\(\int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x (1-k^2 x)} (-a-b x+(a k^2+b k^2) x^2)} \, dx\) [735]

Optimal result

Integrand size = 60, antiderivative size = 57 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a+b} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \]

[Out]

2*arctan((a+b)^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/a^(1/2)/(-1+x))/a^(1/2)/(a+b)^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 5.92 (sec) , antiderivative size = 338, normalized size of antiderivative = 5.93, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6850, 6860, 116, 174, 552, 551} \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (-\frac {2 (a+b) k^2}{-2 a k^2-2 b k^2+b+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (\frac {2 (a+b) k^2}{2 a k^2-b \left (1-2 k^2\right )+\sqrt {b^2+4 a k^2 b+4 a^2 k^2}},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{(a+b) \sqrt {(1-x) x \left (1-k^2 x\right )}} \]

[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 116

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

Rule 174

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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 6850

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 6860

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*} \text {integral}& = \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} \operatorname {EllipticF}\left (\arcsin \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} \operatorname {EllipticF}\left (\arcsin \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} \operatorname {EllipticF}\left (\arcsin \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 ) \text {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 ) \text {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} \operatorname {EllipticF}\left (\arcsin \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 ) \text {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 ) \text {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} \operatorname {EllipticF}\left (\arcsin \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} \operatorname {EllipticPi}\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}},\arcsin \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} \operatorname {EllipticPi}\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}},\arcsin \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*}

Mathematica [A] (verified)

Time = 15.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a+b} \sqrt {(-1+x) x \left (-1+k^2 x\right )}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 2.06 (sec) , antiderivative size = 4588, normalized size of antiderivative = 80.49

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

[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)*((x-1)/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*Ellipt
icF((-(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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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^2-1/(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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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
-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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^2-1/(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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)*x-1/(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/
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)*x-1/(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^2-1/2*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)*x-1/(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^2-1/2*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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (46) = 92\).

Time = 0.50 (sec) , antiderivative size = 345, normalized size of antiderivative = 6.05 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} k^{2} x^{3} + {\left (6 \, {\left (a^{2} + a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} x}{{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (a b + b^{2}\right )} k^{2} x^{3} + 2 \, a b x - {\left (2 \, {\left (a^{2} + a b\right )} k^{2} - b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} + a b}}{2 \, {\left ({\left (a^{2} + a b\right )} k^{2} x^{3} - {\left ({\left (a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} + a b}}\right ] \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (a k^{2} + b k^{2}\right )} x^{2} - b x - a\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (a k^{2} + b k^{2}\right )} x^{2} - b x - a\right )}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.14 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {\ln \left (\frac {a\,\sqrt {a\,\left (a+b\right )}-2\,a\,x\,\sqrt {a\,\left (a+b\right )}-b\,x\,\sqrt {a\,\left (a+b\right )}+a\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+b\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+a\,\left (a+b\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a+b\,x-a\,k^2\,x^2-b\,k^2\,x^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2+b\,a}} \]

[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)