Integrand size = 24, antiderivative size = 144 \[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=\frac {\left (-136+238 x+1555 x^2-453 x^3\right ) \sqrt {-2 x^2-3 x^3+x^4}}{544 x^3 \left (-2-3 x+x^2\right )}+\frac {1}{4} \arctan \left (\frac {-\frac {x}{2}+\frac {x^2}{2}-\frac {1}{2} \sqrt {-2 x^2-3 x^3+x^4}}{x}\right )-\frac {119 \arctan \left (\frac {\frac {x^2}{\sqrt {2}}-\frac {\sqrt {-2 x^2-3 x^3+x^4}}{\sqrt {2}}}{x}\right )}{32 \sqrt {2}} \]
1/544*(-453*x^3+1555*x^2+238*x-136)*(x^4-3*x^3-2*x^2)^(1/2)/x^3/(x^2-3*x-2 )+1/4*arctan((-1/2*x+1/2*x^2-1/2*(x^4-3*x^3-2*x^2)^(1/2))/x)-119/64*arctan ((1/2*2^(1/2)*x^2-1/2*(x^4-3*x^3-2*x^2)^(1/2)*2^(1/2))/x)*2^(1/2)
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=-\frac {272-476 x-3110 x^2+906 x^3+2023 \sqrt {2} x^2 \sqrt {-2-3 x+x^2} \arctan \left (\frac {x-\sqrt {-2-3 x+x^2}}{\sqrt {2}}\right )+272 x^2 \sqrt {-2-3 x+x^2} \arctan \left (\frac {1}{2} \left (1-x+\sqrt {-2-3 x+x^2}\right )\right )}{1088 x \sqrt {x^2 \left (-2-3 x+x^2\right )}} \]
-1/1088*(272 - 476*x - 3110*x^2 + 906*x^3 + 2023*Sqrt[2]*x^2*Sqrt[-2 - 3*x + x^2]*ArcTan[(x - Sqrt[-2 - 3*x + x^2])/Sqrt[2]] + 272*x^2*Sqrt[-2 - 3*x + x^2]*ArcTan[(1 - x + Sqrt[-2 - 3*x + x^2])/2])/(x*Sqrt[x^2*(-2 - 3*x + x^2)])
Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.54, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2467, 25, 1289, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(x-1) \left (x^4-3 x^3-2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x \sqrt {x^2-3 x-2} \int -\frac {1}{(1-x) x^3 \left (x^2-3 x-2\right )^{3/2}}dx}{\sqrt {x^4-3 x^3-2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x \sqrt {x^2-3 x-2} \int \frac {1}{(1-x) x^3 \left (x^2-3 x-2\right )^{3/2}}dx}{\sqrt {x^4-3 x^3-2 x^2}}\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle -\frac {x \sqrt {x^2-3 x-2} \int \left (\frac {1}{x \left (x^2-3 x-2\right )^{3/2}}+\frac {1}{x^2 \left (x^2-3 x-2\right )^{3/2}}+\frac {1}{x^3 \left (x^2-3 x-2\right )^{3/2}}+\frac {1}{\left (x^2-3 x-2\right )^{3/2} (1-x)}\right )dx}{\sqrt {x^4-3 x^3-2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x \sqrt {x^2-3 x-2} \left (-\frac {1}{8} \arctan \left (\frac {x+7}{4 \sqrt {x^2-3 x-2}}\right )+\frac {119 \arctan \left (\frac {3 x+4}{2 \sqrt {2} \sqrt {x^2-3 x-2}}\right )}{64 \sqrt {2}}-\frac {13-3 x}{17 x \sqrt {x^2-3 x-2}}-\frac {13-3 x}{17 x^2 \sqrt {x^2-3 x-2}}-\frac {13-3 x}{17 \sqrt {x^2-3 x-2}}+\frac {373 \sqrt {x^2-3 x-2}}{544 x}-\frac {69 \sqrt {x^2-3 x-2}}{136 x^2}+\frac {10-x}{34 \sqrt {x^2-3 x-2}}\right )}{\sqrt {x^4-3 x^3-2 x^2}}\) |
-((x*Sqrt[-2 - 3*x + x^2]*(-1/17*(13 - 3*x)/Sqrt[-2 - 3*x + x^2] + (10 - x )/(34*Sqrt[-2 - 3*x + x^2]) - (13 - 3*x)/(17*x^2*Sqrt[-2 - 3*x + x^2]) - ( 13 - 3*x)/(17*x*Sqrt[-2 - 3*x + x^2]) - (69*Sqrt[-2 - 3*x + x^2])/(136*x^2 ) + (373*Sqrt[-2 - 3*x + x^2])/(544*x) - ArcTan[(7 + x)/(4*Sqrt[-2 - 3*x + x^2])]/8 + (119*ArcTan[(4 + 3*x)/(2*Sqrt[2]*Sqrt[-2 - 3*x + x^2])])/(64*S qrt[2])))/Sqrt[-2*x^2 - 3*x^3 + x^4])
3.21.24.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.77 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {453 x^{3}-1555 x^{2}-238 x +136}{544 x \sqrt {x^{2} \left (x^{2}-3 x -2\right )}}+\frac {\left (\frac {119 \sqrt {2}\, \arctan \left (\frac {\left (-4-3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}-3 x -2}}\right )}{128}-\frac {\arctan \left (\frac {-7-x}{4 \sqrt {\left (-1+x \right )^{2}-3-x}}\right )}{8}\right ) x \sqrt {x^{2}-3 x -2}}{\sqrt {x^{2} \left (x^{2}-3 x -2\right )}}\) | \(111\) |
| default | \(-\frac {x \left (x^{2}-3 x -2\right ) \left (2023 \sqrt {2}\, \arctan \left (\frac {\left (4+3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}-3 x -2}}\right ) x^{2} \sqrt {x^{2}-3 x -2}-272 \arctan \left (\frac {7+x}{4 \sqrt {x^{2}-3 x -2}}\right ) x^{2} \sqrt {x^{2}-3 x -2}+1812 x^{3}-6220 x^{2}-952 x +544\right )}{2176 \left (x^{4}-3 x^{3}-2 x^{2}\right )^{\frac {3}{2}}}\) | \(113\) |
| pseudoelliptic | \(\frac {-2023 \sqrt {2}\, \arctan \left (\frac {\left (4+3 x \right ) \sqrt {2}\, x}{4 \sqrt {x^{2} \left (x^{2}-3 x -2\right )}}\right ) \sqrt {x^{2} \left (x^{2}-3 x -2\right )}\, x +272 \arctan \left (\frac {x \left (7+x \right )}{4 \sqrt {x^{2} \left (x^{2}-3 x -2\right )}}\right ) \sqrt {x^{2} \left (x^{2}-3 x -2\right )}\, x -1812 x^{3}+6220 x^{2}+952 x -544}{2176 \sqrt {x^{2} \left (x^{2}-3 x -2\right )}\, x}\) | \(119\) |
| trager | \(-\frac {\left (453 x^{3}-1555 x^{2}-238 x +136\right ) \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{544 \left (x^{2}-3 x -2\right ) x^{3}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +4 \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{x \left (-1+x \right )}\right )}{8}+\frac {119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{4}-3 x^{3}-2 x^{2}}}{x^{2}}\right )}{128}\) | \(157\) |
-1/544*(453*x^3-1555*x^2-238*x+136)/x/(x^2*(x^2-3*x-2))^(1/2)+(119/128*2^( 1/2)*arctan(1/4*(-4-3*x)*2^(1/2)/(x^2-3*x-2)^(1/2))-1/8*arctan(1/4*(-7-x)/ ((-1+x)^2-3-x)^(1/2)))*x*(x^2-3*x-2)^(1/2)/(x^2*(x^2-3*x-2))^(1/2)
Time = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=-\frac {906 \, x^{5} - 2718 \, x^{4} - 1812 \, x^{3} - 2023 \, \sqrt {2} {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )} \arctan \left (-\frac {\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}}}{2 \, x}\right ) + 272 \, {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )} \arctan \left (-\frac {x^{2} - x - \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}}}{2 \, x}\right ) + 2 \, \sqrt {x^{4} - 3 \, x^{3} - 2 \, x^{2}} {\left (453 \, x^{3} - 1555 \, x^{2} - 238 \, x + 136\right )}}{1088 \, {\left (x^{5} - 3 \, x^{4} - 2 \, x^{3}\right )}} \]
-1/1088*(906*x^5 - 2718*x^4 - 1812*x^3 - 2023*sqrt(2)*(x^5 - 3*x^4 - 2*x^3 )*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 - 3*x^3 - 2*x^2))/x) + 272*( x^5 - 3*x^4 - 2*x^3)*arctan(-1/2*(x^2 - x - sqrt(x^4 - 3*x^3 - 2*x^2))/x) + 2*sqrt(x^4 - 3*x^3 - 2*x^2)*(453*x^3 - 1555*x^2 - 238*x + 136))/(x^5 - 3 *x^4 - 2*x^3)
\[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (x^{2} \left (x^{2} - 3 x - 2\right )\right )^{\frac {3}{2}} \left (x - 1\right )}\, dx \]
\[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{4} - 3 \, x^{3} - 2 \, x^{2}\right )}^{\frac {3}{2}} {\left (x - 1\right )}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=-\frac {\frac {29 \, x}{\mathrm {sgn}\left (x\right )} - \frac {103}{\mathrm {sgn}\left (x\right )}}{68 \, \sqrt {x^{2} - 3 \, x - 2}} + \frac {119 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}\right )}{64 \, \mathrm {sgn}\left (x\right )} - \frac {\arctan \left (-\frac {1}{2} \, x + \frac {1}{2} \, \sqrt {x^{2} - 3 \, x - 2} + \frac {1}{2}\right )}{4 \, \mathrm {sgn}\left (x\right )} - \frac {47 \, {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{3} + 16 \, {\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{2} + 98 \, x - 98 \, \sqrt {x^{2} - 3 \, x - 2} + 128}{32 \, {\left ({\left (x - \sqrt {x^{2} - 3 \, x - 2}\right )}^{2} + 2\right )}^{2} \mathrm {sgn}\left (x\right )} \]
-1/68*(29*x/sgn(x) - 103/sgn(x))/sqrt(x^2 - 3*x - 2) + 119/64*sqrt(2)*arct an(-1/2*sqrt(2)*(x - sqrt(x^2 - 3*x - 2)))/sgn(x) - 1/4*arctan(-1/2*x + 1/ 2*sqrt(x^2 - 3*x - 2) + 1/2)/sgn(x) - 1/32*(47*(x - sqrt(x^2 - 3*x - 2))^3 + 16*(x - sqrt(x^2 - 3*x - 2))^2 + 98*x - 98*sqrt(x^2 - 3*x - 2) + 128)/( ((x - sqrt(x^2 - 3*x - 2))^2 + 2)^2*sgn(x))
Timed out. \[ \int \frac {1}{(-1+x) \left (-2 x^2-3 x^3+x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (x-1\right )\,{\left (x^4-3\,x^3-2\,x^2\right )}^{3/2}} \,d x \]