + b*x**3)**(2/3)), x )
\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int { -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (b x^{3} - a\right )}} \,d x } \] Input:
integrate(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x, algorithm="maxima")
Output:
-integrate(1/((b*x^3 + a)^(5/3)*(b*x^3 - a)), x)
\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int { -\frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (b x^{3} - a\right )}} \,d x } \] Input:
integrate(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x, algorithm="giac")
Output:
integrate(-1/((b*x^3 + a)^(5/3)*(b*x^3 - a)), x)
Timed out. \[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{5/3}\,\left (a-b\,x^3\right )} \,d x \] Input:
int(1/((a + b*x^3)^(5/3)*(a - b*x^3)),x)
Output:
int(1/((a + b*x^3)^(5/3)*(a - b*x^3)), x)
\[ \int \frac {1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2}-\left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} x^{6}}d x \] Input:
int(1/(-b*x^3+a)/(b*x^3+a)^(5/3),x)
Output:
int(1/((a + b*x**3)**(2/3)*a**2 - (a + b*x**3)**(2/3)*b**2*x**6),x)