Integrand size = 19, antiderivative size = 299 \[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}+\frac {(b c-a d) \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} b^{7/6} d^{5/6}}-\frac {(b c-a d) \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} b^{7/6} d^{5/6}}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{7/6} d^{5/6}}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{6 b^{7/6} d^{5/6}} \] Output:
(b*x+a)^(5/6)*(d*x+c)^(1/6)/b+1/6*(-a*d+b*c)*arctan(1/3*(b^(1/6)-2*d^(1/6) *(b*x+a)^(1/6)/(d*x+c)^(1/6))*3^(1/2)/b^(1/6))*3^(1/2)/b^(7/6)/d^(5/6)-1/6 *(-a*d+b*c)*arctan(1/3*(b^(1/6)+2*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))*3^( 1/2)/b^(1/6))*3^(1/2)/b^(7/6)/d^(5/6)+1/3*(-a*d+b*c)*arctanh(d^(1/6)*(b*x+ a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(7/6)/d^(5/6)+1/6*(-a*d+b*c)*arctanh(b^( 1/6)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6)/(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/(d *x+c)^(1/3)))/b^(7/6)/d^(5/6)
Time = 0.78 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\frac {6 \sqrt [6]{b} d^{5/6} (a+b x)^{5/6} \sqrt [6]{c+d x}+\sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}-\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+\sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+2 (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+(b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 b^{7/6} d^{5/6}} \] Input:
Integrate[(c + d*x)^(1/6)/(a + b*x)^(1/6),x]
Output:
(6*b^(1/6)*d^(5/6)*(a + b*x)^(5/6)*(c + d*x)^(1/6) + Sqrt[3]*(b*c - a*d)*A rcTan[(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))/(2*d^(1/6)*(a + b*x)^(1/6) - b^(1/ 6)*(c + d*x)^(1/6))] + Sqrt[3]*(b*c - a*d)*ArcTan[(Sqrt[3]*b^(1/6)*(c + d* x)^(1/6))/(2*d^(1/6)*(a + b*x)^(1/6) + b^(1/6)*(c + d*x)^(1/6))] + 2*(b*c - a*d)*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + (b*c - a*d)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6)) + (b^( 1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))])/(6*b^(7/6)*d^(5/6))
Time = 0.46 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.34, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {60, 73, 854, 27, 825, 27, 221, 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.
\(\displaystyle \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}}dx}{6 b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{5/6}}d\sqrt [6]{a+b x}}{b^2}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {(b c-a d) \int \frac {b (a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b^2}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 825 |
\(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(b c-a d) \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {(b c-a d) \left (-\frac {\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\) |
Input:
Int[(c + d*x)^(1/6)/(a + b*x)^(1/6),x]
Output:
((a + b*x)^(5/6)*(c + d*x)^(1/6))/b + ((b*c - a*d)*(ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/6))]/(3*b^(1/6)*d^ (5/6)) - (-((Sqrt[3]*ArcTan[(1 - (2*d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/6)))/Sqrt[3]])/d^(1/6)) + Log[b^(1/3) + d^( 1/3)*(a + b*x)^(1/3) - (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c - (a*d)/b + (d *(a + b*x))/b)^(1/6)]/(2*d^(1/6)))/(6*b^(1/6)*d^(2/3)) - ((Sqrt[3]*ArcTan[ (1 + (2*d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d*(a + b*x))/b)^ (1/6)))/Sqrt[3]])/d^(1/6) - Log[b^(1/3) + d^(1/3)*(a + b*x)^(1/3) + (b^(1/ 6)*d^(1/6)*(a + b*x)^(1/6))/(c - (a*d)/b + (d*(a + b*x))/b)^(1/6)]/(2*d^(1 /6)))/(6*b^(1/6)*d^(2/3))))/b
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
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]
\[\int \frac {\left (x d +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {1}{6}}}d x\]
Input:
int((d*x+c)^(1/6)/(b*x+a)^(1/6),x)
Output:
int((d*x+c)^(1/6)/(b*x+a)^(1/6),x)
Leaf count of result is larger than twice the leaf count of optimal. 1443 vs. \(2 (221) = 442\).
Time = 0.12 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.83 \[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(1/6),x, algorithm="fricas")
Output:
1/12*((sqrt(-3)*b + b)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20 *a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5) )^(1/6)*log(-1/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^2*d*x + a*b*d + sqrt(-3)*(b^2*d*x + a*b*d))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2* b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^ 6*d^6)/(b^7*d^5))^(1/6))/(b*x + a)) - (sqrt(-3)*b + b)*((b^6*c^6 - 6*a*b^5 *c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6* a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*log(-1/2*(2*(b*c - a*d)*(b*x + a)^ (5/6)*(d*x + c)^(1/6) - (b^2*d*x + a*b*d + sqrt(-3)*(b^2*d*x + a*b*d))*((b ^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4* b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6))/(b*x + a)) - (sqr t(-3)*b - b)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c ^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^7*d^5))^(1/6)*lo g(-1/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^2*d*x + a*b*d - sqrt(-3)*(b^2*d*x + a*b*d))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^ 2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^ 7*d^5))^(1/6))/(b*x + a)) + (sqrt(-3)*b - b)*((b^6*c^6 - 6*a*b^5*c^5*d + 1 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^ 5 + a^6*d^6)/(b^7*d^5))^(1/6)*log(-1/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^2*d*x + a*b*d - sqrt(-3)*(b^2*d*x + a*b*d))*((b^6*c^6 ...
\[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\int \frac {\sqrt [6]{c + d x}}{\sqrt [6]{a + b x}}\, dx \] Input:
integrate((d*x+c)**(1/6)/(b*x+a)**(1/6),x)
Output:
Integral((c + d*x)**(1/6)/(a + b*x)**(1/6), x)
\[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(1/6),x, algorithm="maxima")
Output:
integrate((d*x + c)^(1/6)/(b*x + a)^(1/6), x)
\[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(1/6),x, algorithm="giac")
Output:
integrate((d*x + c)^(1/6)/(b*x + a)^(1/6), x)
Timed out. \[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{1/6}} \,d x \] Input:
int((c + d*x)^(1/6)/(a + b*x)^(1/6),x)
Output:
int((c + d*x)^(1/6)/(a + b*x)^(1/6), x)
\[ \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx=\int \frac {\left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {1}{6}}}d x \] Input:
int((d*x+c)^(1/6)/(b*x+a)^(1/6),x)
Output:
int((c + d*x)**(1/6)/(a + b*x)**(1/6),x)