3.24.14 \(\int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x (1-k^2 x)} (-1+k^4 x^4)} \, dx\)

Optimal. Leaf size=178 \[ \frac {\left (-2 a k^2-b\right ) \tan ^{-1}\left (\frac {(k-1) x}{\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}\right )}{4 (k-1) k^2}+\frac {\left (-2 a k^2-b\right ) \tan ^{-1}\left (\frac {(k+1) x}{\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}\right )}{4 k^2 (k+1)}+\frac {\left (b-2 a k^2\right ) \tan ^{-1}\left (\frac {\sqrt {k^2+1} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{(x-1) \left (k^2 x-1\right )}\right )}{2 k^2 \sqrt {k^2+1}} \]

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Rubi [C]  time = 7.96, antiderivative size = 495, normalized size of antiderivative = 2.78, number of steps used = 28, number of rules used = 8, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6718, 6725, 115, 6688, 934, 12, 168, 537} \begin {gather*} -\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 a k^2+b\right ) \Pi \left (-\frac {1}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \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 a k^2+b\right ) \Pi \left (\frac {1}{k};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \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 (b-2 a k^2\right ) \Pi \left (-\frac {1}{\sqrt {-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \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 (b-2 a k^2\right ) \Pi \left (\frac {1}{\sqrt {-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{2 \left (-k^2\right )^{3/2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 a \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 )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^2 + a*k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^4*x^4)),x]

[Out]

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

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 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 6725

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

Rubi steps

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

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Mathematica [C]  time = 5.00, size = 282, normalized size = 1.58 \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 (-(k+1) \left (\left (k^2+1\right ) \left (2 a k^2+b\right ) \Pi \left (\frac {k-1}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+(k-1) \left (b-2 a k^2\right ) \left ((1-i k) \Pi \left (\frac {k-i}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+(1+i k) \Pi \left (\frac {k+i}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )\right )+4 k^2 \left (a k^4+a+b\right ) F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (k^3-k^2+k-1\right ) \left (2 a k^2+b\right ) \Pi \left (1+\frac {1}{k};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{2 k^2 \left (k^4-1\right ) \sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + a*k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^4*x^4)),x]

[Out]

((I/2)*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + (1 - k^(-2))/(-1 + x)]*(-1 + x)^(3/2)*(4*k^2*(a + b + a*k^4)*EllipticF
[I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)] + (b + 2*a*k^2)*(-1 + k - k^2 + k^3)*EllipticPi[1 + k^(-1), I*ArcSinh[
1/Sqrt[-1 + x]], 1 - k^(-2)] - (1 + k)*((1 + k^2)*(b + 2*a*k^2)*EllipticPi[(-1 + k)/k, I*ArcSinh[1/Sqrt[-1 + x
]], 1 - k^(-2)] + (-1 + k)*(b - 2*a*k^2)*((1 - I*k)*EllipticPi[(-I + k)/k, I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-
2)] + (1 + I*k)*EllipticPi[(I + k)/k, I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)]))))/(k^2*(-1 + k^4)*Sqrt[(-1 + x)
*x*(-1 + k^2*x)])

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IntegrateAlgebraic [A]  time = 0.41, size = 178, normalized size = 1.00 \begin {gather*} \frac {\left (-b-2 a k^2\right ) \tan ^{-1}\left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 (-1+k) k^2}+\frac {\left (-b-2 a k^2\right ) \tan ^{-1}\left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 k^2 (1+k)}+\frac {\left (b-2 a k^2\right ) \tan ^{-1}\left (\frac {\sqrt {1+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{2 k^2 \sqrt {1+k^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + a*k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^4*x^4)),x]

[Out]

((-b - 2*a*k^2)*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(4*(-1 + k)*k^2) + ((-b - 2*a*k^2)*Ar
cTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(4*k^2*(1 + k)) + ((b - 2*a*k^2)*ArcTan[(Sqrt[1 + k^2]*S
qrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))])/(2*k^2*Sqrt[1 + k^2])

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fricas [B]  time = 1.70, size = 356, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (2 \, a k^{4} - {\left (2 \, a + b\right )} 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 \, {\left (k^{2} + 1\right )} x + 1\right )} \sqrt {k^{2} + 1}}{2 \, {\left ({\left (k^{4} + k^{2}\right )} x^{3} - {\left (k^{4} + 2 \, k^{2} + 1\right )} x^{2} + {\left (k^{2} + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} - 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} - {\left (2 \, a + b\right )} k^{2} + b k - b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} + 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} + {\left (2 \, a + b\right )} k^{2} + b k + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} - k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} - k^{2}\right )} x^{3} - {\left (k^{3} - k^{2} + k - 1\right )} x^{2} + {\left (k - 1\right )} x\right )}}\right )}{8 \, {\left (k^{6} - k^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*k^4*x^4+b*x^2+a)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4-1),x, algorithm="fricas")

[Out]

1/8*(2*(2*a*k^4 - (2*a + b)*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 + 1)*x + 1)*sqrt(k^2 + 1)/((k^4 + k^2)*x^3 - (k^4 + 2*k^2 + 1)*x^2 + (k^2 + 1)*x)) + (2*a*k^5 - 2*a*k^4 +
(2*a + b)*k^3 - (2*a + b)*k^2 + b*k - b)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*(k^2 + k +
1)*x + 1)/((k^3 + k^2)*x^3 - (k^3 + k^2 + k + 1)*x^2 + (k + 1)*x)) + (2*a*k^5 + 2*a*k^4 + (2*a + b)*k^3 + (2*a
+ b)*k^2 + b*k + b)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*(k^2 - k + 1)*x + 1)/((k^3 - k^
2)*x^3 - (k^3 - k^2 + k - 1)*x^2 + (k - 1)*x)))/(k^6 - k^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 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*k^4*x^4+b*x^2+a)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4-1),x, algorithm="giac")

[Out]

integrate((a*k^4*x^4 + b*x^2 + a)/((k^4*x^4 - 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

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maple [C]  time = 0.18, size = 604, normalized size = 3.39

method	result	size
default	\(-\frac {2 a \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^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\left (-2 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}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{2 k^{5} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}-\frac {\left (2 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}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{2 k^{5} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}+\frac {\left (-2 a \,k^{2}+b \right ) \left (\frac {i \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 {i}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {i}{k}\right )}-\frac {i \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 {i}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {i}{k}\right )}\right )}{2 k^{2}}\)	\(604\)
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 {\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}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}+\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}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}-\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}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\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}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\EllipticPi \left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )\right )}{2 k^{2} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\frac {\EllipticPi \left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 k^{4} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}\)	\(813\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*k^4*x^4+b*x^2+a)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4-1),x,method=_RETURNVERBOSE)

[Out]

-2*a/k^2*(-(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)*EllipticF
((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1))^(1/2))-1/2*(-2*a*k^2-b)/k^5*(-(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)/(1/k^2+1/k)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2
+1/k),(1/k^2/(1/k^2-1))^(1/2))-1/2*(2*a*k^2+b)/k^5*(-(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)/(1/k^2-1/k)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-1/k),(1/k^2/(1/k^2
-1))^(1/2))+1/2/k^2*(-2*a*k^2+b)*(I/k^3*(-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)/(1/k^2-I/k)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-I/k),(1/k^2/(1/k^2-1))^(1
/2))-I/k^3*(-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)/(1/
k^2+I/k)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+I/k),(1/k^2/(1/k^2-1))^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 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*k^4*x^4+b*x^2+a)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4-1),x, algorithm="maxima")

[Out]

integrate((a*k^4*x^4 + b*x^2 + a)/((k^4*x^4 - 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + a*k^4*x^4)/((k^4*x^4 - 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a k^{4} x^{4} + a + b x^{2}}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k x - 1\right ) \left (k x + 1\right ) \left (k^{2} x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*k**4*x**4+b*x**2+a)/((1-x)*x*(-k**2*x+1))**(1/2)/(k**4*x**4-1),x)

[Out]

Integral((a*k**4*x**4 + a + b*x**2)/(sqrt(x*(x - 1)*(k**2*x - 1))*(k*x - 1)*(k*x + 1)*(k**2*x**2 + 1)), x)

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