Integrand size = 19, antiderivative size = 258 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {\sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{b^{7/6}} \] Output:
-6*(d*x+c)^(1/6)/b/(b*x+a)^(1/6)+3^(1/2)*d^(1/6)*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))/b^(7/6)-3^(1/2)*d^(1/6) *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 ))/b^(7/6)+2*d^(1/6)*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/ b^(7/6)+d^(1/6)*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)
Time = 0.52 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\frac {-\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{a+b x}}+\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}-\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+\sqrt {3} \sqrt [6]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+2 \sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\sqrt [6]{d} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{b^{7/6}} \] Input:
Integrate[(c + d*x)^(1/6)/(a + b*x)^(7/6),x]
Output:
((-6*b^(1/6)*(c + d*x)^(1/6))/(a + b*x)^(1/6) + Sqrt[3]*d^(1/6)*ArcTan[(Sq rt[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]*d^(1/6)*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*d^(1/6)*ArcTanh[(b ^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + d^(1/6)*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))])/b^(7/6)
Time = 0.43 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.54, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {57, 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.
| \(\displaystyle \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}}dx}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {6 d \int \frac {(a+b x)^{2/3}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{5/6}}d\sqrt [6]{a+b x}}{b^2}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {6 d \int \frac {b (a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b^2}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 d \int \frac {(a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {6 d \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 d \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {6 d \left (-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {6 d \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 d \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 d \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {6 d \left (-\frac {\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {6 d \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {6 d \left (-\frac {\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}\) |
Input:
Int[(c + d*x)^(1/6)/(a + b*x)^(7/6),x]
Output:
(-6*(c + d*x)^(1/6))/(b*(a + b*x)^(1/6)) + (6*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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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]
\[\int \frac {\left (x d +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]
Input:
int((d*x+c)^(1/6)/(b*x+a)^(7/6),x)
Output:
int((d*x+c)^(1/6)/(b*x+a)^(7/6),x)
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (188) = 376\).
Time = 0.12 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="fricas")
Output:
1/2*((b^2*x + a*b + sqrt(-3)*(b^2*x + a*b))*(d/b^7)^(1/6)*log(((b^2*x + a* b + sqrt(-3)*(b^2*x + a*b))*(d/b^7)^(1/6) + 2*(b*x + a)^(5/6)*(d*x + c)^(1 /6))/(b*x + a)) - (b^2*x + a*b + sqrt(-3)*(b^2*x + a*b))*(d/b^7)^(1/6)*log (-((b^2*x + a*b + sqrt(-3)*(b^2*x + a*b))*(d/b^7)^(1/6) - 2*(b*x + a)^(5/6 )*(d*x + c)^(1/6))/(b*x + a)) + (b^2*x + a*b - sqrt(-3)*(b^2*x + a*b))*(d/ b^7)^(1/6)*log(((b^2*x + a*b - sqrt(-3)*(b^2*x + a*b))*(d/b^7)^(1/6) + 2*( b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) - (b^2*x + a*b - sqrt(-3)*(b^2* x + a*b))*(d/b^7)^(1/6)*log(-((b^2*x + a*b - sqrt(-3)*(b^2*x + a*b))*(d/b^ 7)^(1/6) - 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) + 2*(b^2*x + a*b) *(d/b^7)^(1/6)*log(((b^2*x + a*b)*(d/b^7)^(1/6) + (b*x + a)^(5/6)*(d*x + c )^(1/6))/(b*x + a)) - 2*(b^2*x + a*b)*(d/b^7)^(1/6)*log(-((b^2*x + a*b)*(d /b^7)^(1/6) - (b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) - 12*(b*x + a)^( 5/6)*(d*x + c)^(1/6))/(b^2*x + a*b)
\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int \frac {\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \] Input:
integrate((d*x+c)**(1/6)/(b*x+a)**(7/6),x)
Output:
Integral((c + d*x)**(1/6)/(a + b*x)**(7/6), x)
\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="maxima")
Output:
integrate((d*x + c)^(1/6)/(b*x + a)^(7/6), x)
\[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="giac")
Output:
integrate((d*x + c)^(1/6)/(b*x + a)^(7/6), x)
Timed out. \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \] Input:
int((c + d*x)^(1/6)/(a + b*x)^(7/6),x)
Output:
int((c + d*x)^(1/6)/(a + b*x)^(7/6), x)
Time = 0.43 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx=\frac {6 \left (b x +a \right )^{\frac {1}{3}} \left (-d^{2} \left (d x +c \right )^{\frac {1}{6}} \sqrt {b x +a}\, a c -d^{3} \left (d x +c \right )^{\frac {1}{6}} \sqrt {b x +a}\, a x +d \left (d x +c \right )^{\frac {1}{6}} \sqrt {b x +a}\, b \,c^{2}+d^{2} \left (d x +c \right )^{\frac {1}{6}} \sqrt {b x +a}\, b c x -\left (d x +c \right )^{\frac {7}{6}} \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a \,d^{2}-\left (d x +c \right )^{\frac {7}{6}} \sqrt {b x +a}\, \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) b \,d^{2} x +\left (d x +c \right )^{\frac {7}{6}} \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a \,d^{2}+\left (d x +c \right )^{\frac {7}{6}} \sqrt {b x +a}\, \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) b \,d^{2} x \right )}{b d \left (a b \,d^{2} x^{2}-b^{2} c d \,x^{2}+a^{2} d^{2} x -b^{2} c^{2} x +a^{2} c d -a b \,c^{2}\right )} \] Input:
int((d*x+c)^(1/6)/(b*x+a)^(7/6),x)
Output:
(6*(a + b*x)**(1/3)*( - d*(c + d*x)**(1/6)*sqrt(a + b*x)*a*c*d - d*(c + d* x)**(1/6)*sqrt(a + b*x)*a*d**2*x + d*(c + d*x)**(1/6)*sqrt(a + b*x)*b*c**2 + d*(c + d*x)**(1/6)*sqrt(a + b*x)*b*c*d*x - (c + d*x)**(7/6)*sqrt(a + b* x)*log((a + b*x)**(1/6))*a*d**2 - (c + d*x)**(7/6)*sqrt(a + b*x)*log((a + b*x)**(1/6))*b*d**2*x + (c + d*x)**(7/6)*sqrt(a + b*x)*log((c + d*x)**(1/6 ))*a*d**2 + (c + d*x)**(7/6)*sqrt(a + b*x)*log((c + d*x)**(1/6))*b*d**2*x) )/(b*d*(a**2*c*d + a**2*d**2*x - a*b*c**2 + a*b*d**2*x**2 - b**2*c**2*x - b**2*c*d*x**2))