3.29.48 \(\int \frac {-a-b x+(b+a k^2) x^2}{\sqrt {(1-x) x (1-k^2 x)} (1-2 x+k^2 x^2)} \, dx\)

Optimal. Leaf size=293 \[ \frac {\left (2 i a k^2-2 a \sqrt {k^2-1} k^2-i b k^2-2 b \sqrt {k^2-1}+2 i b\right ) \tan ^{-1}\left (\frac {\sqrt {k^2-2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 (x-1) x}\right )}{2 k^2 \sqrt {k^2-1} \sqrt {k^2-2 i \sqrt {k^2-1}-2}}+\frac {\left (-2 i a k^2-2 a \sqrt {k^2-1} k^2+i b k^2-2 b \sqrt {k^2-1}-2 i b\right ) \tan ^{-1}\left (\frac {\sqrt {k^2+2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 (x-1) x}\right )}{2 k^2 \sqrt {k^2-1} \sqrt {k^2+2 i \sqrt {k^2-1}-2}} \]

________________________________________________________________________________________

Rubi [C]  time = 3.76, antiderivative size = 385, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 10, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6718, 6688, 6728, 6, 714, 115, 934, 12, 168, 537} \begin {gather*} -\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (-2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \Pi \left (\frac {1}{1-\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (1-\sqrt {1-k^2}\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \Pi \left (\frac {1}{\sqrt {1-k^2}+1};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \left (\sqrt {1-k^2}+1\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+\frac {b}{k^2}\right ) F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 115

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

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)
])

Rule 714

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c*x)/b]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6718

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 2.81, size = 244, normalized size = 0.83 \begin {gather*} \frac {i \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \left (\left (b \left (\sqrt {1-k^2}-1\right )-2 a k^2\right ) \Pi \left (\frac {k^2-1}{k^2-\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (2 a k^2+b \sqrt {1-k^2}+b\right ) \Pi \left (\frac {k^2-1}{k^2+\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+2 a \sqrt {1-k^2} k^2 F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{k^2 \sqrt {1-k^2} \sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + (1 - k^(-2))/(-1 + x)]*(-1 + x)^(3/2)*(2*a*k^2*Sqrt[1 - k^2]*EllipticF[I*A
rcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)] + (-2*a*k^2 + b*(-1 + Sqrt[1 - k^2]))*EllipticPi[(-1 + k^2)/(-1 + k^2 - Sq
rt[1 - k^2]), I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)] + (b + 2*a*k^2 + b*Sqrt[1 - k^2])*EllipticPi[(-1 + k^2)/(
-1 + k^2 + Sqrt[1 - k^2]), I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)]))/(k^2*Sqrt[1 - k^2]*Sqrt[(-1 + x)*x*(-1 + k
^2*x)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.56, size = 293, normalized size = 1.00 \begin {gather*} \frac {\left (2 i b+2 i a k^2-i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \tan ^{-1}\left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}+\frac {\left (-2 i b-2 i a k^2+i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \tan ^{-1}\left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2+2 i \sqrt {-1+k^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-a - b*x + (b + a*k^2)*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 - 2*x + k^2*x^2)),x]

[Out]

(((2*I)*b + (2*I)*a*k^2 - I*b*k^2 - 2*b*Sqrt[-1 + k^2] - 2*a*k^2*Sqrt[-1 + k^2])*ArcTan[(Sqrt[-2 + k^2 - (2*I)
*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(k^2*(-1 + x)*x)])/(2*k^2*Sqrt[-1 + k^2]*Sqrt[-2 + k^2 -
(2*I)*Sqrt[-1 + k^2]]) + (((-2*I)*b - (2*I)*a*k^2 + I*b*k^2 - 2*b*Sqrt[-1 + k^2] - 2*a*k^2*Sqrt[-1 + k^2])*Arc
Tan[(Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(k^2*(-1 + x)*x)])/(2*k^2*Sqrt[
-1 + k^2]*Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]])

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 491, normalized size = 1.68 \begin {gather*} \left [-\frac {{\left (2 \, a k^{2} + b\right )} \sqrt {-k^{2} + 1} \log \left (\frac {k^{4} x^{4} - 4 \, {\left (2 \, k^{4} - k^{2}\right )} x^{3} + 2 \, {\left (4 \, k^{4} + k^{2} - 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {-k^{2} + 1} - 4 \, {\left (2 \, k^{2} - 1\right )} x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) - {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {2 \, {\left (2 \, a k^{2} + b\right )} \sqrt {k^{2} - 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {k^{2} - 1}}{2 \, {\left ({\left (k^{4} - k^{2}\right )} x^{3} - {\left (k^{4} - 1\right )} x^{2} + {\left (k^{2} - 1\right )} x\right )}}\right ) + {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((2*a*k^2 + b)*sqrt(-k^2 + 1)*log((k^4*x^4 - 4*(2*k^4 - k^2)*x^3 + 2*(4*k^4 + k^2 - 2)*x^2 - 4*sqrt(k^2*
x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*k^2*x + 1)*sqrt(-k^2 + 1) - 4*(2*k^2 - 1)*x + 1)/(k^4*x^4 - 4*k^2*x^3 +
2*(k^2 + 2)*x^2 - 4*x + 1)) - (b*k^2 - b)*log((k^4*x^4 + 4*k^2*x^3 - 2*(3*k^2 + 2)*x^2 - 4*sqrt(k^2*x^3 - (k^2
+ 1)*x^2 + x)*(k^2*x^2 - 1) + 4*x + 1)/(k^4*x^4 - 4*k^2*x^3 + 2*(k^2 + 2)*x^2 - 4*x + 1)))/(k^4 - k^2), 1/4*(
2*(2*a*k^2 + b)*sqrt(k^2 - 1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*k^2*x + 1)*sqrt(k^2 -
1)/((k^4 - k^2)*x^3 - (k^4 - 1)*x^2 + (k^2 - 1)*x)) + (b*k^2 - b)*log((k^4*x^4 + 4*k^2*x^3 - 2*(3*k^2 + 2)*x^2
- 4*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 1) + 4*x + 1)/(k^4*x^4 - 4*k^2*x^3 + 2*(k^2 + 2)*x^2 - 4*x +
1)))/(k^4 - k^2)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a k^{2} + b\right )} x^{2} - b x - a}{{\left (k^{2} x^{2} - 2 \, x + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.35, size = 1907, normalized size = 6.51

method	result	size
default	\(-\frac {2 \left (a \,k^{2}+b \right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a \,k^{2}}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a \,k^{2}}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}}{k^{2}}\)	\(1907\)
elliptic	\(-\frac {2 a \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{4}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{4}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{4}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{2}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{\sqrt {-k^{2}+1}\, \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, k^{4}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {-1+\sqrt {-k^{2}+1}}{k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\left (-k^{2}+1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}\)	\(2014\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*k^2+b)/k^4*(-(x-1/k^2)*k^2)^(1/2)*((-1+x)/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*E
llipticF((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1))^(1/2))+1/k^2*(2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1
/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1
+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*a-2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2
)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2)
)/k^2),(1/k^2/(1/k^2-1))^(1/2))*b+2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)
/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(
1/k^2-1))^(1/2))/k^2*b+2/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2
*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2
-1))^(1/2))*a-1/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*
x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2
))*b+2/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x
)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2*b-
2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*Ell
ipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*a*k^2+2/(-k^2+1)*
(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(
x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*a-2/(-k^2+1)*(-k^2*x+1)^(1/
2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^
(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*b+2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-
1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1
/k^2+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2*b-2/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1
/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2
+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*a+1/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-
1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(-1+(-k
^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*b-2/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1
/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1
/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2*b-2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x
)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1/2))/k^2),
(1/k^2/(1/k^2-1))^(1/2))*a*k^2)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(k-1>0)', see `assume?` for mor
e details)Is k-1 positive, negative or zero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (-a\,k^2-b\right )\,x^2+b\,x+a}{\left (k^2\,x^2-2\,x+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________