\(\int \frac {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{(-2+2 x+x^3)^2} \, dx\) [1083]

Optimal result

Integrand size = 34, antiderivative size = 81 \[ \int \frac {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{\left (-2+2 x+x^3\right )^2} \, dx=\frac {x \sqrt {-2 x+2 x^2+3 x^4}}{2 \left (-2+2 x+x^3\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right )}{2 \sqrt {2}} \]

[Out]

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

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 3.36 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{\left (-2+2 x+x^3\right )^2} \, dx=\frac {\sqrt {x \left (-2+2 x+3 x^3\right )} \left (\frac {2 x^{3/2}}{-2+2 x+x^3}+\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x+\frac {3 x^3}{2}}}\right )}{\sqrt {-1+x+\frac {3 x^3}{2}}}\right )}{4 \sqrt {x}} \]

[In]

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

[Out]

(Sqrt[x*(-2 + 2*x + 3*x^3)]*((2*x^(3/2))/(-2 + 2*x + x^3) + ArcTanh[x^(3/2)/Sqrt[-1 + x + (3*x^3)/2]]/Sqrt[-1
+ x + (3*x^3)/2]))/(4*Sqrt[x])

Maple [A] (verified)

Time = 4.85 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {2}\, \left (x^{3}+2 x -2\right ) \operatorname {arctanh}\left (\frac {\sqrt {3 x^{4}+2 x^{2}-2 x}\, \sqrt {2}}{2 x^{2}}\right )+2 x \sqrt {3 x^{4}+2 x^{2}-2 x}}{4 x^{3}+8 x -8}\)	\(71\)
risch	\(\frac {x^{2} \left (3 x^{3}+2 x -2\right )}{2 \left (x^{3}+2 x -2\right ) \sqrt {x \left (3 x^{3}+2 x -2\right )}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {3 x^{4}+2 x^{2}-2 x}\, \sqrt {2}}{2 x^{2}}\right )}{4}\)	\(71\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (x^{3}+2 x -2\right ) \operatorname {arctanh}\left (\frac {\sqrt {3 x^{4}+2 x^{2}-2 x}\, \sqrt {2}}{2 x^{2}}\right )+2 x \sqrt {3 x^{4}+2 x^{2}-2 x}}{4 x^{3}+8 x -8}\)	\(71\)
trager	\(\frac {x \sqrt {3 x^{4}+2 x^{2}-2 x}}{2 x^{3}+4 x -4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 x \sqrt {3 x^{4}+2 x^{2}-2 x}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{3}+2 x -2}\right )}{8}\)	\(100\)
elliptic	\(\text {Expression too large to display}\)	\(1689\)

[In]

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

[Out]

(2^(1/2)*(x^3+2*x-2)*arctanh(1/2*(3*x^4+2*x^2-2*x)^(1/2)/x^2*2^(1/2))+2*x*(3*x^4+2*x^2-2*x)^(1/2))/(4*x^3+8*x-
8)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.63 \[ \int \frac {(-3+2 x) \sqrt {-2 x+2 x^2+3 x^4}}{\left (-2+2 x+x^3\right )^2} \, dx=\frac {\sqrt {2} {\left (x^{3} + 2 \, x - 2\right )} \log \left (-\frac {49 \, x^{6} + 36 \, x^{4} - 36 \, x^{3} + 4 \, \sqrt {2} {\left (5 \, x^{4} + 2 \, x^{2} - 2 \, x\right )} \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} + 4 \, x^{2} - 8 \, x + 4}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) + 8 \, \sqrt {3 \, x^{4} + 2 \, x^{2} - 2 \, x} x}{16 \, {\left (x^{3} + 2 \, x - 2\right )}} \]

[In]

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

[Out]

1/16*(sqrt(2)*(x^3 + 2*x - 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)) + 8*sqrt(3*x^4 + 2*x^2 - 2*x)*x)/(x^3 +
2*x - 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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