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

Optimal. Leaf size=268 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x} \sqrt {-\sqrt {a} \sqrt {2 a k^2-b}+a k^2+a}}{\sqrt {a} (x-1) \left (k^2 x-1\right )}\right )}{\sqrt {a} \sqrt {-\sqrt {a} \sqrt {2 a k^2-b}+a k^2+a}}-\frac {\sqrt {\sqrt {a} \left (\sqrt {2 a k^2-b}+\sqrt {a} k^2+\sqrt {a}\right )} \tan ^{-1}\left (\frac {\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x} \sqrt {\sqrt {a} \sqrt {2 a k^2-b}+a k^2+a}}{\sqrt {a} (x-1) \left (k^2 x-1\right )}\right )}{a \left (\sqrt {2 a k^2-b}+\sqrt {a} k^2+\sqrt {a}\right )} \]

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Rubi [C]  time = 5.96, antiderivative size = 583, normalized size of antiderivative = 2.18, number of steps used = 28, number of rules used = 10, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6718, 6688, 6728, 714, 115, 6725, 934, 12, 168, 537} \begin {gather*} \frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^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} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {-b-\sqrt {b^2-4 a^2 k^4}}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^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} \Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {\sqrt {b^2-4 a^2 k^4}-b}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^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} \Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {\sqrt {b^2-4 a^2 k^4}-b}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \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} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticF[ArcSin[Sqrt[x]], k^2])/(a*Sqrt[(1 - x)*x*(1 - k^2*x)]) + ((1
- x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[-((Sqrt[2]*Sqrt[a])/Sqrt[-b - Sqrt[b^2 - 4*a^2*k^4]]), ArcSin
[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(a*Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 - x)*Sqrt[-x]*S
qrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(Sqrt[2]*Sqrt[a])/Sqrt[-b - Sqrt[b^2 - 4*a^2*k^4]], ArcSin[Sqrt[-k^2]*Sqrt[-
x]], k^(-2)])/(a*Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^
2*x]*EllipticPi[-((Sqrt[2]*Sqrt[a])/Sqrt[-b + Sqrt[b^2 - 4*a^2*k^4]]), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(
a*Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi
[(Sqrt[2]*Sqrt[a])/Sqrt[-b + Sqrt[b^2 - 4*a^2*k^4]], ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(a*Sqrt[-k^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 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 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]

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

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Mathematica [C]  time = 18.67, size = 13957, normalized size = 52.08 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.06, size = 268, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {a+a k^2-\sqrt {a} \sqrt {-b+2 a k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {a} \sqrt {a+a k^2-\sqrt {a} \sqrt {-b+2 a k^2}}}-\frac {\sqrt {\sqrt {a} \left (\sqrt {a}+\sqrt {a} k^2+\sqrt {-b+2 a k^2}\right )} \tan ^{-1}\left (\frac {\sqrt {a+a k^2+\sqrt {a} \sqrt {-b+2 a k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x) \left (-1+k^2 x\right )}\right )}{a \left (\sqrt {a}+\sqrt {a} k^2+\sqrt {-b+2 a k^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 6.45, size = 2407, normalized size = 8.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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maple [C]  time = 0.17, size = 312, normalized size = 1.16

method	result	size
elliptic	\(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{a \,k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,k^{4} \textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} b +2 a \right ) \left (a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b \right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b}{a \,k^{4}+a +b}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+b \right ) \left (a \,k^{4}+a +b \right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{a}\)	\(312\)
default	\(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{a \,k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,k^{4} \textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} b -2 a \right ) \left (a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b \right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {a \,k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+k^{2} \underline {\hspace {1.25 ex}}\alpha a +k^{2} \underline {\hspace {1.25 ex}}\alpha b +a +b}{a \,k^{4}+a +b}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 a \,k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+b \right ) \left (a \,k^{4}+a +b \right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{a}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k^4*x^4-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(a*k^4*x^4+b*x^2+a),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/a*sum((_alpha^2*b+2*a)/_alpha/(2*_alpha^2*a*k^4+b)*(_alpha^
3*a*k^6+_alpha^2*a*k^4+_alpha*a*k^2+_alpha*b*k^2+a+b)/(a*k^4+a+b)*(-(x-1/k^2)*k^2)^(1/2)*((-1+x)/(1/k^2-1))^(1
/2)*(k^2*x)^(1/2)/(x*(k^2*x^2-k^2*x-x+1))^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),(_alpha^3*a*k^6+_alpha^2*a*k
^4+_alpha*a*k^2+_alpha*b*k^2+a+b)/(a*k^4+a+b),(1/k^2/(1/k^2-1))^(1/2)),_alpha=RootOf(_Z^4*a*k^4+_Z^2*b+a))

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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