\(\int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+(-1-2 a+a^2) x^2+x^3}} \, dx\) [82]

Optimal result

Integrand size = 40, antiderivative size = 71 \[ \int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \, dx=\frac {\log \left (\frac {-a^2+2 \left (-a+a^2\right ) x+x^2-2 a \sqrt {-\left (\left (-2 a+a^2\right ) x\right )+\left (-1-2 a+a^2\right ) x^2+x^3}}{a^2-2 a x+x^2}\right )}{a} \]

[Out]

0

Rubi [C] (warning: unable to verify)

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

Time = 1.06 (sec) , antiderivative size = 529, normalized size of antiderivative = 7.45, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2081, 6865, 1722, 1117, 1720} \[ \int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \, dx=\frac {2 (1-a) \sqrt {x} \sqrt {-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2} \arctan \left (\frac {\sqrt {-a^2+2 a-1} \sqrt {x}}{\sqrt {-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}}\right )}{a \sqrt {-a^2+2 a-1} \sqrt {-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac {((2-a) a)^{3/4} \sqrt {x} \left (\frac {x}{\sqrt {(2-a) a}}+1\right ) \sqrt {\frac {-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac {x}{\sqrt {(2-a) a}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{(2-a) a}}\right ),\frac {1}{4} \left (\frac {-a^2+2 a+1}{\sqrt {(2-a) a}}+2\right )\right )}{a \sqrt {-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac {(2-a) \left (1-\sqrt {(2-a) a}\right ) \sqrt {x} \left (\frac {x}{\sqrt {(2-a) a}}+1\right ) \sqrt {\frac {-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac {x}{\sqrt {(2-a) a}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2-a}+\sqrt {a}\right )^2}{4 \sqrt {(2-a) a}},2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{(2-a) a}}\right ),\frac {1}{4} \left (\frac {-a^2+2 a+1}{\sqrt {(2-a) a}}+2\right )\right )}{((2-a) a)^{3/4} \sqrt {-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}} \]

[In]

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

[Out]

(2*(1 - a)*Sqrt[x]*Sqrt[(2 - a)*a - (1 + 2*a - a^2)*x + x^2]*ArcTan[(Sqrt[-1 + 2*a - a^2]*Sqrt[x])/Sqrt[(2 - a
)*a - (1 + 2*a - a^2)*x + x^2]])/(a*Sqrt[-1 + 2*a - a^2]*Sqrt[(2 - a)*a*x - (1 + 2*a - a^2)*x^2 + x^3]) + (((2
- a)*a)^(3/4)*Sqrt[x]*(1 + x/Sqrt[(2 - a)*a])*Sqrt[((2 - a)*a - (1 + 2*a - a^2)*x + x^2)/((2 - a)*a*(1 + x/Sq
rt[(2 - a)*a])^2)]*EllipticF[2*ArcTan[Sqrt[x]/((2 - a)*a)^(1/4)], (2 + (1 + 2*a - a^2)/Sqrt[(2 - a)*a])/4])/(a
*Sqrt[(2 - a)*a*x - (1 + 2*a - a^2)*x^2 + x^3]) + ((2 - a)*(1 - Sqrt[(2 - a)*a])*Sqrt[x]*(1 + x/Sqrt[(2 - a)*a
])*Sqrt[((2 - a)*a - (1 + 2*a - a^2)*x + x^2)/((2 - a)*a*(1 + x/Sqrt[(2 - a)*a])^2)]*EllipticPi[(Sqrt[2 - a] +
Sqrt[a])^2/(4*Sqrt[(2 - a)*a]), 2*ArcTan[Sqrt[x]/((2 - a)*a)^(1/4)], (2 + (1 + 2*a - a^2)/Sqrt[(2 - a)*a])/4]
)/(((2 - a)*a)^(3/4)*Sqrt[(2 - a)*a*x - (1 + 2*a - a^2)*x^2 + x^3])

Rule 1117

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

Rule 1720

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

Rule 1722

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

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x+x^2}\right ) \int \frac {-2+a+x}{\sqrt {x} (-a+x) \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x+x^2}} \, dx}{\sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x+x^2}\right ) \text {Subst}\left (\int \frac {-2+a+x^2}{\left (-a+x^2\right ) \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \\ & = \frac {\left (2 \sqrt {(2-a) a} \sqrt {x} \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x^2+x^4}} \, dx,x,\sqrt {x}\right )}{a \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}}+\frac {\left (2 \left (1-\frac {\sqrt {a}}{\sqrt {2-a}}\right ) (2-2 a) (2-a) a \sqrt {x} \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x+x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {2-a} \sqrt {a}}}{\left (-a+x^2\right ) \sqrt {(2-a) a+\left (-1-2 a+a^2\right ) x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (-((2-a) a)+a^2\right ) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \\ & = \frac {2 (1-a) \sqrt {x} \sqrt {(2-a) a-\left (1+2 a-a^2\right ) x+x^2} \arctan \left (\frac {\sqrt {-1+2 a-a^2} \sqrt {x}}{\sqrt {(2-a) a-\left (1+2 a-a^2\right ) x+x^2}}\right )}{a \sqrt {-1+2 a-a^2} \sqrt {(2-a) a x-\left (1+2 a-a^2\right ) x^2+x^3}}+\frac {((2-a) a)^{3/4} \sqrt {x} \left (1+\frac {x}{\sqrt {(2-a) a}}\right ) \sqrt {\frac {(2-a) a-\left (1+2 a-a^2\right ) x+x^2}{(2-a) a \left (1+\frac {x}{\sqrt {(2-a) a}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{(2-a) a}}\right ),\frac {1}{4} \left (2+\frac {1+2 a-a^2}{\sqrt {(2-a) a}}\right )\right )}{a \sqrt {(2-a) a x-\left (1+2 a-a^2\right ) x^2+x^3}}+\frac {\sqrt [4]{(2-a) a} \left (1-\sqrt {(2-a) a}\right ) \sqrt {x} \left (1+\frac {x}{\sqrt {(2-a) a}}\right ) \sqrt {\frac {(2-a) a-\left (1+2 a-a^2\right ) x+x^2}{(2-a) a \left (1+\frac {x}{\sqrt {(2-a) a}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2-a}+\sqrt {a}\right )^2}{4 \sqrt {(2-a) a}},2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{(2-a) a}}\right ),\frac {1}{4} \left (2+\frac {1+2 a-a^2}{\sqrt {(2-a) a}}\right )\right )}{a \sqrt {(2-a) a x-\left (1+2 a-a^2\right ) x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65 \[ \int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {\left (2 a-a^2\right ) x+\left (-1-2 a+a^2\right ) x^2+x^3}}{a (-1+x)}\right )}{a} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 1.

Time = 0.51 (sec) , antiderivative size = 317, normalized size of antiderivative = 4.46

method	result	size
default	\(\frac {2 \left (a^{2}-2 a \right ) \sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}\, \sqrt {\frac {-1+x}{-a^{2}+2 a -1}}\, \sqrt {\frac {x}{-a^{2}+2 a}}\, F\left (\sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}, \sqrt {\frac {-a^{2}+2 a}{-a^{2}+2 a -1}}\right )}{\sqrt {a^{2} x^{2}-a^{2} x -2 a \,x^{2}+x^{3}+2 a x -x^{2}}}-\frac {2 \left (-2 a +2\right ) \left (a^{2}-2 a \right ) \sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}\, \sqrt {\frac {-1+x}{-a^{2}+2 a -1}}\, \sqrt {\frac {x}{-a^{2}+2 a}}\, \Pi \left (\sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}, \frac {-a^{2}+2 a}{-a^{2}+a}, \sqrt {\frac {-a^{2}+2 a}{-a^{2}+2 a -1}}\right )}{\sqrt {a^{2} x^{2}-a^{2} x -2 a \,x^{2}+x^{3}+2 a x -x^{2}}\, \left (-a^{2}+a \right )}\)	\(317\)
elliptic	\(\frac {2 \left (a^{2}-2 a \right ) \sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}\, \sqrt {\frac {-1+x}{-a^{2}+2 a -1}}\, \sqrt {\frac {x}{-a^{2}+2 a}}\, F\left (\sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}, \sqrt {\frac {-a^{2}+2 a}{-a^{2}+2 a -1}}\right )}{\sqrt {a^{2} x^{2}-a^{2} x -2 a \,x^{2}+x^{3}+2 a x -x^{2}}}+\frac {2 \left (2 a -2\right ) \left (a^{2}-2 a \right ) \sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}\, \sqrt {\frac {-1+x}{-a^{2}+2 a -1}}\, \sqrt {\frac {x}{-a^{2}+2 a}}\, \Pi \left (\sqrt {\frac {a^{2}-2 a +x}{a^{2}-2 a}}, \frac {-a^{2}+2 a}{-a^{2}+a}, \sqrt {\frac {-a^{2}+2 a}{-a^{2}+2 a -1}}\right )}{\sqrt {a^{2} x^{2}-a^{2} x -2 a \,x^{2}+x^{3}+2 a x -x^{2}}\, \left (-a^{2}+a \right )}\)	\(317\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99 \[ \int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \, dx=\frac {\log \left (-\frac {a^{2} - 2 \, {\left (a^{2} - a\right )} x - x^{2} + 2 \, \sqrt {{\left (a^{2} - 2 \, a - 1\right )} x^{2} + x^{3} - {\left (a^{2} - 2 \, a\right )} x} a}{a^{2} - 2 \, a x + x^{2}}\right )}{a} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.92 \[ \int \frac {-2+a+x}{(-a+x) \sqrt {(2-a) a x+\left (-1-2 a+a^2\right ) x^2+x^3}} \, dx=\frac {2\,\sqrt {\frac {x}{2\,a-a^2}}\,\sqrt {-\frac {x-1}{a^2-2\,a+1}}\,{\left (a-1\right )}^2\,\sqrt {\frac {a^2-2\,a+x}{a^2-2\,a+1}}\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {a^2-2\,a+x}{a^2-2\,a+1}}\right )\middle |-\frac {a^2-2\,a+1}{2\,a-a^2}\right )-2\,\Pi \left (-\frac {a^2-2\,a+1}{a-a^2};\mathrm {asin}\left (\sqrt {\frac {a^2-2\,a+x}{a^2-2\,a+1}}\right )\middle |-\frac {a^2-2\,a+1}{2\,a-a^2}\right )\right )}{a\,\sqrt {x^3+\left (a^2-2\,a-1\right )\,x^2+\left (2\,a-a^2\right )\,x}} \]

[In]

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

[Out]

(2*(x/(2*a - a^2))^(1/2)*(-(x - 1)/(a^2 - 2*a + 1))^(1/2)*(a - 1)^2*((x - 2*a + a^2)/(a^2 - 2*a + 1))^(1/2)*(a
*ellipticF(asin(((x - 2*a + a^2)/(a^2 - 2*a + 1))^(1/2)), -(a^2 - 2*a + 1)/(2*a - a^2)) - 2*ellipticPi(-(a^2 -
2*a + 1)/(a - a^2), asin(((x - 2*a + a^2)/(a^2 - 2*a + 1))^(1/2)), -(a^2 - 2*a + 1)/(2*a - a^2))))/(a*(x*(2*a
- a^2) - x^2*(2*a - a^2 + 1) + x^3)^(1/2))