Integrand size = 19, antiderivative size = 368 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{24 \sqrt {3} b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}\right )}{72 b^{19/6}} \] Output:
91/12*d*(-a*d+b*c)*(b*x+a)^(5/6)*(d*x+c)^(1/6)/b^3+13/2*d*(b*x+a)^(5/6)*(d *x+c)^(7/6)/b^2-6*(d*x+c)^(13/6)/b/(b*x+a)^(1/6)+91/72*d^(1/6)*(-a*d+b*c)^ 2*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^(19/6)-91/72*d^(1/6)*(-a*d+b*c)^2*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^(19/6)+91/36*d ^(1/6)*(-a*d+b*c)^2*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b ^(19/6)+91/72*d^(1/6)*(-a*d+b*c)^2*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^(19/6)
Time = 1.03 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x} \left (-91 a^2 d^2-13 a b d (-13 c+d x)+b^2 \left (-72 c^2+25 c d x+6 d^2 x^2\right )\right )}{\sqrt [6]{a+b x}}-91 \sqrt {3} \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+91 \sqrt {3} \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+182 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{72 b^{19/6}} \] Input:
Integrate[(c + d*x)^(13/6)/(a + b*x)^(7/6),x]
Output:
((6*b^(1/6)*(c + d*x)^(1/6)*(-91*a^2*d^2 - 13*a*b*d*(-13*c + d*x) + b^2*(- 72*c^2 + 25*c*d*x + 6*d^2*x^2)))/(a + b*x)^(1/6) - 91*Sqrt[3]*d^(1/6)*(b*c - a*d)^2*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))] + 91*Sqrt[3]*d^(1/6)*(b*c - a*d)^2*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))] + 182*d^(1/6)*(b*c - a*d)^2*ArcTanh[(b^(1/6)*(c + d*x)^(1/ 6))/(d^(1/6)*(a + b*x)^(1/6))] + 91*d^(1/6)*(b*c - a*d)^2*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))])/(72*b^(19/6))
Time = 0.51 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {57, 60, 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.
| \(\displaystyle \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {13 d \int \frac {(c+d x)^{7/6}}{\sqrt [6]{a+b x}}dx}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}}dx}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}}dx}{6 b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{5/6}}d\sqrt [6]{a+b x}}{b^2}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \int \frac {b (a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b^2}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{b-d (a+b x)}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}+\frac {\int -\frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \left (\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}\right )}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 \sqrt [6]{b} d^{2/3}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{3 d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\int -\frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\int \frac {\sqrt [6]{d} \left (\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {-\frac {3 \int \frac {1}{-\sqrt [3]{a+b x}-3}d\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+1\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{b}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}-\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {1}{2} \int \frac {\sqrt [6]{b}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{\sqrt [3]{b}+\frac {\sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{b}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\sqrt [3]{d} \sqrt [3]{a+b x}}d\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {13 d \left (\frac {7 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}}{\sqrt {3}}\right )}{\sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+1}{\sqrt {3}}\right )}{\sqrt [6]{d}}-\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}+\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b}\right )}{2 \sqrt [6]{d}}}{6 \sqrt [6]{b} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{3 \sqrt [6]{b} d^{5/6}}\right )}{b}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{b}\right )}{12 b}+\frac {(a+b x)^{5/6} (c+d x)^{7/6}}{2 b}\right )}{b}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}\) |
Input:
Int[(c + d*x)^(13/6)/(a + b*x)^(7/6),x]
Output:
(-6*(c + d*x)^(13/6))/(b*(a + b*x)^(1/6)) + (13*d*(((a + b*x)^(5/6)*(c + d *x)^(7/6))/(2*b) + (7*(b*c - a*d)*(((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)))/Sqr t[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))/(12*b)) )/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] :> 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]
\[\int \frac {\left (x d +c \right )^{\frac {13}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]
Input:
int((d*x+c)^(13/6)/(b*x+a)^(7/6),x)
Output:
int((d*x+c)^(13/6)/(b*x+a)^(7/6),x)
Leaf count of result is larger than twice the leaf count of optimal. 2686 vs. \(2 (278) = 556\).
Time = 0.14 (sec) , antiderivative size = 2686, normalized size of antiderivative = 7.30 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="fricas")
Output:
1/144*(91*(b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a* b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c ^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^ 11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(91/2*(2*(b^2*c^2 - 2*a*b*c*d + a^ 2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220* a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6* c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(b*x + a)) - 91*(b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a *b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8* c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a ^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(91/2*(2*(b^2*c^2 - 2*a*b*c*d + a ^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220 *a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6 *c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^1 0 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(...
Timed out. \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(13/6)/(b*x+a)**(7/6),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {13}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="maxima")
Output:
integrate((d*x + c)^(13/6)/(b*x + a)^(7/6), x)
\[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {13}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \] Input:
integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="giac")
Output:
integrate((d*x + c)^(13/6)/(b*x + a)^(7/6), x)
Timed out. \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{13/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \] Input:
int((c + d*x)^(13/6)/(a + b*x)^(7/6),x)
Output:
int((c + d*x)^(13/6)/(a + b*x)^(7/6), x)
\[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(13/6)/(b*x+a)^(7/6),x)
Output:
( - 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*a*c**3*d - 6*d*(c + d*x)**(1/6)* (a + b*x)**(5/6)*a*c**2*d**2*x + 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*b*c **4 + 6*d*(c + d*x)**(1/6)*(a + b*x)**(5/6)*b*c**3*d*x - 6*(c + d*x)**(7/6 )*(a + b*x)**(5/6)*log((a + b*x)**(1/6))*a*c**2*d**2 - 6*(c + d*x)**(7/6)* (a + b*x)**(5/6)*log((a + b*x)**(1/6))*b*c**2*d**2*x + 6*(c + d*x)**(7/6)* (a + b*x)**(5/6)*log((c + d*x)**(1/6))*a*c**2*d**2 + 6*(c + d*x)**(7/6)*(a + b*x)**(5/6)*log((c + d*x)**(1/6))*b*c**2*d**2*x + int(((c + d*x)**(1/6) *x**2)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a**2*b*c*d**4 + int( ((c + d*x)**(1/6)*x**2)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a** 2*b*d**5*x - int(((c + d*x)**(1/6)*x**2)/((a + b*x)**(1/6)*a + (a + b*x)** (1/6)*b*x),x)*a*b**2*c**2*d**3 + int(((c + d*x)**(1/6)*x**2)/((a + b*x)**( 1/6)*a + (a + b*x)**(1/6)*b*x),x)*a*b**2*d**5*x**2 - int(((c + d*x)**(1/6) *x**2)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*b**3*c**2*d**3*x - i nt(((c + d*x)**(1/6)*x**2)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)* b**3*c*d**4*x**2 + 2*int(((c + d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + (a + b *x)**(1/6)*b*x),x)*a**2*b*c**2*d**3 + 2*int(((c + d*x)**(1/6)*x)/((a + b*x )**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a**2*b*c*d**4*x - 2*int(((c + d*x)** (1/6)*x)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x)*a*b**2*c**3*d**2 + 2*int(((c + d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + (a + b*x)**(1/6)*b*x),x) *a*b**2*c*d**4*x**2 - 2*int(((c + d*x)**(1/6)*x)/((a + b*x)**(1/6)*a + ...