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

Optimal. Leaf size=57 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{\sqrt {a} (x-1)}\right )}{\sqrt {a} \sqrt {a+b}} \]

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Rubi [C]  time = 8.51, antiderivative size = 338, normalized size of antiderivative = 5.93, number of steps used = 12, number of rules used = 6, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6718, 6728, 115, 168, 538, 537} \begin {gather*} \frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (-\frac {2 (a+b) k^2}{-2 a k^2-2 b k^2+b+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (\frac {2 (a+b) k^2}{2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{(a+b) \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+b) \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticF[ArcSin[Sqrt[x]], k^2])/((a + b)*Sqrt[(1 - x)*x*(1 - k^2*x)])
+ (2*Sqrt[1 + (k^2*(1 - x))/(1 - k^2)]*Sqrt[1 - x]*Sqrt[x]*EllipticPi[(-2*(a + b)*k^2)/(b - 2*a*k^2 - 2*b*k^2
+ Sqrt[b^2 + 4*a^2*k^2 + 4*a*b*k^2]), ArcSin[Sqrt[1 - x]], -(k^2/(1 - k^2))])/((a + b)*Sqrt[(1 - x)*x*(1 - k^2
*x)]) + (2*Sqrt[1 + (k^2*(1 - x))/(1 - k^2)]*Sqrt[1 - x]*Sqrt[x]*EllipticPi[(2*(a + b)*k^2)/(2*a*k^2 - b*(1 -
2*k^2) + Sqrt[b^2 + 4*a^2*k^2 + 4*a*b*k^2]), ArcSin[Sqrt[1 - x]], -(k^2/(1 - k^2))])/((a + b)*Sqrt[(1 - x)*x*(
1 - k^2*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 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

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

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Mathematica [C]  time = 4.22, size = 227, normalized size = 3.98 \begin {gather*} -\frac {2 i \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \left (-\Pi \left (-\frac {2 (a+b) \left (k^2-1\right )}{-2 a k^2-2 b k^2+b+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )-\Pi \left (\frac {2 (a+b) \left (k^2-1\right )}{2 a k^2+2 b k^2-b+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{(a+b) \sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.33, size = 57, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.14, size = 345, normalized size = 6.05 \begin {gather*} \left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} k^{2} x^{3} + {\left (6 \, {\left (a^{2} + a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} x}{{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (a b + b^{2}\right )} k^{2} x^{3} + 2 \, a b x - {\left (2 \, {\left (a^{2} + a b\right )} k^{2} - b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} + a b}}{2 \, {\left ({\left (a^{2} + a b\right )} k^{2} x^{3} - {\left ({\left (a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} + a b}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a^2 - a*b)*log(((a^2 + 2*a*b + b^2)*k^4*x^4 - 2*(4*a^2 + 5*a*b + b^2)*k^2*x^3 + (6*(a^2 + a*b)*k^2
+ 8*a^2 + 8*a*b + b^2)*x^2 - 4*((a + b)*k^2*x^2 - (2*a + b)*x + a)*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*sqrt(-a^
2 - a*b) + a^2 - 2*(4*a^2 + 3*a*b)*x)/((a^2 + 2*a*b + b^2)*k^4*x^4 - 2*(a*b + b^2)*k^2*x^3 + 2*a*b*x - (2*(a^2
+ a*b)*k^2 - b^2)*x^2 + a^2))/(a^2 + a*b), arctan(1/2*((a + b)*k^2*x^2 - (2*a + b)*x + a)*sqrt(k^2*x^3 - (k^2
+ 1)*x^2 + x)*sqrt(a^2 + a*b)/((a^2 + a*b)*k^2*x^3 - ((a^2 + a*b)*k^2 + a^2 + a*b)*x^2 + (a^2 + a*b)*x))/sqrt
(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} - 2 \, k^{2} x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (a k^{2} + b k^{2}\right )} x^{2} - b x - a\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.30, size = 4588, normalized size = 80.49

method	result	size
default	\(\text {Expression too large to display}\)	\(4588\)
elliptic	\(\text {Expression too large to display}\)	\(4605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a+b)/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)*Ellip
ticF((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1))^(1/2))+1/(a+b)*(2/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)
^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*
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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+
b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*a*b+2/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/
(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2))*Ellipti
cPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-1/2*(b+(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(
1/2))/(a+b)*b^2-1/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*
k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*b^2+2/(1/(a
+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/
k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+
2/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b
-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(
1/2)*b-1/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^
(1/2)*b-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a*b*k^2
+b^2)^(1/2)*b-4/(1/(a+b)*a*b+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^
2+b^2)^(1/2)*b-b)/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/k^2/(a+b)-1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))*a-2/(1/(a+b)*a*b+1/
(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2+1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)-
1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))*b+2/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)
^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*
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+4*a*b*k^2+b^2)^(1/2))/k^2/(a
+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*a*b+2/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1
/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2))*Ellipt
icPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+1/2*(-b+(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))
^(1/2))/(a+b)*b^2-1/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*
b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*b^2-2/(1
/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/
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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2
)*a-2/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/
2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a*b*k^2+b
^2)^(1/2)*b+1/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+
b^2)^(1/2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))/(a+b)*(4*a^2*k^2+4*a
*b*k^2+b^2)^(1/2)*b-4/(1/(a+b)*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*
a*b*k^2+b^2)^(1/2)*b-b)/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/k^2/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))*a-2/(1/(a+b)
*a*b-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*a+1/(a+b)*b^2-1/(a+b)*(4*a^2*k^2+4*a*b*k^2+b^2)^(1/2)*b-b)/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/k^2
/(a+b)+1/2/k^2/(a+b)*(4*a^2*k^2+4*a*b*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+4*a*b*k^2+b^2)^(1/2))/k^2/(a+b)),(1/k^2/(1/k^2-1))^(1/2))*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.84, size = 122, normalized size = 2.14 \begin {gather*} \frac {\ln \left (\frac {a\,\sqrt {a\,\left (a+b\right )}-2\,a\,x\,\sqrt {a\,\left (a+b\right )}-b\,x\,\sqrt {a\,\left (a+b\right )}+a\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+b\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+a\,\left (a+b\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a+b\,x-a\,k^2\,x^2-b\,k^2\,x^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2+b\,a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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