Integrand size = 53, antiderivative size = 293 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\frac {\left (2 i b+2 i a k^2-i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \arctan \left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}+\frac {\left (-2 i b-2 i a k^2+i b k^2-2 b \sqrt {-1+k^2}-2 a k^2 \sqrt {-1+k^2}\right ) \arctan \left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{k^2 (-1+x) x}\right )}{2 k^2 \sqrt {-1+k^2} \sqrt {-2+k^2+2 i \sqrt {-1+k^2}}} \]
1/2*(2*I*b+2*I*a*k^2-I*b*k^2-2*b*(k^2-1)^(1/2)-2*a*k^2*(k^2-1)^(1/2))*arct an((-2+k^2-2*I*(k^2-1)^(1/2))^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/k^2/(-1 +x)/x)/k^2/(k^2-1)^(1/2)/(-2+k^2-2*I*(k^2-1)^(1/2))^(1/2)+1/2*(-2*I*b-2*I* a*k^2+I*b*k^2-2*b*(k^2-1)^(1/2)-2*a*k^2*(k^2-1)^(1/2))*arctan((-2+k^2+2*I* (k^2-1)^(1/2))^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/k^2/(-1+x)/x)/k^2/(k^2 -1)^(1/2)/(-2+k^2+2*I*(k^2-1)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 28.01 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.83 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\frac {i \sqrt {1+\frac {1}{-1+x}} \sqrt {1+\frac {1-\frac {1}{k^2}}{-1+x}} (-1+x)^{3/2} \left (2 a k^2 \sqrt {1-k^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )+\left (-2 a k^2+b \left (-1+\sqrt {1-k^2}\right )\right ) \operatorname {EllipticPi}\left (\frac {-1+k^2}{-1+k^2-\sqrt {1-k^2}},i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )+\left (b+2 a k^2+b \sqrt {1-k^2}\right ) \operatorname {EllipticPi}\left (\frac {-1+k^2}{-1+k^2+\sqrt {1-k^2}},i \text {arcsinh}\left (\frac {1}{\sqrt {-1+x}}\right ),1-\frac {1}{k^2}\right )\right )}{k^2 \sqrt {1-k^2} \sqrt {(-1+x) x \left (-1+k^2 x\right )}} \]
(I*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + (1 - k^(-2))/(-1 + x)]*(-1 + x)^(3/2)* (2*a*k^2*Sqrt[1 - k^2]*EllipticF[I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)] + (-2*a*k^2 + b*(-1 + Sqrt[1 - k^2]))*EllipticPi[(-1 + k^2)/(-1 + k^2 - Sqrt [1 - k^2]), I*ArcSinh[1/Sqrt[-1 + x]], 1 - k^(-2)] + (b + 2*a*k^2 + b*Sqrt [1 - k^2])*EllipticPi[(-1 + k^2)/(-1 + k^2 + Sqrt[1 - k^2]), I*ArcSinh[1/S qrt[-1 + x]], 1 - k^(-2)]))/(k^2*Sqrt[1 - k^2]*Sqrt[(-1 + x)*x*(-1 + k^2*x )])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.56 (sec) , antiderivative size = 1022, normalized size of antiderivative = 3.49, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {2467, 25, 2035, 7279, 6, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \left (a k^2+b\right )-a-b x}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^2 x^2-2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {-\left (\left (a k^2+b\right ) x^2\right )+b x+a}{\sqrt {x} \left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {-\left (\left (a k^2+b\right ) x^2\right )+b x+a}{\sqrt {x} \left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {-\left (\left (a k^2+b\right ) x^2\right )+b x+a}{\left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (-\frac {a}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {2 a k^2+b-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {b}{k^2 \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a k^2+b-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {a+\frac {b}{k^2}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\frac {\left (2 a k^2+b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (a k^2+b\right )\right ) \text {arctanh}\left (\frac {\sqrt {1-k^2} \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 k^2 \left (-k^2-\sqrt {1-k^2}+1\right )}+\frac {\left (2 a k^2+b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (a k^2+b\right )\right ) \text {arctanh}\left (\frac {\sqrt {1-k^2} \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 k^2 \left (-k^2+\sqrt {1-k^2}+1\right )}-\frac {\left (a k^2+b\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{2 k^{5/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\left (2 a k^2+b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (a k^2+b\right )\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 k^{5/2} \left (k-\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\left (2 a k^2+b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (a k^2+b\right )\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 k^{5/2} \left (k+\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (-k+\sqrt {1-k^2}+1\right ) \left (2 a k^2+b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (a k^2+b\right )\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {k+1}{2 k},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{5/2} \left (\sqrt {1-k^2}+1\right ) \left (k+\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (-k-\sqrt {1-k^2}+1\right ) \left (2 a \left (1-\sqrt {1-k^2}\right ) k^2+b \left (-k^2-2 \sqrt {1-k^2}+2\right )\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {k+1}{2 k},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{5/2} \left (1-\sqrt {1-k^2}\right ) \left (k-\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[x]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]*(-1/4*((2*a*k^2 + b*(2 - k^2) - 2*Sqrt[1 - k^2]*(b + a*k^2))*ArcTanh[(Sqrt[1 - k^2]*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(k^2*(1 - k^2 - Sqrt[1 - k^2])) + ((2*a*k^2 + b*(2 - k^2) + 2*Sqrt[1 - k^2]*(b + a*k^2))*ArcTanh[(Sqrt[1 - k^2]*Sqrt[x])/Sqrt [1 - (1 + k^2)*x + k^2*x^2]])/(4*k^2*(1 - k^2 + Sqrt[1 - k^2])) - ((b + a* k^2)*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*A rcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(2*k^(5/2)*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((2*a*k^2 + b*(2 - k^2) - 2*Sqrt[1 - k^2]*(b + a*k^2))*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt [k]*Sqrt[x]], (1 + k)^2/(4*k)])/(4*k^(5/2)*(1 + k - Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((2*a*k^2 + b*(2 - k^2) + 2*Sqrt[1 - k^2]*(b + a*k^2))*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF [2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(4*k^(5/2)*(1 + k + Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) - ((1 - k + Sqrt[1 - k^2])*(2*a*k^ 2 + b*(2 - k^2) + 2*Sqrt[1 - k^2]*(b + a*k^2))*(1 + k*x)*Sqrt[(1 - (1 + k^ 2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticPi[(1 + k)/(2*k), 2*ArcTan[Sqrt[k]*Sq rt[x]], (1 + k)^2/(4*k)])/(8*k^(5/2)*(1 + Sqrt[1 - k^2])*(1 + k + Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) - ((1 - k - Sqrt[1 - k^2])*(b*(2 - k^2 - 2*Sqrt[1 - k^2]) + 2*a*k^2*(1 - Sqrt[1 - k^2]))*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticPi[(1 + k)/(2*k), 2*ArcTan...
3.29.48.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 1907, normalized size of antiderivative = 6.51
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1907\) |
| elliptic | \(\text {Expression too large to display}\) | \(2014\) |
int((-a-b*x+(a*k^2+b)*x^2)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1),x,me thod=_RETURNVERBOSE)
-2*(a*k^2+b)/k^4*(-(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/k^2*(2/(-k^2+1)*(-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)*EllipticPi((-(x -1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^ (1/2))*a-2/(-k^2+1)*(-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)*EllipticPi((-(x-1/k^2)*k^2)^(1/2 ),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*b+2/(-k^2+ 1)*(-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)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1 +(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2*b+2/(-k^2+1)^(1/2)*(-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)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+ 1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))*a-1/(-k^2+1)^(1/2)*(-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)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k ^2),(1/k^2/(1/k^2-1))^(1/2))*b+2/(-k^2+1)^(1/2)*(-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)*Elli pticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/ (1/k^2-1))^(1/2))/k^2*b-2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/...
Time = 0.66 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.68 \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\left [-\frac {{\left (2 \, a k^{2} + b\right )} \sqrt {-k^{2} + 1} \log \left (\frac {k^{4} x^{4} - 4 \, {\left (2 \, k^{4} - k^{2}\right )} x^{3} + 2 \, {\left (4 \, k^{4} + k^{2} - 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {-k^{2} + 1} - 4 \, {\left (2 \, k^{2} - 1\right )} x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) - {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {2 \, {\left (2 \, a k^{2} + b\right )} \sqrt {k^{2} - 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {k^{2} - 1}}{2 \, {\left ({\left (k^{4} - k^{2}\right )} x^{3} - {\left (k^{4} - 1\right )} x^{2} + {\left (k^{2} - 1\right )} x\right )}}\right ) + {\left (b k^{2} - b\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}\right ] \]
integrate((-a-b*x+(a*k^2+b)*x^2)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1 ),x, algorithm="fricas")
[-1/4*((2*a*k^2 + b)*sqrt(-k^2 + 1)*log((k^4*x^4 - 4*(2*k^4 - k^2)*x^3 + 2 *(4*k^4 + k^2 - 2)*x^2 - 4*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2* k^2*x + 1)*sqrt(-k^2 + 1) - 4*(2*k^2 - 1)*x + 1)/(k^4*x^4 - 4*k^2*x^3 + 2* (k^2 + 2)*x^2 - 4*x + 1)) - (b*k^2 - b)*log((k^4*x^4 + 4*k^2*x^3 - 2*(3*k^ 2 + 2)*x^2 - 4*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 1) + 4*x + 1)/ (k^4*x^4 - 4*k^2*x^3 + 2*(k^2 + 2)*x^2 - 4*x + 1)))/(k^4 - k^2), 1/4*(2*(2 *a*k^2 + b)*sqrt(k^2 - 1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^ 2*x^2 - 2*k^2*x + 1)*sqrt(k^2 - 1)/((k^4 - k^2)*x^3 - (k^4 - 1)*x^2 + (k^2 - 1)*x)) + (b*k^2 - b)*log((k^4*x^4 + 4*k^2*x^3 - 2*(3*k^2 + 2)*x^2 - 4*s qrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 1) + 4*x + 1)/(k^4*x^4 - 4*k^2 *x^3 + 2*(k^2 + 2)*x^2 - 4*x + 1)))/(k^4 - k^2)]
Timed out. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]
integrate((-a-b*x+(a*k^2+b)*x^2)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1 ),x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(k-1>0)', see `assume?` for more details)Is
\[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int { \frac {{\left (a k^{2} + b\right )} x^{2} - b x - a}{{\left (k^{2} x^{2} - 2 \, x + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
integrate((-a-b*x+(a*k^2+b)*x^2)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1 ),x, algorithm="giac")
Timed out. \[ \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int -\frac {\left (-a\,k^2-b\right )\,x^2+b\,x+a}{\left (k^2\,x^2-2\,x+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \]