Integrand size = 24, antiderivative size = 72 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\frac {\sqrt [3]{(1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-4 \text {$\#$1}^3+3 \text {$\#$1}^6}\&\right ]}{(1+x)^{2/3}} \] Output:
Unintegrable
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\frac {\sqrt [3]{(1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-4 \text {$\#$1}^3+3 \text {$\#$1}^6}\&\right ]}{(1+x)^{2/3}} \] Input:
Integrate[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]
Output:
(((1 + x)^2)^(1/3)*RootSum[3 + 2*#1^3 - 2*#1^6 + #1^9 & , (Log[(1 + x)^(1/ 3) - #1]*#1^2)/(2 - 4*#1^3 + 3*#1^6) & ])/(1 + x)^(2/3)
Result contains complex when optimal does not.
Time = 9.99 (sec) , antiderivative size = 3748, normalized size of antiderivative = 52.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2008, 2490, 2485, 27, 1199, 2009}
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.
\(\displaystyle \int \frac {\sqrt [3]{x^2+2 x+1}}{x^3+x^2+x+4} \, dx\) |
\(\Big \downarrow \) 2008 |
\(\displaystyle \frac {\sqrt [3]{(x+1)^2} \int \frac {(x+1)^{2/3}}{x^3+x^2+x+4}dx}{(x+1)^{2/3}}\) |
\(\Big \downarrow \) 2490 |
\(\displaystyle \frac {\sqrt [3]{(x+1)^2} \int \frac {(x+1)^{2/3}}{\left (x+\frac {1}{3}\right )^3+\frac {2}{3} \left (x+\frac {1}{3}\right )+\frac {101}{27}}d\left (x+\frac {1}{3}\right )}{(x+1)^{2/3}}\) |
\(\Big \downarrow \) 2485 |
\(\displaystyle \frac {\sqrt [3]{(x+1)^2} \int \frac {36 \sqrt [3]{3} \left (3 \left (x+\frac {1}{3}\right )+2\right )^{2/3}}{\left (6 \left (x+\frac {1}{3}\right )+\sqrt [3]{2} \left (\frac {4}{\sqrt [3]{-101+3 \sqrt {1137}}}-\sqrt [3]{-202+6 \sqrt {1137}}\right )\right ) \left (18 \left (x+\frac {1}{3}\right )^2-3 \sqrt [3]{2} \left (\frac {4}{\sqrt [3]{-101+3 \sqrt {1137}}}-\sqrt [3]{-202+6 \sqrt {1137}}\right ) \left (x+\frac {1}{3}\right )+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+4\right )}d\left (x+\frac {1}{3}\right )}{(x+1)^{2/3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {36 \sqrt [3]{3} \sqrt [3]{(x+1)^2} \int \frac {\left (3 \left (x+\frac {1}{3}\right )+2\right )^{2/3}}{\left (6 \left (x+\frac {1}{3}\right )+\sqrt [3]{2} \left (\frac {4}{\sqrt [3]{-101+3 \sqrt {1137}}}-\sqrt [3]{-202+6 \sqrt {1137}}\right )\right ) \left (18 \left (x+\frac {1}{3}\right )^2-3 \sqrt [3]{2} \left (\frac {4}{\sqrt [3]{-101+3 \sqrt {1137}}}-\sqrt [3]{-202+6 \sqrt {1137}}\right ) \left (x+\frac {1}{3}\right )+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+4\right )}d\left (x+\frac {1}{3}\right )}{(x+1)^{2/3}}\) |
\(\Big \downarrow \) 1199 |
\(\displaystyle \frac {36 \sqrt [3]{3} \sqrt [3]{(x+1)^2} \int \left (\frac {\left (4-2^{2/3} \left (\frac {2\ 2^{2/3}}{\sqrt [3]{-101+3 \sqrt {1137}}}-\sqrt [3]{-101+3 \sqrt {1137}}\right )\right ) \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}}{6 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \left (-2 \left (3 \left (x+\frac {1}{3}\right )+2\right )+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+4\right )}-\frac {\sqrt [3]{3 \left (x+\frac {1}{3}\right )+2} \left (-\left (\left (4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}\right ) \left (3 \left (x+\frac {1}{3}\right )+2\right )\right )+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+8 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+12\right )}{6 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \left (2 \left (3 \left (x+\frac {1}{3}\right )+2\right )^2-\left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}\right ) \left (3 \left (x+\frac {1}{3}\right )+2\right )+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+8 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+12\right )}\right )d\sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}}{(x+1)^{2/3}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {36 \sqrt [3]{3} \sqrt [3]{(x+1)^2} \left (\frac {\left (2 \sqrt [3]{2}-\frac {2\ 2^{2/3}}{\sqrt [3]{-101+3 \sqrt {1137}}}+\sqrt [3]{-101+3 \sqrt {1137}}\right ) \sqrt [3]{-\frac {1}{4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}}} \arctan \left (\frac {1-2 \sqrt [3]{-\frac {2}{4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}}} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}}{\sqrt {3}}\right )}{6 \sqrt {3} \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}+\frac {\left (4 \sqrt {3} \left (-101+3 \sqrt {1137}\right )^{2/3}+\frac {9 i 2^{5/6} \left (31-\sqrt {1137}\right ) \sqrt [3]{-101+3 \sqrt {1137}}}{\sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}}-\sqrt [6]{2} \left (101 \sqrt {6}-9 \sqrt {758}+4 \sqrt [6]{2} \sqrt {3} \sqrt [3]{-101+3 \sqrt {1137}}-36 i \sqrt {\frac {1137}{8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}}+\frac {1260 i}{\sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}}\right )\right ) \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}}{\sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}}}+1}{\sqrt {3}}\right )}{36 \left (-101+3 \sqrt {1137}\right )^{2/3} \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}+\frac {\left (4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+\frac {i \sqrt {\frac {6}{8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}} \left (6\ 2^{2/3} \left (35-\sqrt {1137}\right )-3 \left (31-\sqrt {1137}\right ) \sqrt [3]{-202+6 \sqrt {1137}}\right )}{\left (-101+3 \sqrt {1137}\right )^{2/3}}\right ) \arctan \left (\frac {\frac {2\ 2^{2/3} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}}{\sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}}}+1}{\sqrt {3}}\right )}{12 \sqrt {3} \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}+\frac {\left (2 \sqrt [3]{2}-\frac {2\ 2^{2/3}}{\sqrt [3]{-101+3 \sqrt {1137}}}+\sqrt [3]{-101+3 \sqrt {1137}}\right ) \sqrt [3]{-\frac {1}{4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}}} \log \left (\sqrt [3]{2} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}+\sqrt [3]{-4+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}}\right )}{18 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}-\frac {\left (\frac {16 i \sqrt [6]{2} \sqrt {3}}{\left (-101+3 \sqrt {1137}\right )^{2/3}}+\frac {8 i 2^{5/6} \sqrt {3}}{\sqrt [3]{-101+3 \sqrt {1137}}}+i 2^{5/6} \sqrt {3} \left (-101+3 \sqrt {1137}\right )^{2/3}-4 \left (1-\sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}\right ) \sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}-\sqrt [6]{2} \sqrt [3]{-101+3 \sqrt {1137}} \left (4 i \sqrt {3}+\sqrt {2 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )\right ) \log \left (\sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}}-2^{2/3} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}\right )}{36 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}} \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}+\frac {\left (4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+\frac {i \sqrt {\frac {6}{8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}} \left (6\ 2^{2/3} \left (35-\sqrt {1137}\right )-3 \left (31-\sqrt {1137}\right ) \sqrt [3]{-202+6 \sqrt {1137}}\right )}{\left (-101+3 \sqrt {1137}\right )^{2/3}}\right ) \log \left (\sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}}-2^{2/3} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}\right )}{36 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}+\frac {\left (\frac {16 i \sqrt [6]{2} \sqrt {3}}{\left (-101+3 \sqrt {1137}\right )^{2/3}}+\frac {8 i 2^{5/6} \sqrt {3}}{\sqrt [3]{-101+3 \sqrt {1137}}}+i 2^{5/6} \sqrt {3} \left (-101+3 \sqrt {1137}\right )^{2/3}-4 \left (1-\sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}\right ) \sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}-\sqrt [6]{2} \sqrt [3]{-101+3 \sqrt {1137}} \left (4 i \sqrt {3}+\sqrt {2 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )\right ) \log \left (2 \sqrt [3]{2} \left (3 \left (x+\frac {1}{3}\right )+2\right )^{2/3}+2^{2/3} \sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}+\left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )^{2/3}\right )}{72 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt {8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}} \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}-i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}-\frac {\left (4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+\frac {i \sqrt {\frac {6}{8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}}} \left (6\ 2^{2/3} \left (35-\sqrt {1137}\right )-3 \left (31-\sqrt {1137}\right ) \sqrt [3]{-202+6 \sqrt {1137}}\right )}{\left (-101+3 \sqrt {1137}\right )^{2/3}}\right ) \log \left (2 \sqrt [3]{2} \left (3 \left (x+\frac {1}{3}\right )+2\right )^{2/3}+2^{2/3} \sqrt [3]{8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}+\left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )^{2/3}\right )}{72 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right ) \sqrt [3]{2 \left (8+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}+i \sqrt {6 \left (8+8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}+\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}}-\frac {\left (2 \sqrt [3]{2}-\frac {2\ 2^{2/3}}{\sqrt [3]{-101+3 \sqrt {1137}}}+\sqrt [3]{-101+3 \sqrt {1137}}\right ) \sqrt [3]{-\frac {1}{4-4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}+2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}}} \log \left (2^{2/3} \left (3 \left (x+\frac {1}{3}\right )+2\right )^{2/3}-\sqrt [3]{2 \left (-4+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}\right )} \sqrt [3]{3 \left (x+\frac {1}{3}\right )+2}+\left (-4+4 \sqrt [3]{\frac {2}{-101+3 \sqrt {1137}}}-2^{2/3} \sqrt [3]{-101+3 \sqrt {1137}}\right )^{2/3}\right )}{36 \left (4-8 \left (\frac {2}{-101+3 \sqrt {1137}}\right )^{2/3}-\sqrt [3]{2} \left (-101+3 \sqrt {1137}\right )^{2/3}\right )}\right )}{(x+1)^{2/3}}\) |
Input:
Int[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]
Output:
(36*3^(1/3)*((1 + x)^2)^(1/3)*(((2*2^(1/3) - (2*2^(2/3))/(-101 + 3*Sqrt[11 37])^(1/3) + (-101 + 3*Sqrt[1137])^(1/3))*(-(4 - 4*(2/(-101 + 3*Sqrt[1137] ))^(1/3) + 2^(2/3)*(-101 + 3*Sqrt[1137])^(1/3))^(-1))^(1/3)*ArcTan[(1 - 2* (-2/(4 - 4*(2/(-101 + 3*Sqrt[1137]))^(1/3) + 2^(2/3)*(-101 + 3*Sqrt[1137]) ^(1/3)))^(1/3)*(2 + 3*(1/3 + x))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(4 - 8*(2/(-1 01 + 3*Sqrt[1137]))^(2/3) - 2^(1/3)*(-101 + 3*Sqrt[1137])^(2/3))) + ((4*Sq rt[3]*(-101 + 3*Sqrt[1137])^(2/3) + ((9*I)*2^(5/6)*(31 - Sqrt[1137])*(-101 + 3*Sqrt[1137])^(1/3))/Sqrt[8 + 8*(2/(-101 + 3*Sqrt[1137]))^(2/3) + 2^(1/ 3)*(-101 + 3*Sqrt[1137])^(2/3)] - 2^(1/6)*(101*Sqrt[6] - 9*Sqrt[758] + 4*2 ^(1/6)*Sqrt[3]*(-101 + 3*Sqrt[1137])^(1/3) - (36*I)*Sqrt[1137/(8 + 8*(2/(- 101 + 3*Sqrt[1137]))^(2/3) + 2^(1/3)*(-101 + 3*Sqrt[1137])^(2/3))] + (1260 *I)/Sqrt[8 + 8*(2/(-101 + 3*Sqrt[1137]))^(2/3) + 2^(1/3)*(-101 + 3*Sqrt[11 37])^(2/3)]))*ArcTan[(1 + (2*2^(2/3)*(2 + 3*(1/3 + x))^(1/3))/(8 + 4*(2/(- 101 + 3*Sqrt[1137]))^(1/3) - 2^(2/3)*(-101 + 3*Sqrt[1137])^(1/3) - I*Sqrt[ 6*(8 + 8*(2/(-101 + 3*Sqrt[1137]))^(2/3) + 2^(1/3)*(-101 + 3*Sqrt[1137])^( 2/3))])^(1/3))/Sqrt[3]])/(36*(-101 + 3*Sqrt[1137])^(2/3)*(4 - 8*(2/(-101 + 3*Sqrt[1137]))^(2/3) - 2^(1/3)*(-101 + 3*Sqrt[1137])^(2/3))*(2*(8 + 4*(2/ (-101 + 3*Sqrt[1137]))^(1/3) - 2^(2/3)*(-101 + 3*Sqrt[1137])^(1/3) - I*Sqr t[6*(8 + 8*(2/(-101 + 3*Sqrt[1137]))^(2/3) + 2^(1/3)*(-101 + 3*Sqrt[1137]) ^(2/3))]))^(1/3)) + ((4 - 4*(2/(-101 + 3*Sqrt[1137]))^(1/3) + 2^(2/3)*(...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Simp[((a + b*x)^Exp on[Px, x])^p/(a + b*x)^(Expon[Px, x]*p) Int[u*(a + b*x)^(Expon[Px, x]*p), x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; !IntegerQ[p] && PolyQ[Px, x ] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} , Simp[1/d^(2*p) Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d *(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Fre eQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Timed out.
\[\int \frac {\left (x^{2}+2 x +1\right )^{\frac {1}{3}}}{x^{3}+x^{2}+x +4}d x\]
Input:
int((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x)
Output:
int((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.94 (sec) , antiderivative size = 3149, normalized size of antiderivative = 43.74 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\text {Too large to display} \] Input:
integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="fricas")
Output:
Too large to include
Not integrable
Time = 2.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\int \frac {\sqrt [3]{\left (x + 1\right )^{2}}}{x^{3} + x^{2} + x + 4}\, dx \] Input:
integrate((x**2+2*x+1)**(1/3)/(x**3+x**2+x+4),x)
Output:
Integral(((x + 1)**2)**(1/3)/(x**3 + x**2 + x + 4), x)
Not integrable
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\int { \frac {{\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{3} + x^{2} + x + 4} \,d x } \] Input:
integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="maxima")
Output:
integrate((x^2 + 2*x + 1)^(1/3)/(x^3 + x^2 + x + 4), x)
Not integrable
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\int { \frac {{\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{3} + x^{2} + x + 4} \,d x } \] Input:
integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="giac")
Output:
integrate((x^2 + 2*x + 1)^(1/3)/(x^3 + x^2 + x + 4), x)
Not integrable
Time = 4.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\int \frac {{\left (x^2+2\,x+1\right )}^{1/3}}{x^3+x^2+x+4} \,d x \] Input:
int((2*x + x^2 + 1)^(1/3)/(x + x^2 + x^3 + 4),x)
Output:
int((2*x + x^2 + 1)^(1/3)/(x + x^2 + x^3 + 4), x)
Not integrable
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\int \frac {\left (x^{2}+2 x +1\right )^{\frac {1}{3}}}{x^{3}+x^{2}+x +4}d x \] Input:
int((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x)
Output:
int((x**2 + 2*x + 1)**(1/3)/(x**3 + x**2 + x + 4),x)