\(\int \frac {-1-2 x+2 x^2}{(1-x+x^2) \sqrt {x+x^4}} \, dx\) [197]

Optimal result

Integrand size = 30, antiderivative size = 21 \[ \int \frac {-1-2 x+2 x^2}{\left (1-x+x^2\right ) \sqrt {x+x^4}} \, dx=-\frac {2 \sqrt {x+x^4}}{1-x+x^2} \]

[Out]

-2*(x^4+x)^(1/2)/(x^2-x+1)

Rubi [F]

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

[In]

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

[Out]

(2*x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3])*x)^2]*EllipticF[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt
[3])*x)], (2 + Sqrt[3])/4])/(3^(1/4)*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) + ((2*I)*Sqrt[3]
*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqr
t[x]])/(Sqrt[1 - I*Sqrt[3]]*Sqrt[x + x^4]) - ((2*I)*Sqrt[3]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((
Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(Sqrt[1 + I*Sqrt[3]]*Sqrt[x + x^4]) + ((2*I)
*Sqrt[3]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[1 + x^6]), x]
, x, Sqrt[x]])/(Sqrt[1 - I*Sqrt[3]]*Sqrt[x + x^4]) - ((2*I)*Sqrt[3]*Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[I
nt][1/((Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(Sqrt[1 + I*Sqrt[3]]*Sqrt[x + x^4])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {-1-2 x+2 x^2}{\sqrt {x} \left (1-x+x^2\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \left (\frac {2}{\sqrt {x} \sqrt {1+x^3}}-\frac {3}{\sqrt {x} \left (1-x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (1-x+x^2\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = -\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \int \left (\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt {x} \sqrt {1+x^3}}+\frac {2 i}{\sqrt {3} \sqrt {x} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx}{\sqrt {x+x^4}}+\frac {\left (4 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {2 x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt {x} \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}}-\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \left (-1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = \frac {2 x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (4 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (4 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {2 x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (4 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}-\frac {\left (4 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {1+i \sqrt {3}} \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1+x^6}}+\frac {1}{2 \sqrt {1+i \sqrt {3}} \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {2 x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-i \sqrt {3}} \sqrt {x+x^4}}+\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-i \sqrt {3}} \sqrt {x+x^4}}-\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {1+i \sqrt {3}} \sqrt {x+x^4}}-\frac {\left (2 i \sqrt {3} \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {1+i \sqrt {3}} \sqrt {x+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.89 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-1-2 x+2 x^2}{\left (1-x+x^2\right ) \sqrt {x+x^4}} \, dx=-\frac {2 x (1+x)}{\sqrt {x+x^4}} \]

[In]

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

[Out]

(-2*x*(1 + x))/Sqrt[x + x^4]

Maple [A] (verified)

Time = 4.75 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
gosper	\(-\frac {2 x \left (1+x \right )}{\sqrt {x^{4}+x}}\)	\(14\)
trager	\(-\frac {2 \sqrt {x^{4}+x}}{x^{2}-x +1}\)	\(20\)
default	\(-\frac {2 \left (x^{2}+x \right )}{\sqrt {\left (x^{2}-x +1\right ) \left (x^{2}+x \right )}}\)	\(24\)
elliptic	\(-\frac {2 \left (x^{2}+x \right )}{\sqrt {\left (x^{2}-x +1\right ) \left (x^{2}+x \right )}}\)	\(24\)

[In]

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

[Out]

-2*x*(1+x)/(x^4+x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-1-2 x+2 x^2}{\left (1-x+x^2\right ) \sqrt {x+x^4}} \, dx=-\frac {2 \, \sqrt {x^{4} + x}}{x^{2} - x + 1} \]

[In]

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

[Out]

-2*sqrt(x^4 + x)/(x^2 - x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-1-2 x+2 x^2}{\left (1-x+x^2\right ) \sqrt {x+x^4}} \, dx=-\frac {2\,\sqrt {x^4+x}}{x^2-x+1} \]

[In]

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

[Out]

-(2*(x + x^4)^(1/2))/(x^2 - x + 1)