Integrand size = 22, antiderivative size = 398 \[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}} \] Output:
-1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))* 3^(1/2))*3^(1/2)/a^(1/3)/b^(1/3)-1/6*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^(1/3)/b^(1/3)-1/6*ln(2^(2 /3)-(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)+1/6*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^(1/3)/b^(1/3)-1/3*2^(1/3)*ln(1+2^(1/3)*(a^(1/3) +b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(1/3)/b^(1/3)+1/12*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^(1/3)/b^(1/3)
Time = 2.76 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=\frac {4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{-2 \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}}\right )-4 \log \left (\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-2 \log \left (-\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}\right )+\log \left (2^{2/3} a^{2/3}+2^{2/3} b^{2/3} x^2+2 \sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+4 \left (a+b x^3\right )^{2/3}+2 \sqrt [3]{2} \sqrt [3]{a} \left (\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )\right )+2 \log \left (2^{2/3} a^{2/3}+2^{2/3} b^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+\sqrt [3]{a} \left (2\ 2^{2/3} \sqrt [3]{b} x-\sqrt [3]{2} \sqrt [3]{a+b x^3}\right )\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}} \] Input:
Integrate[(a + b*x^3)^(1/3)/(a - b*x^3),x]
Output:
(4*Sqrt[3]*ArcTan[(Sqrt[3]*(a + b*x^3)^(1/3))/(-2*2^(1/3)*a^(1/3) - 2*2^(1 /3)*b^(1/3)*x + (a + b*x^3)^(1/3))] + 2*Sqrt[3]*ArcTan[(Sqrt[3]*(a + b*x^3 )^(1/3))/(2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))] - 4*Lo g[2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3)] - 2*Log[-(2^(1/ 3)*a^(1/3)) - 2^(1/3)*b^(1/3)*x + 2*(a + b*x^3)^(1/3)] + Log[2^(2/3)*a^(2/ 3) + 2^(2/3)*b^(2/3)*x^2 + 2*2^(1/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 4*(a + b*x^3)^(2/3) + 2*2^(1/3)*a^(1/3)*(2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))] + 2*Log[2^(2/3)*a^(2/3) + 2^(2/3)*b^(2/3)*x^2 - 2^(1/3)*b^(1/3)*x*(a + b*x ^3)^(1/3) + (a + b*x^3)^(2/3) + a^(1/3)*(2*2^(2/3)*b^(1/3)*x - 2^(1/3)*(a + b*x^3)^(1/3))])/(6*2^(2/3)*a^(1/3)*b^(1/3))
Time = 0.78 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right ) \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {1}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {2}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {-\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {9 \sqrt [3]{a} \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{\sqrt [3]{b}}\) |
Input:
Int[(a + b*x^3)^(1/3)/(a - b*x^3),x]
Output:
(9*a^(1/3)*((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))/b^(1/3)
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[(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[((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_) + (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]
\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{3}+a}d x\]
Input:
int((b*x^3+a)^(1/3)/(-b*x^3+a),x)
Output:
int((b*x^3+a)^(1/3)/(-b*x^3+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (284) = 568\).
Time = 16.03 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x^3+a)^(1/3)/(-b*x^3+a),x, algorithm="fricas")
Output:
-1/18*sqrt(3)*2^(1/3)*(-1/(a*b))^(1/3)*arctan(1/3*(6*sqrt(3)*2^(2/3)*(a*b^ 6*x^16 + 33*a^2*b^5*x^13 + 110*a^3*b^4*x^10 + 110*a^4*b^3*x^7 + 33*a^5*b^2 *x^4 + a^6*b*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(2/3) + 24*sqrt(3)*2^(1/3)*(a *b^5*x^14 + 2*a^2*b^4*x^11 - 6*a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*(b *x^3 + a)^(2/3)*(-1/(a*b))^(1/3) - sqrt(3)*(b^6*x^18 - 42*a*b^5*x^15 - 417 *a^2*b^4*x^12 - 812*a^3*b^3*x^9 - 417*a^4*b^2*x^6 - 42*a^5*b*x^3 + a^6))/( b^6*x^18 + 102*a*b^5*x^15 + 447*a^2*b^4*x^12 + 628*a^3*b^3*x^9 + 447*a^4*b ^2*x^6 + 102*a^5*b*x^3 + a^6)) - 1/36*2^(1/3)*(-1/(a*b))^(1/3)*log((12*2^( 2/3)*(a*b^3*x^8 + 4*a^2*b^2*x^5 + a^3*b*x^2)*(b*x^3 + a)^(2/3)*(-1/(a*b))^ (2/3) - 2^(1/3)*(b^4*x^12 + 32*a*b^3*x^9 + 78*a^2*b^2*x^6 + 32*a^3*b*x^3 + a^4)*(-1/(a*b))^(1/3) + 6*(b^3*x^10 + 11*a*b^2*x^7 + 11*a^2*b*x^4 + a^3*x )*(b*x^3 + a)^(1/3))/(b^4*x^12 - 4*a*b^3*x^9 + 6*a^2*b^2*x^6 - 4*a^3*b*x^3 + a^4)) + 1/18*2^(1/3)*(-1/(a*b))^(1/3)*log(-(12*(b*x^3 + a)^(2/3)*x^2 + 2^(2/3)*(b^2*x^6 - 2*a*b*x^3 + a^2)*(-1/(a*b))^(2/3) + 6*2^(1/3)*(b*x^4 + a*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(1/3))/(b^2*x^6 - 2*a*b*x^3 + a^2))
\[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=- \int \frac {\sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx \] Input:
integrate((b*x**3+a)**(1/3)/(-b*x**3+a),x)
Output:
-Integral((a + b*x**3)**(1/3)/(-a + b*x**3), x)
\[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b x^{3} - a} \,d x } \] Input:
integrate((b*x^3+a)^(1/3)/(-b*x^3+a),x, algorithm="maxima")
Output:
-integrate((b*x^3 + a)^(1/3)/(b*x^3 - a), x)
\[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b x^{3} - a} \,d x } \] Input:
integrate((b*x^3+a)^(1/3)/(-b*x^3+a),x, algorithm="giac")
Output:
integrate(-(b*x^3 + a)^(1/3)/(b*x^3 - a), x)
Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{a-b\,x^3} \,d x \] Input:
int((a + b*x^3)^(1/3)/(a - b*x^3),x)
Output:
int((a + b*x^3)^(1/3)/(a - b*x^3), x)
\[ \int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx=\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{3}+a}d x \] Input:
int((b*x^3+a)^(1/3)/(-b*x^3+a),x)
Output:
int((a + b*x**3)**(1/3)/(a - b*x**3),x)