Integrand size = 60, antiderivative size = 57 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a+b} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \]
Time = 15.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a+b} \sqrt {(-1+x) x \left (-1+k^2 x\right )}}{\sqrt {a} (-1+x)}\right )}{\sqrt {a} \sqrt {a+b}} \]
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]
(2*ArcTan[(Sqrt[a + b]*Sqrt[(-1 + x)*x*(-1 + k^2*x)])/(Sqrt[a]*(-1 + x))]) /(Sqrt[a]*Sqrt[a + b])
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.
| \(\displaystyle \int \frac {k^2 x^2-2 k^2 x+1}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (x^2 \left (a k^2+b k^2\right )-a-b x\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {x^2 k^2-2 x k^2+1}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
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]
3.8.35.3.1 Defintions of rubi rules used
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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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 = 2.06 (sec) , antiderivative size = 4588, normalized size of antiderivative = 80.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4588\) |
| elliptic | \(\text {Expression too large to display}\) | \(4605\) |
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)
-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)*EllipticF((-(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...
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 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} k^{2} x^{3} + {\left (6 \, {\left (a^{2} + a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} x}{{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (a b + b^{2}\right )} k^{2} x^{3} + 2 \, a b x - {\left (2 \, {\left (a^{2} + a b\right )} k^{2} - b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} + a b}}{2 \, {\left ({\left (a^{2} + a b\right )} k^{2} x^{3} - {\left ({\left (a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} + a b}}\right ] \]
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")
[-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)]
Timed out. \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\text {Timed out} \]
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)
\[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (a k^{2} + b k^{2}\right )} x^{2} - b x - a\right )}} \,d x } \]
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")
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)
\[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (a k^{2} + b k^{2}\right )} x^{2} - b x - a\right )}} \,d x } \]
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")
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)
Time = 9.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.14 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {\ln \left (\frac {a\,\sqrt {a\,\left (a+b\right )}-2\,a\,x\,\sqrt {a\,\left (a+b\right )}-b\,x\,\sqrt {a\,\left (a+b\right )}+a\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+b\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+a\,\left (a+b\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a+b\,x-a\,k^2\,x^2-b\,k^2\,x^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2+b\,a}} \]
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)