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

Optimal. Leaf size=76 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x} \sqrt {a k^2+a+b}}{\sqrt {a} (x-1) \left (k^2 x-1\right )}\right )}{\sqrt {a} \sqrt {a k^2+a+b}} \]

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Rubi [C]  time = 3.22, antiderivative size = 299, normalized size of antiderivative = 3.93, number of steps used = 16, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {6718, 6688, 6728, 714, 115, 934, 12, 168, 537} \begin {gather*} \frac {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b-\sqrt {b^2-4 a^2 k^2}};\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 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b+\sqrt {b^2-4 a^2 k^2}};\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^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(a + b*x + a*k^2*x^2)),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)]) + (2*(
1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(-2*a)/(b - Sqrt[b^2 - 4*a^2*k^2]), ArcSin[Sqrt[-k^2]*Sqrt[
-x]], k^(-2)])/(a*Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + (2*(1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 -
k^2*x]*EllipticPi[(-2*a)/(b + Sqrt[b^2 - 4*a^2*k^2]), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(a*Sqrt[-k^2]*Sqr
t[(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 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^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (a+b x+a k^2 x^2\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+b x+a 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 {-1+k^2 x^2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a 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 {1}{a \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {2 a+b x}{a \sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a 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 \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}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a k^2 x^2\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^2}}{\sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}}+\frac {b+\sqrt {b^2-4 a^2 k^2}}{\sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 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^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}} \, 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 (\sqrt {2} \left (b-\sqrt {b^2-4 a^2 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 (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (b+\sqrt {b^2-4 a^2 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 (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \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^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 (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \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^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 (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \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 (2 \left (b-\sqrt {b^2-4 a^2 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 (b-\sqrt {b^2-4 a^2 k^2}-2 a k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (b+\sqrt {b^2-4 a^2 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 (b+\sqrt {b^2-4 a^2 k^2}-2 a k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \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 {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b-\sqrt {b^2-4 a^2 k^2}};\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 {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b+\sqrt {b^2-4 a^2 k^2}};\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 = 6.04, size = 345, normalized size = 4.54 \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 (2 a \left (k^2-1\right ) \sqrt {b^2-4 a^2 k^2} F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (b \sqrt {b^2-4 a^2 k^2}+2 a \left (\sqrt {b^2-4 a^2 k^2}-2 a k^2\right )+b^2\right ) \Pi \left (\frac {2 \left (a k^2+a+b\right )}{2 a k^2+b-\sqrt {b^2-4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (b \sqrt {b^2-4 a^2 k^2}+2 a \left (\sqrt {b^2-4 a^2 k^2}+2 a k^2\right )-b^2\right ) \Pi \left (\frac {2 \left (a k^2+a+b\right )}{2 a k^2+b+\sqrt {b^2-4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{a \sqrt {(x-1) x \left (k^2 x-1\right )} \left (a k^2+a+b\right ) \sqrt {b^2-4 a^2 k^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 76, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a+b+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+b+a k^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTan[(Sqrt[a + b + 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 + b + a*k^2])

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fricas [B]  time = 0.91, size = 413, normalized size = 5.43 \begin {gather*} \left [-\frac {\sqrt {-a^{2} k^{2} - a^{2} - a b} \log \left (\frac {a^{2} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} k^{4} + {\left (4 \, a^{2} + 3 \, a b\right )} k^{2}\right )} x^{3} + {\left (8 \, a^{2} k^{4} + 2 \, {\left (9 \, a^{2} + 4 \, a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left (a k^{2} x^{2} - {\left (2 \, a k^{2} + 2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} k^{2} - a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} k^{2} + 4 \, a^{2} + 3 \, a b\right )} x}{a^{2} k^{4} x^{4} + 2 \, a b k^{2} x^{3} + 2 \, a b x + {\left (2 \, a^{2} k^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} k^{2} + a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left (a k^{2} x^{2} - {\left (2 \, a k^{2} + 2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} k^{2} + a^{2} + a b}}{2 \, {\left ({\left (a^{2} k^{4} + {\left (a^{2} + a b\right )} k^{2}\right )} x^{3} - {\left (a^{2} k^{4} + {\left (2 \, a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} k^{2} + a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} k^{2} + a^{2} + a b}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a^2*k^2 - a^2 - a*b)*log((a^2*k^4*x^4 - 2*(4*a^2*k^4 + (4*a^2 + 3*a*b)*k^2)*x^3 + (8*a^2*k^4 + 2*(
9*a^2 + 4*a*b)*k^2 + 8*a^2 + 8*a*b + b^2)*x^2 - 4*(a*k^2*x^2 - (2*a*k^2 + 2*a + b)*x + a)*sqrt(k^2*x^3 - (k^2
+ 1)*x^2 + x)*sqrt(-a^2*k^2 - a^2 - a*b) + a^2 - 2*(4*a^2*k^2 + 4*a^2 + 3*a*b)*x)/(a^2*k^4*x^4 + 2*a*b*k^2*x^3
+ 2*a*b*x + (2*a^2*k^2 + b^2)*x^2 + a^2))/(a^2*k^2 + a^2 + a*b), arctan(1/2*(a*k^2*x^2 - (2*a*k^2 + 2*a + b)*
x + a)*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*sqrt(a^2*k^2 + a^2 + a*b)/((a^2*k^4 + (a^2 + a*b)*k^2)*x^3 - (a^2*k^4
+ (2*a^2 + a*b)*k^2 + a^2 + a*b)*x^2 + (a^2*k^2 + a^2 + a*b)*x))/sqrt(a^2*k^2 + a^2 + a*b)]

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

[Out]

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

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maple [C]  time = 0.27, size = 1175, normalized size = 15.46

method	result	size
elliptic	\(-\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 )}{a \,k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {4 \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 {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\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}}-\frac {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b^{2}}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a^{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 {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \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}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a^{2}}-\frac {4 \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 {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\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}}+\frac {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b^{2}}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a^{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 {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \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}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a^{2}}\)	\(1175\)
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 {-\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 {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b^{2}}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a}+\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 {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \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}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a}+\frac {4 \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 {-b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}-\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\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}}+\frac {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b^{2}}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a}+\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 {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \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}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right ) a}-\frac {4 \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 {b +\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{\sqrt {-4 a^{2} k^{2}+b^{2}}\, k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {b}{2 a \,k^{2}}+\frac {\sqrt {-4 a^{2} k^{2}+b^{2}}}{2 a \,k^{2}}\right )}}{a}\)	\(1178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/k^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)*Ellip
ticF((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1))^(1/2))+4/(-4*a^2*k^2+b^2)^(1/2)/k^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)/(1/k^2+1/2*b/a/k^2-1/2/a/k^2*(-4*a^2*k^2+b
^2)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-1/2/a/k^2*(-b+(-4*a^2*k^2+b^2)^(1/2))),(1/k^2/(1/k^2
-1))^(1/2))-1/(-4*a^2*k^2+b^2)^(1/2)/k^4*(-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+1/2*b/a/k^2-1/2/a/k^2*(-4*a^2*k^2+b^2)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1
/2),1/k^2/(1/k^2-1/2/a/k^2*(-b+(-4*a^2*k^2+b^2)^(1/2))),(1/k^2/(1/k^2-1))^(1/2))*b^2/a^2+1/k^4*(-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+1/2*b/a/k^2-1/2/a/k^2*
(-4*a^2*k^2+b^2)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-1/2/a/k^2*(-b+(-4*a^2*k^2+b^2)^(1/2))),
(1/k^2/(1/k^2-1))^(1/2))*b/a^2-4/(-4*a^2*k^2+b^2)^(1/2)/k^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)/(1/k^2+1/2*b/a/k^2+1/2/a/k^2*(-4*a^2*k^2+b^2)^(1/2))*EllipticPi(
(-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+1/2*(b+(-4*a^2*k^2+b^2)^(1/2))/a/k^2),(1/k^2/(1/k^2-1))^(1/2))+1/(-4*a^2*k
^2+b^2)^(1/2)/k^4*(-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+1/2*b/a/k^2+1/2/a/k^2*(-4*a^2*k^2+b^2)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+1/2*(b
+(-4*a^2*k^2+b^2)^(1/2))/a/k^2),(1/k^2/(1/k^2-1))^(1/2))*b^2/a^2+1/k^4*(-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+1/2*b/a/k^2+1/2/a/k^2*(-4*a^2*k^2+b^2)^(1/2))*
EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+1/2*(b+(-4*a^2*k^2+b^2)^(1/2))/a/k^2),(1/k^2/(1/k^2-1))^(1/2))*
b/a^2

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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((k^2*x^2-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(a*k^2*x^2+b*x+a),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(2*a*k-b>0)', see `assume?` for
more details)Is 2*a*k-b positive, negative or zero?

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mupad [B]  time = 3.76, size = 90, normalized size = 1.18 \begin {gather*} \frac {\ln \left (\frac {a-2\,a\,x-b\,x-2\,a\,k^2\,x+a\,k^2\,x^2+\sqrt {a\,\left (a\,k^2+a+b\right )}\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a\,k^2\,x^2+b\,x+a}\right )\,1{}\mathrm {i}}{\sqrt {a^2\,k^2+a^2+b\,a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((a - 2*a*x - b*x + (a*(a + b + a*k^2))^(1/2)*(x*(k^2*x - 1)*(x - 1))^(1/2)*2i - 2*a*k^2*x + a*k^2*x^2)/(a
+ b*x + a*k^2*x^2))*1i)/(a*b + a^2 + a^2*k^2)^(1/2)

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

[Out]

Timed out

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