\(\int \frac {1+x^2+x^3}{(-1+x^2+x^3) \sqrt [3]{x^2+x^3}} \, dx\) [2826]

Optimal result

Integrand size = 30, antiderivative size = 285 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]+2 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {2 \log (x) \text {$\#$1}-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

(Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(x^2 + x^3)^(1/3) - (3*x^(2/3)
*(1 + x)^(1/3)*Log[x^(1/3) - (1 + x)^(1/3)])/(2*(x^2 + x^3)^(1/3)) + (6*x^(2/3)*(1 + x)^(1/3)*Defer[Subst][Def
er[Int][1/((1 + x^3)^(1/3)*(-1 + x^6 + x^9)), x], x, x^(1/3)])/(x^2 + x^3)^(1/3)

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-2 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+\log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )+4 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]-\frac {4}{3} \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-2 \log (x) \text {$\#$1}+6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}+\log (x) \text {$\#$1}^2-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2+2 \log (x) \text {$\#$1}^4-6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ]\right )}{2 \sqrt [3]{x^2 (1+x)}} \]

[In]

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

[Out]

(x^(2/3)*(1 + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] - 2*Log[-x^(1/3) + (1
+ x)^(1/3)] + Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)] + 4*RootSum[-1 - #1 + #1^3 & , (-(Log[x^(1/
3)]*#1) + Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1)/(-1 + 3*#1^2) & ] - (4*RootSum[1 - #1 + #1^2 - 2*#1^3 + #1^4 + #
1^6 & , (-2*Log[x]*#1 + 6*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1 + Log[x]*#1^2 - 3*Log[(1 + x)^(1/3) - x^(1/3)*#1]
*#1^2 + 2*Log[x]*#1^4 - 6*Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^4)/(-1 + 2*#1 - 6*#1^2 + 4*#1^3 + 6*#1^5) & ])/3)
)/(2*(x^2*(1 + x))^(1/3))

Maple [N/A] (verified)

Time = 114.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-\textit {\_Z} -1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{3 \textit {\_R}^{2}-1}\right )+\frac {\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R}^{3}+\textit {\_R} -2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{6 \textit {\_R}^{5}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )-\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )\)	\(200\)
trager	\(\text {Expression too large to display}\)	\(400598\)

[In]

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

[Out]

2*sum(_R*ln((-_R*x+(x^2*(1+x))^(1/3))/x)/(3*_R^2-1),_R=RootOf(_Z^3-_Z-1))+1/2*ln(((x^2*(1+x))^(2/3)+(x^2*(1+x)
)^(1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3*(2*(x^2*(1+x))^(1/3)+x)*3^(1/2)/x)+2*sum(_R*(2*_R^3+_R-2)*ln((-_R*x+(x^
2*(1+x))^(1/3))/x)/(6*_R^5+4*_R^3-6*_R^2+2*_R-1),_R=RootOf(_Z^6+_Z^4-2*_Z^3+_Z^2-_Z+1))-ln(((x^2*(1+x))^(1/3)-
x)/x)

Fricas [C] (verification not implemented)

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

Time = 5.60 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.69 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/138*(3*sqrt(23)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16
) - 69*(100/4761*sqrt(69) + 4/23)^(1/3) + 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log(1/69*(3*x*(69*(100/4761*sqrt
(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 46*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(10
0/4761*sqrt(69) + 4/23)^(1/3)) + 3*(3*sqrt(23)*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) +
4/23)^(1/3)) - 46*sqrt(23)*x)*sqrt(-1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^
(1/3))^2 - 16) + 1656*x + 13800*(x^3 + x^2)^(1/3))/x) - 1/138*(3*sqrt(23)*sqrt(-1/69*(69*(100/4761*sqrt(69) +
4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16) + 69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*
sqrt(69) + 4/23)^(1/3))*log(1/69*(3*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3
))^2 + 46*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) - 3*(3*sqrt(23)*x*(69*(
100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) - 46*sqrt(23)*x)*sqrt(-1/69*(69*(100/476
1*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 - 16) + 1656*x + 13800*(x^3 + x^2)^(1/3))/x)
+ 1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log(-1/69*(3*x*(69*(100/4761
*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 46*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) -
4/(100/4761*sqrt(69) + 4/23)^(1/3)) + 1656*x - 6900*(x^3 + x^2)^(1/3))/x) + 1/46*(sqrt(23)*sqrt(-0.46834333995
83197? + 0.?e-71*I) + 34.26254145273487? + 0.?e-71*I)*log(-((71.2918670130777? + 0.?e-71*I)*sqrt(23)*sqrt(-0.4
683433399583197? + 0.?e-71*I)*x + (654.502264627727? + 0.?e-69*I)*x - 800*(x^3 + x^2)^(1/3))/x) - 1/46*(sqrt(2
3)*sqrt(-0.4683433399583197? + 0.?e-71*I) - 34.26254145273487? + 0.?e-71*I)*log(-(-(71.2918670130777? + 0.?e-3
3*I)*sqrt(23)*sqrt(-0.4683433399583197? + 0.?e-71*I)*x + (654.502264627727? + 0.?e-69*I)*x - 800*(x^3 + x^2)^(
1/3))/x) - sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) - (0.3106288296404671? + 0.53802491
52329462?*I)*log(((264.9435914489492? - 458.8957615293512?*I)*x + 400*(x^3 + x^2)^(1/3))/x) - (0.4342090280276
823? + 0.6093739760383123?*I)*log(((62.30754086491403? - 341.9037831771358?*I)*x + 400*(x^3 + x^2)^(1/3))/x) -
log(-(x - (x^3 + x^2)^(1/3))/x) - (0.4342090280276823? - 0.6093739760383123?*I)*log(-(-(62.30754086491403? +
341.9037831771358?*I)*x - 400*(x^3 + x^2)^(1/3))/x) - (0.3106288296404671? - 0.5380249152329462?*I)*log(-(-(26
4.9435914489492? + 458.8957615293512?*I)*x - 400*(x^3 + x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 + x^2)^(1/3)*x +
(x^3 + x^2)^(2/3))/x^2)

Sympy [N/A]

Not integrable

Time = 13.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^{3} + x^{2} + 1}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{3} + x^{2} - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.85 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{3} + x^{2} + 1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^3+x^2+1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+x^2-1\right )} \,d x \]

[In]

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

[Out]

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