\(\int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx\) [1315]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=-\frac {3 \sqrt [3]{1+x+x^3}}{x}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x+x^3}}\right )-\log \left (-x+\sqrt [3]{1+x+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x+x^3}+\left (1+x+x^3\right )^{2/3}\right ) \]

[Out]

-3*(x^3+x+1)^(1/3)/x-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3+x+1)^(1/3)))-ln(-x+(x^3+x+1)^(1/3))+1/2*ln(x^2+x*(x^3+
x+1)^(1/3)+(x^3+x+1)^(2/3))

Rubi [F]

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

[In]

Int[((3 + 2*x)*(1 + x + x^3)^(1/3))/(x^2*(1 + x)),x]

[Out]

(9*2^(2/3)*(1 + x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3)
+ x)^(1/3)*((6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt
[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/x^2, x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(
1/3) - (2*(-9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqr
t[93]))^(2/3) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3)) - (3*2^(2
/3)*(1 + x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3) + x)^(1
/3)*((6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt[93]))^
(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/x, x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(1/3) - (2
*(-9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(
2/3) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3)) + (3*2^(2/3)*(1 +
x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3) + x)^(1/3)*((6 +
6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt[93]))^(1/3) - (
(-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/(1 + x), x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-
9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3
) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \sqrt [3]{1+x+x^3}}{x^2}-\frac {\sqrt [3]{1+x+x^3}}{x}+\frac {\sqrt [3]{1+x+x^3}}{1+x}\right ) \, dx \\ & = 3 \int \frac {\sqrt [3]{1+x+x^3}}{x^2} \, dx-\int \frac {\sqrt [3]{1+x+x^3}}{x} \, dx+\int \frac {\sqrt [3]{1+x+x^3}}{1+x} \, dx \\ & = -\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{1+x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\left (3 \sqrt [3]{1+x+x^3}\right ) \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x^2} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=-\frac {3 \sqrt [3]{1+x+x^3}}{x}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x+x^3}}\right )-\log \left (-x+\sqrt [3]{1+x+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x+x^3}+\left (1+x+x^3\right )^{2/3}\right ) \]

[In]

Integrate[((3 + 2*x)*(1 + x + x^3)^(1/3))/(x^2*(1 + x)),x]

[Out]

(-3*(1 + x + x^3)^(1/3))/x - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x + x^3)^(1/3))] - Log[-x + (1 + x + x^3)^
(1/3)] + Log[x^2 + x*(1 + x + x^3)^(1/3) + (1 + x + x^3)^(2/3)]/2

Maple [A] (verified)

Time = 9.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+x +1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \ln \left (\frac {-x +\left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {x^{2}+x \left (x^{3}+x +1\right )^{\frac {1}{3}}+\left (x^{3}+x +1\right )^{\frac {2}{3}}}{x^{2}}\right ) x -6 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{2 x}\)	\(93\)
trager	\(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}-3 \ln \left (-\frac {198 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-279 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x -279 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-345 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +111 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +111 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+133 x^{3}-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-42 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -42 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+57 x +57}{1+x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {30006 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+37665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x +37665 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+27663 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-15003 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -1119 \left (x^{3}+x +1\right )^{\frac {2}{3}} x -1119 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+2215 x^{3}-15003 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+18675 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +18675 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1772 x +1772}{1+x}\right )+\ln \left (-\frac {198 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-279 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x -279 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-345 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +111 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +111 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+133 x^{3}-99 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-42 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -42 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+57 x +57}{1+x}\right )\)	\(581\)
risch	\(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-4 x^{6}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-12 x^{2} \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}-8 x^{4}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-8 x^{3}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -4 x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-8 x -4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-8 x^{6}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}-12 x^{4}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x +12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-12 x^{3}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -4 x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-8 x -4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )}{2}\right ) {\left (\left (x^{3}+x +1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{3}+x +1\right )^{\frac {2}{3}}}\)	\(755\)

[In]

int((2*x+3)*(x^3+x+1)^(1/3)/x^2/(1+x),x,method=_RETURNVERBOSE)

[Out]

1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3+x+1)^(1/3)))*x-2*ln((-x+(x^3+x+1)^(1/3))/x)*x+ln((x^2+x*(x^3+x+1
)^(1/3)+(x^3+x+1)^(2/3))/x^2)*x-6*(x^3+x+1)^(1/3))/x

Fricas [F(-2)]

Exception generated. \[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

\[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=\int \frac {\left (2 x + 3\right ) \sqrt [3]{x^{3} + x + 1}}{x^{2} \left (x + 1\right )}\, dx \]

[In]

integrate((3+2*x)*(x**3+x+1)**(1/3)/x**2/(1+x),x)

[Out]

Integral((2*x + 3)*(x**3 + x + 1)**(1/3)/(x**2*(x + 1)), x)

Maxima [F]

\[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=\int { \frac {{\left (x^{3} + x + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 3\right )}}{{\left (x + 1\right )} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((x^3 + x + 1)^(1/3)*(2*x + 3)/((x + 1)*x^2), x)

Giac [F]

\[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=\int { \frac {{\left (x^{3} + x + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 3\right )}}{{\left (x + 1\right )} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((x^3 + x + 1)^(1/3)*(2*x + 3)/((x + 1)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx=\int \frac {\left (2\,x+3\right )\,{\left (x^3+x+1\right )}^{1/3}}{x^2\,\left (x+1\right )} \,d x \]

[In]

int(((2*x + 3)*(x + x^3 + 1)^(1/3))/(x^2*(x + 1)),x)

[Out]

int(((2*x + 3)*(x + x^3 + 1)^(1/3))/(x^2*(x + 1)), x)