\(\int \frac {3+x}{\sqrt [3]{-1+x^2} (5-x+2 x^2)} \, dx\) [2504]

Optimal result

Integrand size = 25, antiderivative size = 209 \[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\frac {\arctan \left (\frac {3^{5/6} \sqrt [3]{-1+x^2}}{2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{3} \sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{2} \sqrt [6]{3}}+\frac {\log \left (-\sqrt [3]{2} 3^{2/3}+\sqrt [3]{2} 3^{2/3} x+3 \sqrt [3]{-1+x^2}\right )}{\sqrt [3]{2} 3^{2/3}}-\frac {\log \left (2^{2/3} \sqrt [3]{3}-2\ 2^{2/3} \sqrt [3]{3} x+2^{2/3} \sqrt [3]{3} x^2+\left (\sqrt [3]{2} 3^{2/3}-\sqrt [3]{2} 3^{2/3} x\right ) \sqrt [3]{-1+x^2}+3 \left (-1+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2} 3^{2/3}} \]

[Out]

1/6*arctan(3^(5/6)*(x^2-1)^(1/3)/(2*2^(1/3)-2*2^(1/3)*x+3^(1/3)*(x^2-1)^(1/3)))*2^(2/3)*3^(5/6)+1/6*ln(-2^(1/3
)*3^(2/3)+2^(1/3)*3^(2/3)*x+3*(x^2-1)^(1/3))*2^(2/3)*3^(1/3)-1/12*ln(2^(2/3)*3^(1/3)-2*2^(2/3)*3^(1/3)*x+2^(2/
3)*3^(1/3)*x^2+(2^(1/3)*3^(2/3)-2^(1/3)*3^(2/3)*x)*(x^2-1)^(1/3)+3*(x^2-1)^(2/3))*2^(2/3)*3^(1/3)

Rubi [F]

\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \]

[In]

Int[(3 + x)/((-1 + x^2)^(1/3)*(5 - x + 2*x^2)),x]

[Out]

Defer[Int][(3 + x)/((-1 + x^2)^(1/3)*(5 - x + 2*x^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.89 \[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {3^{5/6} \sqrt [3]{-1+x^2}}{2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{3} \sqrt [3]{-1+x^2}}\right )+2 \log \left (-\sqrt [3]{2} 3^{2/3}+\sqrt [3]{2} 3^{2/3} x+3 \sqrt [3]{-1+x^2}\right )-\log \left (2^{2/3} \sqrt [3]{3}-2\ 2^{2/3} \sqrt [3]{3} x+2^{2/3} \sqrt [3]{3} x^2-\sqrt [3]{2} 3^{2/3} (-1+x) \sqrt [3]{-1+x^2}+3 \left (-1+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2} 3^{2/3}} \]

[In]

Integrate[(3 + x)/((-1 + x^2)^(1/3)*(5 - x + 2*x^2)),x]

[Out]

(2*Sqrt[3]*ArcTan[(3^(5/6)*(-1 + x^2)^(1/3))/(2*2^(1/3) - 2*2^(1/3)*x + 3^(1/3)*(-1 + x^2)^(1/3))] + 2*Log[-(2
^(1/3)*3^(2/3)) + 2^(1/3)*3^(2/3)*x + 3*(-1 + x^2)^(1/3)] - Log[2^(2/3)*3^(1/3) - 2*2^(2/3)*3^(1/3)*x + 2^(2/3
)*3^(1/3)*x^2 - 2^(1/3)*3^(2/3)*(-1 + x)*(-1 + x^2)^(1/3) + 3*(-1 + x^2)^(2/3)])/(2*2^(1/3)*3^(2/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.49 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.35

method	result	size
trager	\(\text {Expression too large to display}\)	\(910\)

[In]

int((3+x)/(x^2-1)^(1/3)/(2*x^2-x+5),x,method=_RETURNVERBOSE)

[Out]

1/6*RootOf(_Z^3-12)*ln(-(1818*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x^2+3
*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x^2-5454*RootOf(RootOf(_Z^3-12)^2+6*
_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x-9*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*Roo
tOf(_Z^3-12)^3*x-1215*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^2*(x^2-1)^(2/3)-3
006*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)*(x^2-1)^(1/3)*x-405*RootOf(_Z^3-12)
^2*(x^2-1)^(1/3)*x+3006*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)*(x^2-1)^(1/3)+1
212*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x^2+405*RootOf(_Z^3-12)^2*(x^2-1)^(1/3)+2*RootOf(_Z
^3-12)*x^2-7878*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x-13*RootOf(_Z^3-12)*x+576*(x^2-1)^(2/3
)-4242*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)-7*RootOf(_Z^3-12))/(2*x^2-x+5))+RootOf(RootOf(_Z
^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*ln((18*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(
_Z^3-12)^2*x^2+303*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x^2-54*RootOf(Root
Of(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x-909*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^
3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x-1215*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)
^2*(x^2-1)^(2/3)+576*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)*(x^2-1)^(1/3)*x-40
5*RootOf(_Z^3-12)^2*(x^2-1)^(1/3)*x-576*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)
*(x^2-1)^(1/3)+24*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x^2+405*RootOf(_Z^3-12)^2*(x^2-1)^(1/
3)+404*RootOf(_Z^3-12)*x^2-30*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x-505*RootOf(_Z^3-12)*x-3
006*(x^2-1)^(2/3)+42*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)+707*RootOf(_Z^3-12))/(2*x^2-x+5))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (154) = 308\).

Time = 5.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.50 \[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=-\frac {1}{18} \cdot 18^{\frac {1}{6}} \sqrt {6} \arctan \left (\frac {18^{\frac {1}{6}} {\left (6 \cdot 18^{\frac {2}{3}} \sqrt {6} {\left (8 \, x^{4} - 26 \, x^{3} + 33 \, x^{2} - 56 \, x + 5\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} \sqrt {6} {\left (8 \, x^{6} + 96 \, x^{5} - 582 \, x^{4} + 155 \, x^{3} + 1029 \, x^{2} - 399 \, x - 91\right )} + 36 \, \sqrt {6} {\left (4 \, x^{5} - 62 \, x^{4} + 133 \, x^{3} - 31 \, x^{2} - 73 \, x + 29\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}\right )}}{18 \, {\left (8 \, x^{6} - 336 \, x^{5} + 1038 \, x^{4} - 709 \, x^{3} - 483 \, x^{2} + 897 \, x - 199\right )}}\right ) - \frac {1}{108} \cdot 18^{\frac {2}{3}} \log \left (\frac {3 \cdot 18^{\frac {2}{3}} {\left (4 \, x^{2} - 11 \, x + 1\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} {\left (4 \, x^{4} - 58 \, x^{3} + 75 \, x^{2} + 44 \, x - 29\right )} - 36 \, {\left (x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{4 \, x^{4} - 4 \, x^{3} + 21 \, x^{2} - 10 \, x + 25}\right ) + \frac {1}{54} \cdot 18^{\frac {2}{3}} \log \left (\frac {18^{\frac {2}{3}} {\left (2 \, x^{2} - x + 5\right )} + 18 \cdot 18^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 54 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2} - x + 5}\right ) \]

[In]

integrate((3+x)/(x^2-1)^(1/3)/(2*x^2-x+5),x, algorithm="fricas")

[Out]

-1/18*18^(1/6)*sqrt(6)*arctan(1/18*18^(1/6)*(6*18^(2/3)*sqrt(6)*(8*x^4 - 26*x^3 + 33*x^2 - 56*x + 5)*(x^2 - 1)
^(2/3) + 18^(1/3)*sqrt(6)*(8*x^6 + 96*x^5 - 582*x^4 + 155*x^3 + 1029*x^2 - 399*x - 91) + 36*sqrt(6)*(4*x^5 - 6
2*x^4 + 133*x^3 - 31*x^2 - 73*x + 29)*(x^2 - 1)^(1/3))/(8*x^6 - 336*x^5 + 1038*x^4 - 709*x^3 - 483*x^2 + 897*x
- 199)) - 1/108*18^(2/3)*log((3*18^(2/3)*(4*x^2 - 11*x + 1)*(x^2 - 1)^(2/3) + 18^(1/3)*(4*x^4 - 58*x^3 + 75*x
^2 + 44*x - 29) - 36*(x^3 - 6*x^2 + 3*x + 2)*(x^2 - 1)^(1/3))/(4*x^4 - 4*x^3 + 21*x^2 - 10*x + 25)) + 1/54*18^
(2/3)*log((18^(2/3)*(2*x^2 - x + 5) + 18*18^(1/3)*(x^2 - 1)^(1/3)*(x - 1) + 54*(x^2 - 1)^(2/3))/(2*x^2 - x + 5
))

Sympy [F]

\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\int \frac {x + 3}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (2 x^{2} - x + 5\right )}\, dx \]

[In]

integrate((3+x)/(x**2-1)**(1/3)/(2*x**2-x+5),x)

[Out]

Integral((x + 3)/(((x - 1)*(x + 1))**(1/3)*(2*x**2 - x + 5)), x)

Maxima [F]

\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\int { \frac {x + 3}{{\left (2 \, x^{2} - x + 5\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((3+x)/(x^2-1)^(1/3)/(2*x^2-x+5),x, algorithm="maxima")

[Out]

integrate((x + 3)/((2*x^2 - x + 5)*(x^2 - 1)^(1/3)), x)

Giac [F]

\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\int { \frac {x + 3}{{\left (2 \, x^{2} - x + 5\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((3+x)/(x^2-1)^(1/3)/(2*x^2-x+5),x, algorithm="giac")

[Out]

integrate((x + 3)/((2*x^2 - x + 5)*(x^2 - 1)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx=\int \frac {x+3}{{\left (x^2-1\right )}^{1/3}\,\left (2\,x^2-x+5\right )} \,d x \]

[In]

int((x + 3)/((x^2 - 1)^(1/3)*(2*x^2 - x + 5)),x)

[Out]

int((x + 3)/((x^2 - 1)^(1/3)*(2*x^2 - x + 5)), x)