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]

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

[Out]

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)

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.96, 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 = {2081, 6865, 1724, 1118, 430, 1234, 551} \begin {gather*} \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-\left ((1-2 a) x^3\right )}}-\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-\left ((1-2 a) x^3\right )}} \end {gather*}

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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 1118

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 1234

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 1724

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 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*} \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 20.53, size = 133, normalized size = 2.89 \begin {gather*} \frac {2 i (-1+x)^{3/2} \sqrt {\frac {x}{-1+x}} \sqrt {-\frac {a^2+x-2 a x}{(-1+2 a) (-1+x)}} \left (-F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|-\frac {(-1+a)^2}{-1+2 a}\right )+2 a \Pi \left (1-a;i \sinh ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )|-\frac {(-1+a)^2}{-1+2 a}\right )\right )}{\sqrt {-\left ((-1+x) x \left (a^2+x-2 a x\right )\right )}} \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.10, size = 536, normalized size = 11.65

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

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,method=_RETURNVERBOSE)

[Out]

-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))+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*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.32, size = 63, normalized size = 1.37 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

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]

Could not integrate

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

\text{Hanged}

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