Integrand size = 30, antiderivative size = 73 \[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (-6+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (-1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right )+\frac {1}{4} \left (6+\sqrt {2}\right ) \text {arctanh}\left (\frac {-1+x}{\left (1+\sqrt {2}\right ) \sqrt {-x+x^3}}\right ) \]
1/4*(-6+2^(1/2))*arctanh((-1+x)/(2^(1/2)-1)/(x^3-x)^(1/2))+1/4*(6+2^(1/2)) *arctanh((-1+x)/(1+2^(1/2))/(x^3-x)^(1/2))
Time = 11.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{4} \left (\left (6+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (-1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )+\left (-6+\sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) (-1+x)}{\sqrt {x \left (-1+x^2\right )}}\right )\right ) \]
((6 + Sqrt[2])*ArcTanh[((-1 + Sqrt[2])*(-1 + x))/Sqrt[x*(-1 + x^2)]] + (-6 + Sqrt[2])*ArcTanh[((1 + Sqrt[2])*(-1 + x))/Sqrt[x*(-1 + x^2)]])/4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.03 (sec) , antiderivative size = 398, normalized size of antiderivative = 5.45, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2467, 2035, 7279, 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.
| \(\displaystyle \int \frac {x^2+3 x-2}{\left (x^2-2 x-1\right ) \sqrt {x^3-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int \frac {-x^2-3 x+2}{\sqrt {x} \left (-x^2+2 x+1\right ) \sqrt {x^2-1}}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {-x^2-3 x+2}{\left (-x^2+2 x+1\right ) \sqrt {x^2-1}}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \left (\frac {1-5 x}{\left (-x^2+2 x+1\right ) \sqrt {x^2-1}}+\frac {1}{\sqrt {x^2-1}}\right )d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \left (\frac {\left (4+5 \sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{4 \sqrt {x^2-1}}+\frac {\left (4-5 \sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{4 \sqrt {x^2-1}}-\frac {\left (5+2 \sqrt {2}\right ) \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {x^2-1}}+\frac {\left (5-2 \sqrt {2}\right ) \sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {x^2-1}}+\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2-1}}+\frac {\left (1+3 \sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-1-\sqrt {2},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}+\frac {\left (1-3 \sqrt {2}\right ) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-1+\sqrt {2},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}\) |
(2*Sqrt[x]*Sqrt[-1 + x^2]*(((4 - 5*Sqrt[2])*Sqrt[1 - x]*Sqrt[1 + x]*Ellipt icF[ArcSin[Sqrt[x]], -1])/(4*Sqrt[-1 + x^2]) + ((4 + 5*Sqrt[2])*Sqrt[1 - x ]*Sqrt[1 + x]*EllipticF[ArcSin[Sqrt[x]], -1])/(4*Sqrt[-1 + x^2]) + (Sqrt[- 1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2]) /(Sqrt[2]*Sqrt[-1 + x^2]) + ((5 - 2*Sqrt[2])*Sqrt[-1 + x]*Sqrt[1 + x]*Elli pticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(4*Sqrt[-1 + x^2]) - ( (5 + 2*Sqrt[2])*Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x] )/Sqrt[-1 + x]], 1/2])/(4*Sqrt[-1 + x^2]) + ((1 + 3*Sqrt[2])*Sqrt[1 - x]*S qrt[1 + x]*EllipticPi[-1 - Sqrt[2], ArcSin[Sqrt[x]], -1])/(2*Sqrt[-1 + x^2 ]) + ((1 - 3*Sqrt[2])*Sqrt[1 - x]*Sqrt[1 + x]*EllipticPi[-1 + Sqrt[2], Arc Sin[Sqrt[x]], -1])/(2*Sqrt[-1 + x^2])))/Sqrt[-x + x^3]
3.10.57.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 3.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.74
| method | result | size |
| default | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(273\) |
| elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2-\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}+\frac {5 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{-2+\sqrt {2}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{3}-x}\, \left (-2+\sqrt {2}\right )}\) | \(273\) |
| trager | \(\frac {\operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) \ln \left (\frac {-278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x^{2}+1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x -621337 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )-1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2}+5650808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x -193794 x^{2}+1924943 \sqrt {x^{3}-x}-6272145 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+5684624 x -5878418}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+7 x -15\right )}^{2}}\right )}{2}-\frac {\ln \left (-\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x +1051511 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2}-2713432 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x +839055 x^{2}+1190599 \sqrt {x^{3}-x}+3764943 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )-1278560 x +2117615}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+5 x -9\right )}^{2}}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )}{2}-\frac {3 \ln \left (-\frac {278808 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x^{2}-1394040 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2} x +1051511 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x^{2}+1038514 \sqrt {x^{3}-x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+1672848 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )^{2}-2713432 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x +839055 x^{2}+1190599 \sqrt {x^{3}-x}+3764943 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )-1278560 x +2117615}{{\left (4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right ) x -8 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+24 \textit {\_Z} +17\right )+5 x -9\right )}^{2}}\right )}{2}\) | \(545\) |
(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticF((1+x)^(1/2),1 /2*2^(1/2))+2^(1/2)*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2 -2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2-2^(1/2)),1/2*2^(1/2))+5/2*(1+x)^(1 /2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2-2^(1/2))*EllipticPi((1+x)^( 1/2),-1/(-2-2^(1/2)),1/2*2^(1/2))-2^(1/2)*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^( 1/2)/(x^3-x)^(1/2)/(-2+2^(1/2))*EllipticPi((1+x)^(1/2),-1/(-2+2^(1/2)),1/2 *2^(1/2))+5/2*(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)/(-2+2^(1/ 2))*EllipticPi((1+x)^(1/2),-1/(-2+2^(1/2)),1/2*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.73 \[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 12 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )} + 2 \, x^{2} - 12 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) + \frac {3}{4} \, \log \left (\frac {x^{4} + 4 \, x^{3} + 2 \, x^{2} - 4 \, \sqrt {x^{3} - x} {\left (x^{2} + 1\right )} - 4 \, x + 1}{x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1}\right ) \]
1/8*sqrt(2)*log((x^4 + 12*x^3 + 4*sqrt(2)*sqrt(x^3 - x)*(x^2 + 2*x - 1) + 2*x^2 - 12*x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1)) + 3/4*log((x^4 + 4*x^3 + 2*x^2 - 4*sqrt(x^3 - x)*(x^2 + 1) - 4*x + 1)/(x^4 - 4*x^3 + 2*x^2 + 4*x + 1))
\[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int \frac {x^{2} + 3 x - 2}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} - 2 x - 1\right )}\, dx \]
\[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x^{2} + 3 \, x - 2}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
\[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\int { \frac {x^{2} + 3 \, x - 2}{\sqrt {x^{3} - x} {\left (x^{2} - 2 \, x - 1\right )}} \,d x } \]
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.18 \[ \int \frac {-2+3 x+x^2}{\left (-1-2 x+x^2\right ) \sqrt {-x+x^3}} \, dx=\frac {\sqrt {2}\,\sqrt {-x}\,\left (5\,\sqrt {2}+4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {\sqrt {2}\,\sqrt {-x}\,\left (5\,\sqrt {2}-4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}-1\right )}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}} \]
(2^(1/2)*(-x)^(1/2)*(5*2^(1/2) + 4)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi (-1/(2^(1/2) + 1), asin((-x)^(1/2)), -1))/(2*(x^3 - x)^(1/2)*(2^(1/2) + 1) ) - (2^(1/2)*(-x)^(1/2)*(5*2^(1/2) - 4)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipt icPi(1/(2^(1/2) - 1), asin((-x)^(1/2)), -1))/(2*(x^3 - x)^(1/2)*(2^(1/2) - 1)) - (2*(-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF(asin((-x)^(1/2) ), -1))/(x^3 - x)^(1/2)