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

Optimal. Leaf size=46 \[ \log \left (\frac{-2 \left (\sqrt{(1-x) x \left (a^2-2 a x+x\right )}+x\right )-a^2+2 a x+x^2}{(a-x)^2}\right ) \]

[Out]

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

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Rubi [C]  time = 1.49236, antiderivative size = 180, normalized size of antiderivative = 3.91, number of steps used = 7, number of rules used = 7, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.137, Rules used = {2056, 6733, 1710, 1104, 419, 1220, 537} \[ \frac{4 (1-a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} \Pi \left (\frac{1}{a};\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}}-\frac{2 (1-2 a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} F\left (\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(-2*(1 - 2*a)*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*EllipticF[ArcSin[Sqrt[x]], -((1 - 2*a)/a^2)])/Sq
rt[a^2*x + (1 - 2*a - a^2)*x^2 - (1 - 2*a)*x^3] + (4*(1 - a)*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + ((1 - 2*a)*x)/a^2]*E
llipticPi[a^(-1), ArcSin[Sqrt[x]], -((1 - 2*a)/a^2)])/Sqrt[a^2*x + (1 - 2*a - a^2)*x^2 - (1 - 2*a)*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 1710

Int[((A_.) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist
[B/e, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[(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[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] && NegQ[c/a]

Rule 1104

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), 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 1220

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(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)*Sqrt[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] && NegQ[c/a]

Rule 537

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

Rubi steps

\begin{align*} \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 &=\frac{\left (\sqrt{x} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \int \frac{-a+(-1+2 a) x}{\sqrt{x} (-a+x) \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}} \, dx}{\sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=\frac{\left (2 \sqrt{x} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname{Subst}\left (\int \frac{-a+(-1+2 a) x^2}{\left (-a+x^2\right ) \sqrt{a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac{\left (4 (1-a) a \sqrt{x} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a+x^2\right ) \sqrt{a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}+\frac{\left (2 (-1+2 a) \sqrt{x} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac{\left (4 (1-a) a \sqrt{1-x} \sqrt{x} \sqrt{1+\frac{(1-2 a) x}{a^2}} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a+x^2\right ) \sqrt{1+\frac{2 (-1+2 a) x^2}{1-(-1+a)^2-2 a-a^2}} \sqrt{1+\frac{2 (-1+2 a) x^2}{1+(-1+a)^2-2 a-a^2}}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2+\left (1-2 a-a^2\right ) x+(-1+2 a) x^2} \sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}+\frac{\left (2 (-1+2 a) \sqrt{1-x} \sqrt{x} \sqrt{1+\frac{(1-2 a) x}{a^2}} \sqrt{a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{2 (-1+2 a) x^2}{1-(-1+a)^2-2 a-a^2}} \sqrt{1+\frac{2 (-1+2 a) x^2}{1+(-1+a)^2-2 a-a^2}}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2+\left (1-2 a-a^2\right ) x+(-1+2 a) x^2} \sqrt{a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac{2 (1-2 a) \sqrt{1-x} \sqrt{x} \sqrt{1+\frac{(1-2 a) x}{a^2}} F\left (\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{a^2 x+\left (1-2 a-a^2\right ) x^2-(1-2 a) x^3}}+\frac{4 (1-a) \sqrt{1-x} \sqrt{x} \sqrt{1+\frac{(1-2 a) x}{a^2}} \Pi \left (\frac{1}{a};\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{a^2 x+\left (1-2 a-a^2\right ) x^2-(1-2 a) x^3}}\\ \end{align*}

Mathematica [C]  time = 1.07234, size = 133, normalized size = 2.89 \[ \frac{2 i (x-1)^{3/2} \sqrt{\frac{x}{x-1}} \sqrt{-\frac{a^2-2 a x+x}{(2 a-1) (x-1)}} \left (2 a \Pi \left (1-a;i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|-\frac{(a-1)^2}{2 a-1}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right ),-\frac{(a-1)^2}{2 a-1}\right )\right )}{\sqrt{-(x-1) x \left (a^2-2 a x+x\right )}} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

((2*I)*(-1 + x)^(3/2)*Sqrt[x/(-1 + x)]*Sqrt[-((a^2 + x - 2*a*x)/((-1 + 2*a)*(-1 + x)))]*(-EllipticF[I*ArcSinh[
1/Sqrt[-1 + x]], -((-1 + a)^2/(-1 + 2*a))] + 2*a*EllipticPi[1 - a, I*ArcSinh[1/Sqrt[-1 + x]], -((-1 + a)^2/(-1
+ 2*a))]))/Sqrt[-((-1 + x)*x*(a^2 + x - 2*a*x))]

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Maple [C]  time = 0.047, size = 536, normalized size = 11.7 \begin{align*} 2\,{\frac{{a}^{2}}{ \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}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3}}{ \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}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3} \left ( a-1 \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}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

2*a^2/(-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)*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/(-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)*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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

-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)

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Fricas [A]  time = 2.31541, size = 144, normalized size = 3.13 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)