\(\int \frac {-3 x+2 x^2}{(-2+2 x+x^3) \sqrt {-2 x+2 x^2+3 x^4}} \, dx\) [571]

Optimal result

Integrand size = 38, antiderivative size = 44 \[ \int \frac {-3 x+2 x^2}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right )}{\sqrt {2}} \]

[Out]

1/2*arctanh(2^(1/2)*x*(3*x^4+2*x^2-2*x)^(1/2)/(3*x^3+2*x-2))*2^(1/2)

Rubi [F]

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

[In]

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

[Out]

(-6*Sqrt[x]*Sqrt[-2 + 2*x + 3*x^3]*Defer[Subst][Defer[Int][x^2/((-2 + 2*x^2 + x^6)*Sqrt[-2 + 2*x^2 + 3*x^6]),
x], x, Sqrt[x]])/Sqrt[-2*x + 2*x^2 + 3*x^4] + (4*Sqrt[x]*Sqrt[-2 + 2*x + 3*x^3]*Defer[Subst][Defer[Int][x^4/((
-2 + 2*x^2 + x^6)*Sqrt[-2 + 2*x^2 + 3*x^6]), x], x, Sqrt[x]])/Sqrt[-2*x + 2*x^2 + 3*x^4]

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

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.41 \[ \int \frac {-3 x+2 x^2}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx=\frac {\sqrt {x} \sqrt {-2+2 x+3 x^3} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x+\frac {3 x^3}{2}}}\right )}{\sqrt {2} \sqrt {x \left (-2+2 x+3 x^3\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {3 x^{4}+2 x^{2}-2 x}\, \sqrt {2}}{2 x^{2}}\right )}{2}\)	\(31\)
pseudoelliptic	\(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {3 x^{4}+2 x^{2}-2 x}\, \sqrt {2}}{2 x^{2}}\right )}{2}\)	\(31\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+4 x \sqrt {3 x^{4}+2 x^{2}-2 x}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{3}+2 x -2}\right )}{4}\)	\(70\)
elliptic	\(\text {Expression too large to display}\)	\(1659\)

[In]

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

[Out]

1/2*2^(1/2)*arctanh(1/2*(3*x^4+2*x^2-2*x)^(1/2)/x^2*2^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (39) = 78\).

Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.11 \[ \int \frac {-3 x+2 x^2}{\left (-2+2 x+x^3\right ) \sqrt {-2 x+2 x^2+3 x^4}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (-\frac {49 \, x^{6} + 36 \, x^{4} - 36 \, x^{3} + 4 \, \sqrt {2} {\left (5 \, x^{4} + 2 \, x^{2} - 2 \, x\right )} \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} + 4 \, x^{2} - 8 \, x + 4}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) \]

[In]

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

[Out]

1/8*sqrt(2)*log(-(49*x^6 + 36*x^4 - 36*x^3 + 4*sqrt(2)*(5*x^4 + 2*x^2 - 2*x)*sqrt(3*x^4 + 2*x^2 - 2*x) + 4*x^2
- 8*x + 4)/(x^6 + 4*x^4 - 4*x^3 + 4*x^2 - 8*x + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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