\(\int \frac {1}{(1+x) (-2+3 x-2 x^2+3 x^3-2 x^4)^{3/2}} \, dx\) [2273]

Optimal result

Integrand size = 30, antiderivative size = 173 \[ \int \frac {1}{(1+x) \left (-2+3 x-2 x^2+3 x^3-2 x^4\right )^{3/2}} \, dx=\frac {i (-1+x) \sqrt {2+x+2 x^2} \left (\frac {i \left (-18-107 x+73 x^2+22 x^3\right )}{600 (-1+x)^2 \sqrt {2+x+2 x^2}}-\frac {i \text {arctanh}\left (\sqrt {\frac {2}{3}}+\sqrt {\frac {2}{3}} x-\frac {\sqrt {2+x+2 x^2}}{\sqrt {3}}\right )}{12 \sqrt {3}}-\frac {91 i \text {arctanh}\left (\sqrt {\frac {2}{5}}-\sqrt {\frac {2}{5}} x+\frac {\sqrt {2+x+2 x^2}}{\sqrt {5}}\right )}{200 \sqrt {5}}\right )}{\sqrt {-(-1+x)^2 \left (2+x+2 x^2\right )}} \]

[Out]

I*(-1+x)*(2*x^2+x+2)^(1/2)*(1/600*I*(22*x^3+73*x^2-107*x-18)/(-1+x)^2/(2*x^2+x+2)^(1/2)-1/36*I*arctanh(1/3*6^(
1/2)+1/3*x*6^(1/2)-1/3*(2*x^2+x+2)^(1/2)*3^(1/2))*3^(1/2)+91/1000*I*arctanh(-1/5*10^(1/2)+1/5*10^(1/2)*x-1/5*(
2*x^2+x+2)^(1/2)*5^(1/2))*5^(1/2))/(-(-1+x)^2*(2*x^2+x+2))^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(1+x) \left (-2+3 x-2 x^2+3 x^3-2 x^4\right )^{3/2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1+x) \left (-2+3 x-2 x^2+3 x^3-2 x^4\right )^{3/2}} \, dx=\frac {819 \sqrt {5} (-1+x)^2 \sqrt {2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {2} x+\sqrt {2+x+2 x^2}}{\sqrt {5}}\right )-5 \left (-54-321 x+219 x^2+66 x^3-50 \sqrt {3} (-1+x)^2 \sqrt {2+x+2 x^2} \text {arctanh}\left (\frac {1}{3} \left (\sqrt {6}+\sqrt {6} x-\sqrt {3} \sqrt {2+x+2 x^2}\right )\right )\right )}{9000 (-1+x) \sqrt {-(-1+x)^2 \left (2+x+2 x^2\right )}} \]

[In]

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

[Out]

(819*Sqrt[5]*(-1 + x)^2*Sqrt[2 + x + 2*x^2]*ArcTanh[(Sqrt[2] - Sqrt[2]*x + Sqrt[2 + x + 2*x^2])/Sqrt[5]] - 5*(
-54 - 321*x + 219*x^2 + 66*x^3 - 50*Sqrt[3]*(-1 + x)^2*Sqrt[2 + x + 2*x^2]*ArcTanh[(Sqrt[6] + Sqrt[6]*x - Sqrt
[3]*Sqrt[2 + x + 2*x^2])/3]))/(9000*(-1 + x)*Sqrt[-((-1 + x)^2*(2 + x + 2*x^2))])

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {22 x^{3}+73 x^{2}-107 x -18}{600 \left (-1+x \right ) \sqrt {-\left (-1+x \right )^{2} \left (2 x^{2}+x +2\right )}}-\frac {\left (-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-3+3 x \right ) \sqrt {3}}{6 \sqrt {-2 \left (1+x \right )^{2}+3 x}}\right )}{72}+\frac {91 \sqrt {5}\, \arctan \left (\frac {\left (-5-5 x \right ) \sqrt {5}}{10 \sqrt {-2 \left (-1+x \right )^{2}-5 x}}\right )}{2000}\right ) \left (-1+x \right ) \sqrt {-2 x^{2}-x -2}}{\sqrt {-\left (-1+x \right )^{2} \left (2 x^{2}+x +2\right )}}\)	\(134\)
trager	\(\frac {\left (22 x^{3}+73 x^{2}-107 x -18\right ) \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}}{600 \left (-1+x \right )^{3} \left (2 x^{2}+x +2\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +2 \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (1+x \right ) \left (-1+x \right )}\right )}{72}-\frac {91 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right ) x^{2}+2 \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right )}{\left (-1+x \right )^{2}}\right )}{2000}\)	\(188\)
default	\(-\frac {\left (250 \sqrt {3}\, \sqrt {-2 x^{2}-x -2}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) x^{2}+819 \sqrt {5}\, \sqrt {-2 x^{2}-x -2}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) x^{2}-500 \sqrt {3}\, \sqrt {-2 x^{2}-x -2}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) x -1638 \sqrt {5}\, \sqrt {-2 x^{2}-x -2}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) x +250 \sqrt {3}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}+819 \sqrt {5}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}-660 x^{3}-2190 x^{2}+3210 x +540\right ) \left (-1+x \right ) \left (2 x^{2}+x +2\right )}{18000 \left (-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2\right )^{\frac {3}{2}}}\)	\(287\)

[In]

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

[Out]

-1/600*(22*x^3+73*x^2-107*x-18)/(-1+x)/(-(-1+x)^2*(2*x^2+x+2))^(1/2)-(-1/72*3^(1/2)*arctan(1/6*(-3+3*x)*3^(1/2
)/(-2*(1+x)^2+3*x)^(1/2))+91/2000*5^(1/2)*arctan(1/10*(-5-5*x)*5^(1/2)/(-2*(-1+x)^2-5*x)^(1/2)))*(-1+x)*(-2*x^
2-x-2)^(1/2)/(-(-1+x)^2*(2*x^2+x+2))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(1+x) \left (-2+3 x-2 x^2+3 x^3-2 x^4\right )^{3/2}} \, dx=-\frac {819 \, \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2} {\left (x + 1\right )}}{2 \, {\left (2 \, x^{3} - x^{2} + x - 2\right )}}\right ) + 250 \, \sqrt {3} {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\frac {\sqrt {3} \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2}}{2 \, {\left (2 \, x^{2} + x + 2\right )}}\right ) - 30 \, \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2} {\left (22 \, x^{3} + 73 \, x^{2} - 107 \, x - 18\right )}}{18000 \, {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )}} \]

[In]

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

[Out]

-1/18000*(819*sqrt(5)*(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)*arctan(1/2*sqrt(5)*sqrt(-2*x^4 + 3*x^3 - 2*x^2
+ 3*x - 2)*(x + 1)/(2*x^3 - x^2 + x - 2)) + 250*sqrt(3)*(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)*arctan(1/2*
sqrt(3)*sqrt(-2*x^4 + 3*x^3 - 2*x^2 + 3*x - 2)/(2*x^2 + x + 2)) - 30*sqrt(-2*x^4 + 3*x^3 - 2*x^2 + 3*x - 2)*(2
2*x^3 + 73*x^2 - 107*x - 18))/(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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