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

Optimal result

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

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)
 

Mathematica [A] (verified)

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

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]
 

Output:

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

Rubi [F]

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

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2 a+b+\left (-2 a k^2-2 b k^2+b\right ) x}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (-\left ((a+b) k^2 x^2\right )+b x+a\right )}-\frac {1}{(a+b) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\)

Input:

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]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 1.74 (sec) , antiderivative size = 4587, normalized size of antiderivative = 80.47

method	result	size
default	\(\text {Expression too large to display}\)	\(4587\)
elliptic	\(\text {Expression too large to display}\)	\(4605\)

Input:

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)
 

Output:

-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)*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)+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))*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)+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...
 

Fricas [B] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 345, normalized size of antiderivative = 6.05 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} k^{2} x^{3} + {\left (6 \, {\left (a^{2} + a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} x}{{\left (a^{2} + 2 \, a b + b^{2}\right )} k^{4} x^{4} - 2 \, {\left (a b + b^{2}\right )} k^{2} x^{3} + 2 \, a b x - {\left (2 \, {\left (a^{2} + a b\right )} k^{2} - b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left ({\left (a + b\right )} k^{2} x^{2} - {\left (2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} + a b}}{2 \, {\left ({\left (a^{2} + a b\right )} k^{2} x^{3} - {\left ({\left (a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} + a b}}\right ] \] Input:

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")
 

Output:

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

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)
 

Output:

Timed out
 

Maxima [F]

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

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")
 

Output:

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]

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

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")
 

Output:

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 = 11.38 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.14 \[ \int \frac {1-2 k^2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-a-b x+\left (a k^2+b k^2\right ) x^2\right )} \, dx=\frac {\ln \left (\frac {a\,\sqrt {a\,\left (a+b\right )}-2\,a\,x\,\sqrt {a\,\left (a+b\right )}-b\,x\,\sqrt {a\,\left (a+b\right )}+a\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+b\,k^2\,x^2\,\sqrt {a\,\left (a+b\right )}+a\,\left (a+b\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a+b\,x-a\,k^2\,x^2-b\,k^2\,x^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2+b\,a}} \] Input:

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)
 

Output:

(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)
 

Reduce [F]

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

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)
 

Output:

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)