Integrand size = 19, antiderivative size = 332 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}-\frac {\sqrt {3} \sqrt [6]{b} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}+\frac {\sqrt {3} \sqrt [6]{b} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}+\frac {2 \sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{7/6}} \]
-6*(b*x+a)^(1/6)/d/(d*x+c)^(1/6)+2*b^(1/6)*arctanh(d^(1/6)*(b*x+a)^(1/6)/b ^(1/6)/(d*x+c)^(1/6))/d^(7/6)-1/2*b^(1/6)*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3) /(d*x+c)^(1/3)-b^(1/6)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))/d^(7/6)+1/2*b^ (1/6)*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/(d*x+c)^(1/3)+b^(1/6)*d^(1/6)*(b*x+ a)^(1/6)/(d*x+c)^(1/6))/d^(7/6)+b^(1/6)*arctan(-1/3*3^(1/2)+2/3*d^(1/6)*(b *x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6)*3^(1/2))*3^(1/2)/d^(7/6)+b^(1/6)*arctan( 1/3*3^(1/2)+2/3*d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6)*3^(1/2))*3^(1/ 2)/d^(7/6)
Time = 0.53 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\frac {-\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\sqrt {3} \sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-\sqrt {3} \sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+2 \sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{d^{7/6}} \]
((-6*d^(1/6)*(a + b*x)^(1/6))/(c + d*x)^(1/6) + Sqrt[3]*b^(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]*b^(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*b^(1/6)*ArcTanh[( b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + b^(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))])/d^(7/6)
Time = 0.43 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {57, 73, 770, 754, 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.
| \(\displaystyle \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {b \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}}dx}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {6 \int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [6]{a+b x}}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {6 \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {6 \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {6 \left (\frac {1}{6} \sqrt [6]{b} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{d}-\frac {6 \sqrt [6]{a+b x}}{d \sqrt [6]{c+d x}}\) |
(-6*(a + b*x)^(1/6))/(d*(c + d*x)^(1/6)) + (6*((b^(1/6)*ArcTanh[(d^(1/6)*( a + b*x)^(1/6))/(b^(1/6)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/6))])/(3*d^(1/ 6)) + (b^(1/6)*(-((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)*((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))/d
3.18.74.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 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]
\[\int \frac {\left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {7}{6}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (239) = 478\).
Time = 0.24 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\frac {{\left (d^{2} x + c d + \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (d^{2} x + c d + \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - {\left (d^{2} x + c d + \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (d^{2} x + c d + \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + {\left (d^{2} x + c d - \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (d^{2} x + c d - \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - {\left (d^{2} x + c d - \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (d^{2} x + c d - \sqrt {-3} {\left (d^{2} x + c d\right )}\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) + 2 \, {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - 2 \, {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{7}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - 12 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{2 \, {\left (d^{2} x + c d\right )}} \]
1/2*((d^2*x + c*d + sqrt(-3)*(d^2*x + c*d))*(b/d^7)^(1/6)*log(((d^2*x + c* d + sqrt(-3)*(d^2*x + c*d))*(b/d^7)^(1/6) + 2*(b*x + a)^(1/6)*(d*x + c)^(5 /6))/(d*x + c)) - (d^2*x + c*d + sqrt(-3)*(d^2*x + c*d))*(b/d^7)^(1/6)*log (-((d^2*x + c*d + sqrt(-3)*(d^2*x + c*d))*(b/d^7)^(1/6) - 2*(b*x + a)^(1/6 )*(d*x + c)^(5/6))/(d*x + c)) + (d^2*x + c*d - sqrt(-3)*(d^2*x + c*d))*(b/ d^7)^(1/6)*log(((d^2*x + c*d - sqrt(-3)*(d^2*x + c*d))*(b/d^7)^(1/6) + 2*( b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c)) - (d^2*x + c*d - sqrt(-3)*(d^2* x + c*d))*(b/d^7)^(1/6)*log(-((d^2*x + c*d - sqrt(-3)*(d^2*x + c*d))*(b/d^ 7)^(1/6) - 2*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c)) + 2*(d^2*x + c*d) *(b/d^7)^(1/6)*log(((d^2*x + c*d)*(b/d^7)^(1/6) + (b*x + a)^(1/6)*(d*x + c )^(5/6))/(d*x + c)) - 2*(d^2*x + c*d)*(b/d^7)^(1/6)*log(-((d^2*x + c*d)*(b /d^7)^(1/6) - (b*x + a)^(1/6)*(d*x + c)^(5/6))/(d*x + c)) - 12*(b*x + a)^( 1/6)*(d*x + c)^(5/6))/(d^2*x + c*d)
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\int \frac {\sqrt [6]{a + b x}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/6}}{{\left (c+d\,x\right )}^{7/6}} \,d x \]