3.10.46 \(\int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx\) [946]

3.10.46.1 Optimal result

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
 

3.10.46.2 Mathematica [A] (verified)

Time = 0.07 (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)
 

3.10.46.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.73 (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)*(...
 

3.10.46.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1199

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]
 

rule 2008

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2485

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]
 

rule 2490

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]
 

3.10.46.4 Maple [F(-1)]

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)
 

3.10.46.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.91 (sec) , antiderivative size = 3331, normalized size of antiderivative = 46.26 \[ \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx=\text {Too large to display} \]

input

integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="fricas")
 

output

-1/574564*287282^(2/3)*(sqrt(379)*sqrt(-379/3*((1/287282)^(1/3)*(1867629*s 
qrt(1137) + 2438107)^(1/3) - 10439176*(1/287282)^(2/3)/(1867629*sqrt(1137) 
 + 2438107)^(1/3) + 2)^2 + 1516/3*(1/287282)^(1/3)*(1867629*sqrt(1137) + 2 
438107)^(1/3) - 15825790816/3*(1/287282)^(2/3)/(1867629*sqrt(1137) + 24381 
07)^(1/3) - 53572/3) - 379/3*(1/287282)^(1/3)*(1867629*sqrt(1137) + 243810 
7)^(1/3) + 3956447704/3*(1/287282)^(2/3)/(1867629*sqrt(1137) + 2438107)^(1 
/3) + 1516/3)^(1/3)*(sqrt(-3) + 1)*log(1/9*((1509557*287282^(2/3)*(sqrt(-3 
)*(x + 1) + x + 1)*((1/287282)^(1/3)*(1867629*sqrt(1137) + 2438107)^(1/3) 
- 10439176*(1/287282)^(2/3)/(1867629*sqrt(1137) + 2438107)^(1/3) + 2)^2 - 
11724744*287282^(2/3)*(sqrt(-3)*(x + 1) + x + 1)*((1/287282)^(1/3)*(186762 
9*sqrt(1137) + 2438107)^(1/3) - 10439176*(1/287282)^(2/3)/(1867629*sqrt(11 
37) + 2438107)^(1/3) + 2) + 44943336*287282^(2/3)*(sqrt(-3)*(x + 1) + x + 
1) + 3*(3983*287282^(2/3)*sqrt(379)*(sqrt(-3)*(x + 1) + x + 1)*((1/287282) 
^(1/3)*(1867629*sqrt(1137) + 2438107)^(1/3) - 10439176*(1/287282)^(2/3)/(1 
867629*sqrt(1137) + 2438107)^(1/3) + 2) + 7038*287282^(2/3)*sqrt(379)*(sqr 
t(-3)*(x + 1) + x + 1))*sqrt(-379/3*((1/287282)^(1/3)*(1867629*sqrt(1137) 
+ 2438107)^(1/3) - 10439176*(1/287282)^(2/3)/(1867629*sqrt(1137) + 2438107 
)^(1/3) + 2)^2 + 1516/3*(1/287282)^(1/3)*(1867629*sqrt(1137) + 2438107)^(1 
/3) - 15825790816/3*(1/287282)^(2/3)/(1867629*sqrt(1137) + 2438107)^(1/3) 
- 53572/3))*(sqrt(379)*sqrt(-379/3*((1/287282)^(1/3)*(1867629*sqrt(1137...
 

3.10.46.6 Sympy [N/A]

Not integrable

Time = 2.45 (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)
 

3.10.46.7 Maxima [N/A]

Not integrable

Time = 0.22 (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)
 

3.10.46.8 Giac [N/A]

Not integrable

Time = 0.28 (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)
 

3.10.46.9 Mupad [N/A]

Not integrable

Time = 5.51 (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)