\(\int \frac {-1+2 x+2 x^2}{(1+2 x) \sqrt {-x+x^4}} \, dx\) [219]

Optimal result

Integrand size = 29, antiderivative size = 23 \[ \int \frac {-1+2 x+2 x^2}{(1+2 x) \sqrt {-x+x^4}} \, dx=\text {arctanh}\left (\frac {2 \sqrt {-x+x^4}}{1+2 x^2}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(2*Sqrt[x]*Sqrt[-1 + x^3]*ArcTanh[x^(3/2)/Sqrt[-1 + x^3]])/(3*Sqrt[-x + x^4]) + ((1 - x)*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])/(2*3
^(1/4)*Sqrt[-(((1 - x)*x)/(1 - (1 + Sqrt[3])*x)^2)]*Sqrt[-x + x^4]) - (((3*I)/2)*Sqrt[x]*Sqrt[-1 + x^3]*Defer[
Subst][Defer[Int][1/((I - Sqrt[2]*x)*Sqrt[-1 + x^6]), x], x, Sqrt[x]])/Sqrt[-x + x^4] - (((3*I)/2)*Sqrt[x]*Sqr
t[-1 + x^3]*Defer[Subst][Defer[Int][1/((I + Sqrt[2]*x)*Sqrt[-1 + x^6]), x], x, Sqrt[x]])/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} (1+2 x) \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {-1+x^6}}+\frac {x^2}{\sqrt {-1+x^6}}-\frac {3}{2 \left (1+2 x^2\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {\left (\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 {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {(1-x) 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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 \sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {i}{2 \left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}}+\frac {i}{2 \left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {(1-x) 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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {-x+x^4}} \\ & = \frac {2 \sqrt {x} \sqrt {-1+x^3} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {-x+x^4}}+\frac {(1-x) 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 )}{2 \sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}}-\frac {\left (3 i \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-x+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.68 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {-1+2 x+2 x^2}{(1+2 x) \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {-1+x^3} \text {arctanh}\left (\frac {(-1+x) \sqrt {x}}{\sqrt {-1+x^3}}\right )}{\sqrt {x \left (-1+x^3\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35

method	result	size
trager	\(-\ln \left (-\frac {-2 x^{2}+2 \sqrt {x^{4}-x}-1}{2 x +1}\right )\)	\(31\)
default	\(\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{3}+\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(509\)
elliptic	\(\text {Expression too large to display}\)	\(774\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-1+2 x+2 x^2}{(1+2 x) \sqrt {-x+x^4}} \, dx=\log \left (-\frac {2 \, x^{2} + 2 \, \sqrt {x^{4} - x} + 1}{2 \, x + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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