\(\int \frac {(-1-2 x+x^2) (-1+2 x+x^2)}{(1-x+2 x^2+x^3+x^4) \sqrt [3]{-x+x^5}} \, dx\) [1560]

Optimal result

Integrand size = 46, antiderivative size = 107 \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2-2 x^2+\sqrt [3]{-x+x^5}}\right )+\log \left (1+x^2+\sqrt [3]{-x+x^5}\right )-\frac {1}{2} \log \left (1+2 x^2+x^4+\left (-1-x^2\right ) \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]

[Out]

3^(1/2)*arctan(3^(1/2)*(x^5-x)^(1/3)/(-2-2*x^2+(x^5-x)^(1/3)))+ln(1+x^2+(x^5-x)^(1/3))-1/2*ln(1+2*x^2+x^4+(-x^
2-1)*(x^5-x)^(1/3)+(x^5-x)^(2/3))

Rubi [F]

\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \]

[In]

Int[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4)*(-x + x^5)^(1/3)),x]

[Out]

(3*x*(1 - x^4)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^4])/(2*(-x + x^5)^(1/3)) + (3*x^(1/3)*(-1 + x^4)^(1/3)
*Defer[Subst][Defer[Int][x^4/((-1 + x^12)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^12)), x], x, x^(1/3)])/(-x + x^5)^(
1/3) - (24*x^(1/3)*(-1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^7/((-1 + x^12)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^
12)), x], x, x^(1/3)])/(-x + x^5)^(1/3) - (3*x^(1/3)*(-1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^10/((-1 + x^12
)^(1/3)*(1 - x^3 + 2*x^6 + x^9 + x^12)), x], x, x^(1/3)])/(-x + x^5)^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (1-x+2 x^2+x^3+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {1-6 x^2+x^4}{\sqrt [3]{x} \sqrt [3]{-1+x^4} \left (1-x+2 x^2+x^3+x^4\right )} \, dx}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x \left (1-6 x^6+x^{12}\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \left (\frac {x}{\sqrt [3]{-1+x^{12}}}+\frac {x^4 \left (1-8 x^3-x^6\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \left (1-8 x^3-x^6\right )}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}-\frac {8 x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}-\frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (24 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ & = \frac {3 x \sqrt [3]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^4\right )}{2 \sqrt [3]{-x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}-\frac {\left (24 \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-x^3+2 x^6+x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \]

[In]

Integrate[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4)*(-x + x^5)^(1/3)),x]

[Out]

Integrate[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4)*(-x + x^5)^(1/3)), x]

Maple [C] (verified)

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

Time = 3.64 (sec) , antiderivative size = 782, normalized size of antiderivative = 7.31

method	result	size
trager	\(\text {Expression too large to display}\)	\(782\)

[In]

int((x^2-2*x-1)*(x^2+2*x-1)/(x^4+x^3+2*x^2-x+1)/(x^5-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2+_Z+1)*ln((-215628532170574272*RootOf(_Z^2+_Z+1)^2*x^4+898452217377392800*RootOf(_Z^2+_Z+1)^2*x^3+8
12973192552633838*RootOf(_Z^2+_Z+1)*x^4-2253715023870923763*(x^5-x)^(1/3)*RootOf(_Z^2+_Z+1)*x^2-43125706434114
8544*RootOf(_Z^2+_Z+1)^2*x^2-326661081770322853*RootOf(_Z^2+_Z+1)*x^3-412140106595081815*x^4+22537150238709237
63*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-832090150298700567*(x^5-x)^(1/3)*x^2-898452217377392800*RootOf(_Z^2+_Z+1)^2
*x+1625946385105267676*RootOf(_Z^2+_Z+1)*x^2-130149507345815310*x^3+832090150298700567*(x^5-x)^(2/3)-225371502
3870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)-215628532170574272*RootOf(_Z^2+_Z+1)^2+326661081770322853*RootOf(_Z
^2+_Z+1)*x-824280213190163630*x^2-832090150298700567*(x^5-x)^(1/3)+812973192552633838*RootOf(_Z^2+_Z+1)+130149
507345815310*x-412140106595081815)/(x^4+x^3+2*x^2-x+1))-ln(-(215628532170574272*RootOf(_Z^2+_Z+1)^2*x^4-898452
217377392800*RootOf(_Z^2+_Z+1)^2*x^3+1244230256893782382*RootOf(_Z^2+_Z+1)*x^4-2253715023870923763*(x^5-x)^(1/
3)*RootOf(_Z^2+_Z+1)*x^2+431257064341148544*RootOf(_Z^2+_Z+1)^2*x^2-2123565516525108453*RootOf(_Z^2+_Z+1)*x^3+
1440741831318289925*x^4+2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-1421624873572223196*(x^5-x)^(1/3)*
x^2+898452217377392800*RootOf(_Z^2+_Z+1)^2*x+2488460513787564764*RootOf(_Z^2+_Z+1)*x^2-1094963791801900343*x^3
+1421624873572223196*(x^5-x)^(2/3)-2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)+215628532170574272*Root
Of(_Z^2+_Z+1)^2+2123565516525108453*RootOf(_Z^2+_Z+1)*x+2881483662636579850*x^2-1421624873572223196*(x^5-x)^(1
/3)+1244230256893782382*RootOf(_Z^2+_Z+1)+1094963791801900343*x+1440741831318289925)/(x^4+x^3+2*x^2-x+1))*Root
Of(_Z^2+_Z+1)-ln(-(215628532170574272*RootOf(_Z^2+_Z+1)^2*x^4-898452217377392800*RootOf(_Z^2+_Z+1)^2*x^3+12442
30256893782382*RootOf(_Z^2+_Z+1)*x^4-2253715023870923763*(x^5-x)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+43125706434114854
4*RootOf(_Z^2+_Z+1)^2*x^2-2123565516525108453*RootOf(_Z^2+_Z+1)*x^3+1440741831318289925*x^4+225371502387092376
3*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-1421624873572223196*(x^5-x)^(1/3)*x^2+898452217377392800*RootOf(_Z^2+_Z+1)^2
*x+2488460513787564764*RootOf(_Z^2+_Z+1)*x^2-1094963791801900343*x^3+1421624873572223196*(x^5-x)^(2/3)-2253715
023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3)+215628532170574272*RootOf(_Z^2+_Z+1)^2+2123565516525108453*RootOf
(_Z^2+_Z+1)*x+2881483662636579850*x^2-1421624873572223196*(x^5-x)^(1/3)+1244230256893782382*RootOf(_Z^2+_Z+1)+
1094963791801900343*x+1440741831318289925)/(x^4+x^3+2*x^2-x+1))

Fricas [A] (verification not implemented)

none

Time = 1.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {541310 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + \sqrt {3} {\left (311575 \, x^{4} + 193471 \, x^{3} + 623150 \, x^{2} - 193471 \, x + 311575\right )} + 777518 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{3 \, {\left (166375 \, x^{4} - 493039 \, x^{3} + 332750 \, x^{2} + 493039 \, x + 166375\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + x^{3} + 2 \, x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} - x + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{3} + 2 \, x^{2} - x + 1}\right ) \]

[In]

integrate((x^2-2*x-1)*(x^2+2*x-1)/(x^4+x^3+2*x^2-x+1)/(x^5-x)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan(1/3*(541310*sqrt(3)*(x^5 - x)^(1/3)*(x^2 + 1) + sqrt(3)*(311575*x^4 + 193471*x^3 + 623150*x^2
- 193471*x + 311575) + 777518*sqrt(3)*(x^5 - x)^(2/3))/(166375*x^4 - 493039*x^3 + 332750*x^2 + 493039*x + 1663
75)) + 1/2*log((x^4 + x^3 + 2*x^2 + 3*(x^5 - x)^(1/3)*(x^2 + 1) - x + 3*(x^5 - x)^(2/3) + 1)/(x^4 + x^3 + 2*x^
2 - x + 1))

Sympy [F]

\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {\left (x^{2} - 2 x - 1\right ) \left (x^{2} + 2 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + x^{3} + 2 x^{2} - x + 1\right )}\, dx \]

[In]

integrate((x**2-2*x-1)*(x**2+2*x-1)/(x**4+x**3+2*x**2-x+1)/(x**5-x)**(1/3),x)

[Out]

Integral((x**2 - 2*x - 1)*(x**2 + 2*x - 1)/((x*(x - 1)*(x + 1)*(x**2 + 1))**(1/3)*(x**4 + x**3 + 2*x**2 - x +
1)), x)

Maxima [F]

\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {{\left (x^{2} + 2 \, x - 1\right )} {\left (x^{2} - 2 \, x - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} + x^{3} + 2 \, x^{2} - x + 1\right )}} \,d x } \]

[In]

integrate((x^2-2*x-1)*(x^2+2*x-1)/(x^4+x^3+2*x^2-x+1)/(x^5-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^2 + 2*x - 1)*(x^2 - 2*x - 1)/((x^5 - x)^(1/3)*(x^4 + x^3 + 2*x^2 - x + 1)), x)

Giac [F]

\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {{\left (x^{2} + 2 \, x - 1\right )} {\left (x^{2} - 2 \, x - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} + x^{3} + 2 \, x^{2} - x + 1\right )}} \,d x } \]

[In]

integrate((x^2-2*x-1)*(x^2+2*x-1)/(x^4+x^3+2*x^2-x+1)/(x^5-x)^(1/3),x, algorithm="giac")

[Out]

integrate((x^2 + 2*x - 1)*(x^2 - 2*x - 1)/((x^5 - x)^(1/3)*(x^4 + x^3 + 2*x^2 - x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\int \frac {\left (x^2+2\,x-1\right )\,\left (-x^2+2\,x+1\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^4+x^3+2\,x^2-x+1\right )} \,d x \]

[In]

int(-((2*x + x^2 - 1)*(2*x - x^2 + 1))/((x^5 - x)^(1/3)*(2*x^2 - x + x^3 + x^4 + 1)),x)

[Out]

-int(((2*x + x^2 - 1)*(2*x - x^2 + 1))/((x^5 - x)^(1/3)*(2*x^2 - x + x^3 + x^4 + 1)), x)