Integrand size = 19, antiderivative size = 324 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}+\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{6 d^{13/6}} \] Output:
-6*(b*x+a)^(7/6)/d/(d*x+c)^(1/6)+7*b*(b*x+a)^(1/6)*(d*x+c)^(5/6)/d^2+7/6*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)/d^(13/6)-7/6*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)/d^(1 3/6)-7/3*b^(1/6)*(-a*d+b*c)*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^ (1/6))/d^(13/6)-7/6*b^(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)))/d^(13/6)
Time = 0.97 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x} (7 b c-6 a d+b d x)}{\sqrt [6]{c+d x}}-7 \sqrt {3} \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+7 \sqrt {3} \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-14 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )-7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 d^{13/6}} \] Input:
Integrate[(a + b*x)^(7/6)/(c + d*x)^(7/6),x]
Output:
((6*d^(1/6)*(a + b*x)^(1/6)*(7*b*c - 6*a*d + b*d*x))/(c + d*x)^(1/6) - 7*S qrt[3]*b^(1/6)*(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))] + 7*Sqrt[3]*b^(1/6)*(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))] - 14*b^(1/6)*(b*c - a*d)*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] - 7*b^(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*d^(13/6))
Time = 0.46 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.31, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {57, 60, 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.
\(\displaystyle \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {7 b \int \frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}dx}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}}dx}{6 d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [6]{a+b x}}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 770 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 754 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{3} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{3} \sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{6} \sqrt [6]{b} \int \frac {2 \sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}+\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {7 b \left (\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{d}-\frac {(b c-a d) \left (\frac {1}{6} \sqrt [6]{b} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {1}{6} \sqrt [6]{b} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}\right )+\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{d}}\right )}{b d}\right )}{d}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}\) |
Input:
Int[(a + b*x)^(7/6)/(c + d*x)^(7/6),x]
Output:
(-6*(a + b*x)^(7/6))/(d*(c + d*x)^(1/6)) + (7*b*(((a + b*x)^(1/6)*(c + d*x )^(5/6))/d - ((b*c - a*d)*((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)*(-((S qrt[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))/(b*d)))/d
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] :> 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[((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]
\[\int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (x d +c \right )^{\frac {7}{6}}}d x\]
Input:
int((b*x+a)^(7/6)/(d*x+c)^(7/6),x)
Output:
int((b*x+a)^(7/6)/(d*x+c)^(7/6),x)
Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (242) = 484\).
Time = 0.13 (sec) , antiderivative size = 1562, normalized size of antiderivative = 4.82 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(7/6),x, algorithm="fricas")
Output:
-1/12*(7*(d^3*x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^ 5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5 *b^2*c*d^5 + a^6*b*d^6)/d^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(1/6 )*(d*x + c)^(5/6) + (d^3*x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2* d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)/d^13)^(1/6))/(d*x + c)) - 7*(d^3*x + c* d^2 + sqrt(-3)*(d^3*x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4 *d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d ^6)/d^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (d^3*x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^5*d + 15* a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^ 5 + a^6*b*d^6)/d^13)^(1/6))/(d*x + c)) + 7*(d^3*x + c*d^2 - sqrt(-3)*(d^3* x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^ 3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)/d^13)^(1/6)*log( -7/2*(2*(b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + (d^3*x + c*d^2 - sqr t(-3)*(d^3*x + c*d^2))*((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20 *a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)/d^13) ^(1/6))/(d*x + c)) - 7*(d^3*x + c*d^2 - sqrt(-3)*(d^3*x + c*d^2))*((b^7*c^ 6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c ^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)/d^13)^(1/6)*log(-7/2*(2*(b*c - a*...
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{6}}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \] Input:
integrate((b*x+a)**(7/6)/(d*x+c)**(7/6),x)
Output:
Integral((a + b*x)**(7/6)/(c + d*x)**(7/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(7/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(7/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((b*x+a)^(7/6)/(d*x+c)^(7/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(7/6)/(d*x + c)^(7/6), x)
Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/6}}{{\left (c+d\,x\right )}^{7/6}} \,d x \] Input:
int((a + b*x)^(7/6)/(c + d*x)^(7/6),x)
Output:
int((a + b*x)^(7/6)/(c + d*x)^(7/6), x)
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\frac {-6 \left (b x +a \right )^{\frac {1}{6}} a^{3} d +6 \left (b x +a \right )^{\frac {1}{6}} a^{2} b c -6 \left (b x +a \right )^{\frac {1}{6}} a^{2} b d x +6 \left (b x +a \right )^{\frac {1}{6}} a \,b^{2} c x -6 \left (d x +c \right ) \left (b x +a \right )^{\frac {7}{6}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}\right ) a b +6 \left (d x +c \right ) \left (b x +a \right )^{\frac {7}{6}} \mathrm {log}\left (\left (d x +c \right )^{\frac {1}{6}}\right ) a b +\left (d x +c \right )^{\frac {1}{6}} \left (\int \frac {\left (b x +a \right )^{\frac {1}{6}} x}{\left (d x +c \right )^{\frac {1}{6}} c +\left (d x +c \right )^{\frac {1}{6}} d x}d x \right ) a^{2} b \,d^{2}-\left (d x +c \right )^{\frac {1}{6}} \left (\int \frac {\left (b x +a \right )^{\frac {1}{6}} x}{\left (d x +c \right )^{\frac {1}{6}} c +\left (d x +c \right )^{\frac {1}{6}} d x}d x \right ) a \,b^{2} c d +\left (d x +c \right )^{\frac {1}{6}} \left (\int \frac {\left (b x +a \right )^{\frac {1}{6}} x}{\left (d x +c \right )^{\frac {1}{6}} c +\left (d x +c \right )^{\frac {1}{6}} d x}d x \right ) a \,b^{2} d^{2} x -\left (d x +c \right )^{\frac {1}{6}} \left (\int \frac {\left (b x +a \right )^{\frac {1}{6}} x}{\left (d x +c \right )^{\frac {1}{6}} c +\left (d x +c \right )^{\frac {1}{6}} d x}d x \right ) b^{3} c d x}{\left (d x +c \right )^{\frac {1}{6}} d \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:
int((b*x+a)^(7/6)/(d*x+c)^(7/6),x)
Output:
( - 6*(a + b*x)**(1/6)*a**3*d + 6*(a + b*x)**(1/6)*a**2*b*c - 6*(a + b*x)* *(1/6)*a**2*b*d*x + 6*(a + b*x)**(1/6)*a*b**2*c*x - 6*(c + d*x)*(a + b*x)* *(7/6)*log((a + b*x)**(1/6))*a*b + 6*(c + d*x)*(a + b*x)**(7/6)*log((c + d *x)**(1/6))*a*b + (c + d*x)**(1/6)*int(((a + b*x)**(1/6)*x)/((c + d*x)**(1 /6)*c + (c + d*x)**(1/6)*d*x),x)*a**2*b*d**2 - (c + d*x)**(1/6)*int(((a + b*x)**(1/6)*x)/((c + d*x)**(1/6)*c + (c + d*x)**(1/6)*d*x),x)*a*b**2*c*d + (c + d*x)**(1/6)*int(((a + b*x)**(1/6)*x)/((c + d*x)**(1/6)*c + (c + d*x) **(1/6)*d*x),x)*a*b**2*d**2*x - (c + d*x)**(1/6)*int(((a + b*x)**(1/6)*x)/ ((c + d*x)**(1/6)*c + (c + d*x)**(1/6)*d*x),x)*b**3*c*d*x)/((c + d*x)**(1/ 6)*d*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))