Integrand size = 22, antiderivative size = 483 \[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=-\frac {7}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a+b x^3\right )^{4/3}-\frac {4 \sqrt [3]{2} a^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} a^{5/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {7 a^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}-\frac {2 \sqrt [3]{2} a^{5/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {2 \sqrt [3]{2} a^{5/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}-\frac {4 \sqrt [3]{2} a^{5/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}}+\frac {\sqrt [3]{2} a^{5/3} \log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{b}} \]
-7/5*a*x*(b*x^3+a)^(1/3)-1/5*x*(b*x^3+a)^(4/3)-7/5*a^2*x*(1+b*x^3/a)^(2/3) *hypergeom([1/3, 2/3],[4/3],-b*x^3/a)/(b*x^3+a)^(2/3)-2/3*2^(1/3)*a^(5/3)* ln(2^(2/3)+(-a^(1/3)-b^(1/3)*x)/(b*x^3+a)^(1/3))/b^(1/3)+2/3*2^(1/3)*a^(5/ 3)*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))/b^(1/3)-4/3*2^(1/3)*a^(5/3)*ln(1+2^(1/3)*(a^(1/3) +b^(1/3)*x)/(b*x^3+a)^(1/3))/b^(1/3)+1/3*2^(1/3)*a^(5/3)*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))/b^(1/3)-4/3*2^(1/3)*a^(5/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)* x)/(b*x^3+a)^(1/3))*3^(1/2))/b^(1/3)*3^(1/2)-2/3*2^(1/3)*a^(5/3)*arctan(1/ 3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/b^(1/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 10.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=\frac {27 a b x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+4 x \left (-8 a^2-9 a b x^3-b^2 x^6+\frac {52 a^4 \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (4 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+b x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )\right )}\right )}{20 \left (a+b x^3\right )^{2/3}} \]
(27*a*b*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a] + 4*x*(-8*a^2 - 9*a*b*x^3 - b^2*x^6 + (52*a^4*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*AppellF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3) /a])))))/(20*(a + b*x^3)^(2/3))
Time = 0.77 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.16, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {933, 27, 1025, 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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle -\frac {\int -\frac {2 a b \sqrt [3]{b x^3+a} \left (7 b x^3+3 a\right )}{a-b x^3}dx}{5 b}-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} a \int \frac {\sqrt [3]{b x^3+a} \left (7 b x^3+3 a\right )}{a-b x^3}dx-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {2}{5} a \left (-\frac {\int -\frac {a b \left (27 b x^3+13 a\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{2 b}-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5} a \left (\frac {\int \frac {a b \left (27 b x^3+13 a\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{2 b}-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \int \frac {27 b x^3+13 a}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-27 \int \frac {1}{\left (b x^3+a\right )^{2/3}}dx\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {27 \left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 928 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {\int \frac {1}{\left (b x^3+a\right )^{2/3}}dx}{2 a}+\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 779 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {9 \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right ) \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {9 \left (\frac {1}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {2}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 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 )-\frac {27 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}}\right )-\frac {7}{2} x \sqrt [3]{b x^3+a}\right )-\frac {1}{5} x \left (b x^3+a\right )^{4/3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 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 )-\frac {27 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}}\right )-\frac {7}{2} x \sqrt [3]{b x^3+a}\right )-\frac {1}{5} x \left (b x^3+a\right )^{4/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 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 {27 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}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 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 {27 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}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 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 {27 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}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{5} a \left (\frac {1}{2} a \left (40 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {27 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )-\frac {7}{2} x \sqrt [3]{a+b x^3}\right )-\frac {1}{5} x \left (a+b x^3\right )^{4/3}\) |
-1/5*(x*(a + b*x^3)^(4/3)) + (2*a*((-7*x*(a + b*x^3)^(1/3))/2 + (a*((-27*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) + 40*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*Log[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))/Sqrt[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)))))/2))/5
3.1.34.3.1 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[((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[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 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_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1)) Int[(a + b*x ^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 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]
\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {7}{3}}}{-b \,x^{3}+a}d x\]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=- \int \frac {a^{2} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx - \int \frac {b^{2} x^{6} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx - \int \frac {2 a b x^{3} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx \]
-Integral(a**2*(a + b*x**3)**(1/3)/(-a + b*x**3), x) - Integral(b**2*x**6* (a + b*x**3)**(1/3)/(-a + b*x**3), x) - Integral(2*a*b*x**3*(a + b*x**3)** (1/3)/(-a + b*x**3), x)
\[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {7}{3}}}{b x^{3} - a} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {7}{3}}}{b x^{3} - a} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{7/3}}{a-b x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{7/3}}{a-b\,x^3} \,d x \]