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

Optimal. Leaf size=1 \[ 0 \]

[Out]

0

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Rubi [C]  time = 1.66858, antiderivative size = 529, normalized size of antiderivative = 529., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2056, 6733, 1708, 1103, 1706} \[ \frac{2 (1-a) \sqrt{x} \sqrt{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2} \tan ^{-1}\left (\frac{\sqrt{-a^2+2 a-1} \sqrt{x}}{\sqrt{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}}\right )}{a \sqrt{-a^2+2 a-1} \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac{((2-a) a)^{3/4} \sqrt{x} \left (\frac{x}{\sqrt{(2-a) a}}+1\right ) \sqrt{\frac{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac{x}{\sqrt{(2-a) a}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt [4]{(2-a) a}}\right )|\frac{1}{4} \left (\frac{-a^2+2 a+1}{\sqrt{(2-a) a}}+2\right )\right )}{a \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}}+\frac{(2-a) \left (1-\sqrt{(2-a) a}\right ) \sqrt{x} \left (\frac{x}{\sqrt{(2-a) a}}+1\right ) \sqrt{\frac{-\left (-a^2+2 a+1\right ) x+(2-a) a+x^2}{(2-a) a \left (\frac{x}{\sqrt{(2-a) a}}+1\right )^2}} \Pi \left (\frac{\left (\sqrt{2-a}+\sqrt{a}\right )^2}{4 \sqrt{(2-a) a}};2 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt [4]{(2-a) a}}\right )|\frac{1}{4} \left (\frac{-a^2+2 a+1}{\sqrt{(2-a) a}}+2\right )\right )}{((2-a) a)^{3/4} \sqrt{-\left (-a^2+2 a+1\right ) x^2+(2-a) a x+x^3}} \]

Warning: Unable to verify antiderivative.

[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 2056

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 6733

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]

Rule 1708

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 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1706

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)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2 - (b*A)/(4*a*B)])/(4*d*e*A*q*Sqrt[
a + b*x^2 + c*x^4]), 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]

Rubi steps

\begin{align*} \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{\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \tan ^{-1}\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}} F\left (2 \tan ^{-1}\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}} \Pi \left (\frac{\left (\sqrt{2-a}+\sqrt{a}\right )^2}{4 \sqrt{(2-a) a}};2 \tan ^{-1}\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 [C]  time = 0.585636, size = 127, normalized size = 127. \[ \frac{2 \sqrt{\frac{1}{x-1}+1} (x-1)^{3/2} \sqrt{\frac{(a-1)^2}{x-1}+1} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-(a-1)^2}}{\sqrt{x-1}}\right ),\frac{1}{(a-1)^2}\right )-2 \Pi \left (\frac{1}{1-a};\sin ^{-1}\left (\frac{\sqrt{-(a-1)^2}}{\sqrt{x-1}}\right )|\frac{1}{(a-1)^2}\right )\right )}{\sqrt{-(a-1)^2} \sqrt{(x-1) x \left (a^2-2 a+x\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.041, size = 317, normalized size = 317. \begin{align*} 2\,{\frac{{a}^{2}-2\,a}{\sqrt{{a}^{2}{x}^{2}-{a}^{2}x-2\,a{x}^{2}+{x}^{3}+2\,ax-{x}^{2}}}\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}}}{\it EllipticF} \left ( \sqrt{{\frac{{a}^{2}-2\,a+x}{{a}^{2}-2\,a}}},\sqrt{{\frac{-{a}^{2}+2\,a}{-{a}^{2}+2\,a-1}}} \right ) }+2\,{\frac{ \left ( 2\,a-2 \right ) \left ({a}^{2}-2\,a \right ) }{\sqrt{{a}^{2}{x}^{2}-{a}^{2}x-2\,a{x}^{2}+{x}^{3}+2\,ax-{x}^{2}} \left ( -{a}^{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}}}{\it EllipticPi} \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C]  time = 2.28269, size = 150, normalized size = 150. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + x - 2}{\sqrt{x \left (x - 1\right ) \left (a^{2} - 2 a + x\right )} \left (- a + x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)