3.1.83 \(\int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-(-1+2 a+a^2) x^2+(-1+2 a) x^3}} \, dx\) [83]

3.1.83.1 Optimal result

Integrand size = 51, antiderivative size = 46 \[ \int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx=\log \left (\frac {-a^2+2 a x+x^2-2 \left (x+\sqrt {(1-x) x \left (a^2+x-2 a x\right )}\right )}{(a-x)^2}\right ) \]

output

ln((-a^2+2*a*x+x^2-2*x-2*((1-x)*x*(a^2-2*a*x+x))^(1/2))/(a-x)^2)
 

3.1.83.2 Mathematica [A] (verified)

Time = 10.88 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {a^2 x+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^3}}{-a^2+(-1+2 a) x}\right ) \]

input

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

output

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

3.1.83.3 Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.82 (sec) , antiderivative size = 233, normalized size of antiderivative = 5.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {2467, 2035, 2228, 1417, 321, 1544, 412}

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.

\(\displaystyle \int \frac {(2 a-1) x-a}{(x-a) \sqrt {-\left (a^2+2 a-1\right ) x^2+a^2 x+(2 a-1) x^3}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \int \frac {a+(1-2 a) x}{(a-x) \sqrt {x} \sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}dx}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \int \frac {a+(1-2 a) x}{(a-x) \sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}d\sqrt {x}}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 2228

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \left (2 (1-a) a \int \frac {1}{(a-x) \sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}d\sqrt {x}-(1-2 a) \int \frac {1}{\sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}d\sqrt {x}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 1417

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \left (2 (1-a) a \int \frac {1}{(a-x) \sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}d\sqrt {x}-\frac {(1-2 a) \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \int \frac {1}{\sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1}}d\sqrt {x}}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \left (2 (1-a) a \int \frac {1}{(a-x) \sqrt {a^2-(1-2 a) x^2+\left (-a^2-2 a+1\right ) x}}d\sqrt {x}-\frac {(1-2 a) \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 1544

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \left (\frac {2 (1-a) a \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \int \frac {1}{\sqrt {1-x} (a-x) \sqrt {\frac {(1-2 a) x}{a^2}+1}}d\sqrt {x}}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}-\frac {(1-2 a) \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 \sqrt {x} \sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2} \left (\frac {2 (1-a) \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \operatorname {EllipticPi}\left (\frac {1}{a},\arcsin \left (\sqrt {x}\right ),-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}-\frac {(1-2 a) \sqrt {1-x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x+a^2-(1-2 a) x^2}}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-\left ((1-2 a) x^3\right )}}\)

input

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

output

(2*Sqrt[x]*Sqrt[a^2 + (1 - 2*a - a^2)*x - (1 - 2*a)*x^2]*(-(((1 - 2*a)*Sqr 
t[1 - x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*EllipticF[ArcSin[Sqrt[x]], -((1 - 2*a 
)/a^2)])/Sqrt[a^2 + (1 - 2*a - a^2)*x - (1 - 2*a)*x^2]) + (2*(1 - a)*Sqrt[ 
1 - x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*EllipticPi[a^(-1), ArcSin[Sqrt[x]], -(( 
1 - 2*a)/a^2)])/Sqrt[a^2 + (1 - 2*a - a^2)*x - (1 - 2*a)*x^2]))/Sqrt[a^2*x 
 + (1 - 2*a - a^2)*x^2 - (1 - 2*a)*x^3]
 

3.1.83.3.1 Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 1417

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt[1 + 2*c*(x^2/(b + q 
))]/Sqrt[a + b*x^2 + c*x^4])   Int[1/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 
2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] 
&& NegQ[c/a]
 

rule 1544

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*( 
Sqrt[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[1/((d + e*x^2)*S 
qrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{ 
a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] 
&& NegQ[c/a]
 

rule 2035

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]
 

rule 2228

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> Simp[B/e   Int[1/Sqrt[a + b*x^2 + c*x^4], x], 
 x] + Simp[(e*A - d*B)/e   Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], 
 x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a 
]
 

rule 2467

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]
 

3.1.83.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.66 (sec) , antiderivative size = 367, normalized size of antiderivative = 7.98

method	result	size
elliptic	\(-\frac {2 a^{2} \sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}\, \sqrt {\frac {-1+x}{\frac {a^{2}}{-1+2 a}-1}}\, \sqrt {\frac {x \left (-1+2 a \right )}{a^{2}}}\, F\left (\sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}, \sqrt {\frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-1\right )}}\right )}{\sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}}-\frac {4 a^{3} \left (a -1\right ) \sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}\, \sqrt {\frac {-1+x}{\frac {a^{2}}{-1+2 a}-1}}\, \sqrt {\frac {x \left (-1+2 a \right )}{a^{2}}}\, \Pi \left (\sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}, \frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-a \right )}, \sqrt {\frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-1\right )}}\right )}{\left (-1+2 a \right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}\, \left (\frac {a^{2}}{-1+2 a}-a \right )}\)	\(367\)
default	\(\frac {2 a^{2} \sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}\, \sqrt {\frac {-1+x}{\frac {a^{2}}{-1+2 a}-1}}\, \sqrt {\frac {x \left (-1+2 a \right )}{a^{2}}}\, F\left (\sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}, \sqrt {\frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-1\right )}}\right )}{\left (-1+2 a \right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}}-\frac {4 a^{3} \sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}\, \sqrt {\frac {-1+x}{\frac {a^{2}}{-1+2 a}-1}}\, \sqrt {\frac {x \left (-1+2 a \right )}{a^{2}}}\, F\left (\sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}, \sqrt {\frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-1\right )}}\right )}{\left (-1+2 a \right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}}-\frac {4 a^{3} \left (a -1\right ) \sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}\, \sqrt {\frac {-1+x}{\frac {a^{2}}{-1+2 a}-1}}\, \sqrt {\frac {x \left (-1+2 a \right )}{a^{2}}}\, \Pi \left (\sqrt {-\frac {\left (x -\frac {a^{2}}{-1+2 a}\right ) \left (-1+2 a \right )}{a^{2}}}, \frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-a \right )}, \sqrt {\frac {a^{2}}{\left (-1+2 a \right ) \left (\frac {a^{2}}{-1+2 a}-1\right )}}\right )}{\left (-1+2 a \right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}\, \left (\frac {a^{2}}{-1+2 a}-a \right )}\)	\(536\)

input

int((-a+(-1+2*a)*x)/(-a+x)/(a^2*x-(a^2+2*a-1)*x^2+(-1+2*a)*x^3)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

output

-2*a^2*(-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2)*((-1+x)/(a^2/(-1+2*a)-1))^(1 
/2)*(x/a^2*(-1+2*a))^(1/2)/(-a^2*x^2+2*a*x^3+a^2*x-2*a*x^2-x^3+x^2)^(1/2)* 
EllipticF((-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2),(a^2/(-1+2*a)/(a^2/(-1+2* 
a)-1))^(1/2))-4*a^3*(a-1)/(-1+2*a)*(-(x-a^2/(-1+2*a))/a^2*(-1+2*a))^(1/2)* 
((-1+x)/(a^2/(-1+2*a)-1))^(1/2)*(x/a^2*(-1+2*a))^(1/2)/(-a^2*x^2+2*a*x^3+a 
^2*x-2*a*x^2-x^3+x^2)^(1/2)/(a^2/(-1+2*a)-a)*EllipticPi((-(x-a^2/(-1+2*a)) 
/a^2*(-1+2*a))^(1/2),a^2/(-1+2*a)/(a^2/(-1+2*a)-a),(a^2/(-1+2*a)/(a^2/(-1+ 
2*a)-1))^(1/2))
 

3.1.83.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx=\log \left (-\frac {a^{2} - 2 \, {\left (a - 1\right )} x - x^{2} + 2 \, \sqrt {{\left (2 \, a - 1\right )} x^{3} + a^{2} x - {\left (a^{2} + 2 \, a - 1\right )} x^{2}}}{a^{2} - 2 \, a x + x^{2}}\right ) \]

input

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

output

log(-(a^2 - 2*(a - 1)*x - x^2 + 2*sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a 
- 1)*x^2))/(a^2 - 2*a*x + x^2))
 

3.1.83.6 Sympy [F]

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

input

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

output

Integral((2*a*x - a - x)/(sqrt(x*(x - 1)*(-a**2 + 2*a*x - x))*(-a + x)), x 
)
 

3.1.83.7 Maxima [F]

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

input

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

output

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

3.1.83.8 Giac [F]

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

input

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

output

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

3.1.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx=\text {Hanged} \]

input

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

output

\text{Hanged}
 

3.1.83.10 Reduce [F]

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

input

int(( - 2*a*x + a + x)/(sqrt( - a**2*x**2 + a**2*x + 2*a*x**3 - 2*a*x**2 - 
 x**3 + x**2)*(a - x)),x)
 

output

int(( - 2*a*x + a + x)/(sqrt( - a**2*x**2 + a**2*x + 2*a*x**3 - 2*a*x**2 - 
 x**3 + x**2)*(a - x)),x)