Integrand size = 20, antiderivative size = 114 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \left (101+52 x+32 x^2\right ) \sqrt [4]{x^3+x^4}-\frac {155}{64} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {155}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right ) \]
1/96*(32*x^2+52*x+101)*(x^4+x^3)^(1/4)-155/64*arctan(x/(x^4+x^3)^(1/4))+2* 2^(1/4)*arctan(2^(1/4)*x/(x^4+x^3)^(1/4))+155/64*arctanh(x/(x^4+x^3)^(1/4) )-2*2^(1/4)*arctanh(2^(1/4)*x/(x^4+x^3)^(1/4))
Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (202 x^{3/4} \sqrt [4]{1+x}+104 x^{7/4} \sqrt [4]{1+x}+64 x^{11/4} \sqrt [4]{1+x}-465 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+384 \sqrt [4]{2} \arctan \left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )+465 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )-384 \sqrt [4]{2} \text {arctanh}\left (\sqrt [4]{2} \sqrt [4]{\frac {x}{1+x}}\right )\right )}{192 \left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(1 + x)^(3/4)*(202*x^(3/4)*(1 + x)^(1/4) + 104*x^(7/4)*(1 + x)^(1 /4) + 64*x^(11/4)*(1 + x)^(1/4) - 465*ArcTan[(x/(1 + x))^(1/4)] + 384*2^(1 /4)*ArcTan[2^(1/4)*(x/(1 + x))^(1/4)] + 465*ArcTanh[(x/(1 + x))^(1/4)] - 3 84*2^(1/4)*ArcTanh[2^(1/4)*(x/(1 + x))^(1/4)]))/(192*(x^3*(1 + x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(114)=228\).
Time = 0.42 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.29, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.050, Rules used = {1948, 25, 112, 27, 173, 105, 105, 104, 827, 216, 219, 1194, 27, 87, 57, 60, 73, 854, 827, 216, 219}
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.
| \(\displaystyle \int \frac {x^2 \sqrt [4]{x^4+x^3}}{x-1} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {x^{11/4} \sqrt [4]{x+1}}{1-x}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{11/4} \sqrt [4]{x+1}}{1-x}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{3} \int \frac {x^{7/4} (13 x+11)}{4 (1-x) (x+1)^{3/4}}dx-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \int \frac {x^{7/4} (13 x+11)}{(1-x) (x+1)^{3/4}}dx-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 173 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \int \frac {x^{7/4}}{(1-x) (x+1)^{11/4}}dx-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \int \frac {x^{3/4}}{(1-x) (x+1)^{7/4}}dx-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{(1-x) \sqrt [4]{x} (x+1)^{3/4}}dx-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \left (2 \int \frac {\sqrt {x}}{\sqrt {x+1} \left (1-\frac {2 x}{x+1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \left (2 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x+1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {x}}{\sqrt {x+1}}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \left (2 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {x}}{\sqrt {x+1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\int \frac {x^{7/4} \left (13 x^2+50 x+85\right )}{(x+1)^{11/4}}dx\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 1194 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {1}{2} \int \frac {5 x^{7/4} (41 x+136)}{4 (x+1)^{11/4}}dx+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \int \frac {x^{7/4} (41 x+136)}{(x+1)^{11/4}}dx+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \int \frac {x^{7/4}}{(x+1)^{7/4}}dx\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \int \frac {x^{3/4}}{(x+1)^{3/4}}dx-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-\frac {3}{4} \int \frac {1}{\sqrt [4]{x} (x+1)^{3/4}}dx\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-3 \int \frac {\sqrt {x}}{(x+1)^{3/4}}d\sqrt [4]{x}\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-3 \int \frac {\sqrt {x}}{1-x}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{12} \left (-\frac {5}{8} \left (\frac {380 x^{11/4}}{7 (x+1)^{7/4}}-\frac {93}{7} \left (\frac {7}{3} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )-\frac {4 x^{7/4}}{3 (x+1)^{3/4}}\right )\right )+96 \left (\frac {1}{2} \left (2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2\ 2^{3/4}}\right )-\frac {2 x^{3/4}}{3 (x+1)^{3/4}}\right )-\frac {2 x^{7/4}}{7 (x+1)^{7/4}}\right )-\frac {13 x^{15/4}}{2 (x+1)^{7/4}}\right )-\frac {1}{3} x^{11/4} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
-(((x^3 + x^4)^(1/4)*(-1/3*(x^(11/4)*(1 + x)^(1/4)) + ((-13*x^(15/4))/(2*( 1 + x)^(7/4)) - (5*((380*x^(11/4))/(7*(1 + x)^(7/4)) - (93*((-4*x^(7/4))/( 3*(1 + x)^(3/4)) + (7*(x^(3/4)*(1 + x)^(1/4) - 3*(-1/2*ArcTan[x^(1/4)/(1 + x)^(1/4)] + ArcTanh[x^(1/4)/(1 + x)^(1/4)]/2)))/3))/7))/8 + 96*((-2*x^(7/ 4))/(7*(1 + x)^(7/4)) + ((-2*x^(3/4))/(3*(1 + x)^(3/4)) + 2*(-1/2*ArcTan[( 2^(1/4)*x^(1/4))/(1 + x)^(1/4)]/2^(3/4) + ArcTanh[(2^(1/4)*x^(1/4))/(1 + x )^(1/4)]/(2*2^(3/4))))/2))/12))/(x^(3/4)*(1 + x)^(1/4)))
3.18.1.3.1 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] :> 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_ )))/((e_.) + (f_.)*(x_)), x_] :> Simp[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/ f^(m + n + 2)) Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Si mp[1/f^(m + n + 2) Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n + 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x )^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 *p + 1)) Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) *(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ [p, 0] && !IntegerQ[m] && !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(90)=180\).
Time = 4.78 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.82
| method | result | size |
| pseudoelliptic | \(-\frac {x^{9} \left (-128 x^{2} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+384 \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}+768 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}-208 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} x -465 \ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )+465 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )-930 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-404 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}{384 {\left (x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}^{3} {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x \right )}^{3} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{3}}\) | \(208\) |
| trager | \(\left (\frac {1}{3} x^{2}+\frac {13}{24} x +\frac {101}{96}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\frac {155 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+x^{3}}\, x +2 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-2 x^{3}-x^{2}}{x^{2}}\right )}{128}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\right )-\frac {155 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-4 x^{2} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{256}\) | \(444\) |
| risch | \(\text {Expression too large to display}\) | \(876\) |
-1/384*x^9*(-128*x^2*(x^3*(1+x))^(1/4)+384*ln((-2^(1/4)*x-(x^3*(1+x))^(1/4 ))/(2^(1/4)*x-(x^3*(1+x))^(1/4)))*2^(1/4)+768*arctan(1/2*2^(3/4)/x*(x^3*(1 +x))^(1/4))*2^(1/4)-208*(x^3*(1+x))^(1/4)*x-465*ln((x+(x^3*(1+x))^(1/4))/x )+465*ln(((x^3*(1+x))^(1/4)-x)/x)-930*arctan((x^3*(1+x))^(1/4)/x)-404*(x^3 *(1+x))^(1/4))/(x+(x^3*(1+x))^(1/4))^3/((x^3*(1+x))^(1/4)-x)^3/(x^2+(x^3*( 1+x))^(1/2))^3
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} + 52 \, x + 101\right )} - 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{64} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {155}{128} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {155}{128} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/96*(x^4 + x^3)^(1/4)*(32*x^2 + 52*x + 101) - 2^(1/4)*log((2^(1/4)*x + (x ^4 + x^3)^(1/4))/x) + 2^(1/4)*log(-(2^(1/4)*x - (x^4 + x^3)^(1/4))/x) - I* 2^(1/4)*log((I*2^(1/4)*x + (x^4 + x^3)^(1/4))/x) + I*2^(1/4)*log((-I*2^(1/ 4)*x + (x^4 + x^3)^(1/4))/x) + 155/64*arctan((x^4 + x^3)^(1/4)/x) + 155/12 8*log((x + (x^4 + x^3)^(1/4))/x) - 155/128*log(-(x - (x^4 + x^3)^(1/4))/x)
\[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (x + 1\right )}}{x - 1}\, dx \]
\[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{2}}{x - 1} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\frac {1}{96} \, {\left (101 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 150 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 81 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} - 2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {155}{64} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {155}{128} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {155}{128} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/96*(101*(1/x + 1)^(9/4) - 150*(1/x + 1)^(5/4) + 81*(1/x + 1)^(1/4))*x^3 - 2*2^(1/4)*arctan(1/2*2^(3/4)*(1/x + 1)^(1/4)) - 2^(1/4)*log(2^(1/4) + (1 /x + 1)^(1/4)) + 2^(1/4)*log(abs(-2^(1/4) + (1/x + 1)^(1/4))) + 155/64*arc tan((1/x + 1)^(1/4)) + 155/128*log((1/x + 1)^(1/4) + 1) - 155/128*log(abs( (1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x} \, dx=\int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x-1} \,d x \]