Integrand size = 45, antiderivative size = 44 \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{\sqrt {d}} \] Output:
2*arctanh(d^(1/2)*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/(a-x))/d^(1/2)
Time = 15.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x)}}{a-x}\right )}{\sqrt {d}} \] Input:
Integrate[(a*b - 2*a*x + x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x + d*x^2)),x]
Output:
(2*ArcTanh[(Sqrt[d]*Sqrt[x*(-a + x)*(-b + x)])/(a - x)])/Sqrt[d]
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.
| \(\displaystyle \int \frac {a b-2 a x+x^2}{\sqrt {x (x-a) (x-b)} \left (a-x (b d+1)+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {1}{d \sqrt {x^2-(a+b) x+a b}}-\frac {-b d a+a-(-2 a d+b d+1) x}{d \sqrt {x^2-(a+b) x+a b} \left (d x^2+(-b d-1) x+a\right )}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {x^2-2 a x+a b}{\sqrt {x^2-(a+b) x+a b} \left (d x^2-(b d+1) x+a\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
Input:
Int[(a*b - 2*a*x + x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x + d*x^ 2)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 1.05 (sec) , antiderivative size = 2492, normalized size of antiderivative = 56.64
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2492\) |
| elliptic | \(\text {Expression too large to display}\) | \(2514\) |
Input:
int((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x,method=_ RETURNVERBOSE)
Output:
-2/d*b*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b* x^2+x^3)^(1/2)*EllipticF((-(-b+x)/b)^(1/2),(b/(-a+b))^(1/2))+1/d*(-1/(b^2* d^2-4*a*d+2*b*d+1)^(1/2)*b^3*(1-x/b)^(1/2)*(1/(-a+b)*x-a/(-a+b))^(1/2)*(x/ b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2 *b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a *d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*d-2/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b^ 2*(1-x/b)^(1/2)*(1/(-a+b)*x-a/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2 +x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi(( -(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+ b))^(1/2))+4/(b^2*d^2-4*a*d+2*b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(1/(-a+b)*x-a/( -a+b))^(1/2)*(x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d* (b^2*d^2-4*a*d+2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b* d+1+(b^2*d^2-4*a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a-1/(b^2*d^2-4*a*d+2 *b*d+1)^(1/2)*b*(1-x/b)^(1/2)*(1/(-a+b)*x-a/(-a+b))^(1/2)*(x/b)^(1/2)/(a*b *x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d+2*b*d+1)^(1/2) )*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4*a*d+2*b*d+1)^( 1/2))),(b/(-a+b))^(1/2))/d+2*b*(1-x/b)^(1/2)*(1/(-a+b)*x-a/(-a+b))^(1/2)*( x/b)^(1/2)/(a*b*x-a*x^2-b*x^2+x^3)^(1/2)/(1/2*b-1/2/d-1/2/d*(b^2*d^2-4*a*d +2*b*d+1)^(1/2))*EllipticPi((-(-b+x)/b)^(1/2),b/(b-1/2/d*(b*d+1+(b^2*d^2-4 *a*d+2*b*d+1)^(1/2))),(b/(-a+b))^(1/2))*a-b^2*(1-x/b)^(1/2)*(1/(-a+b)*x...
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (35) = 70\).
Time = 0.38 (sec) , antiderivative size = 231, normalized size of antiderivative = 5.25 \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\left [\frac {\log \left (\frac {d^{2} x^{4} - 2 \, {\left (b d^{2} - 3 \, d\right )} x^{3} + {\left (b^{2} d^{2} - 6 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 4 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {d} + 2 \, {\left (3 \, a b d - a\right )} x}{d^{2} x^{4} - 2 \, {\left (b d^{2} + d\right )} x^{3} + {\left (b^{2} d^{2} + 2 \, {\left (a + b\right )} d + 1\right )} x^{2} + a^{2} - 2 \, {\left (a b d + a\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d x^{2} - {\left (b d - 1\right )} x - a\right )} \sqrt {-d}}{2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}\right )}{d}\right ] \] Input:
integrate((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, a lgorithm="fricas")
Output:
[1/2*log((d^2*x^4 - 2*(b*d^2 - 3*d)*x^3 + (b^2*d^2 - 6*(a + b)*d + 1)*x^2 + a^2 - 4*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b*d - 1)*x - a)*sqrt(d ) + 2*(3*a*b*d - a)*x)/(d^2*x^4 - 2*(b*d^2 + d)*x^3 + (b^2*d^2 + 2*(a + b) *d + 1)*x^2 + a^2 - 2*(a*b*d + a)*x))/sqrt(d), sqrt(-d)*arctan(1/2*sqrt(a* b*x - (a + b)*x^2 + x^3)*(d*x^2 - (b*d - 1)*x - a)*sqrt(-d)/(a*b*d*x - (a + b)*d*x^2 + d*x^3))/d]
Timed out. \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((a*b-2*a*x+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(a-(b*d+1)*x+d*x**2),x )
Output:
Timed out
Exception generated. \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, a lgorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((b*d+1)^2-4*a*d>0)', see `assume ?` for mor
\[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\int { \frac {a b - 2 \, a x + x^{2}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (d x^{2} - {\left (b d + 1\right )} x + a\right )}} \,d x } \] Input:
integrate((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x, a lgorithm="giac")
Output:
integrate((a*b - 2*a*x + x^2)/(sqrt((a - x)*(b - x)*x)*(d*x^2 - (b*d + 1)* x + a)), x)
Time = 8.21 (sec) , antiderivative size = 437, normalized size of antiderivative = 9.93 \[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx=\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (2\,a\,d-b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}-1\right )}{d^2\,\left (b-\frac {b\,d-\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{d\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {b\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (\frac {b}{b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (b\,d-2\,a\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1\right )}{d^2\,\left (b-\frac {b\,d+\sqrt {b^2\,d^2+2\,b\,d-4\,a\,d+1}+1}{2\,d}\right )\,\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}} \] Input:
int((a*b - 2*a*x + x^2)/((x*(a - x)*(b - x))^(1/2)*(a - x*(b*d + 1) + d*x^ 2)),x)
Output:
(b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^ (1/2)), -b/(a - b))*(2*a*d - b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) - 1 ))/(d^2*(b - (b*d - (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (2*b*ellipticF(asin(((b - x)/b)^(1/2)), -b/ (a - b))*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2))/(d*(x^3 - x^2*(a + b) + a*b*x)^(1/2)) - (b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticPi(b/(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d)), asin(((b - x)/b)^(1/2)), -b/(a - b))*(b*d - 2*a*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1))/(d^2*(b - (b*d + (2*b*d - 4*a*d + b^2*d^2 + 1)^(1/2) + 1)/(2*d))*(x^3 - x^2*(a + b) + a*b*x)^(1/2))
\[ \int \frac {a b-2 a x+x^2}{\sqrt {x (-a+x) (-b+x)} \left (a-(1+b d) x+d x^2\right )} \, dx =\text {Too large to display} \] Input:
int((a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a-(b*d+1)*x+d*x^2),x)
Output:
( - 2*sqrt(x)*sqrt(b - x)*sqrt( - a + x)*i + 2*sqrt(x)*sqrt(b - x)*sqrt(a - x) + 3*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x)/(a**2*b - a**2*x - a*b **2*d*x + 2*a*b*d*x**2 - 2*a*b*x - a*d*x**3 + 2*a*x**2 + b**2*d*x**2 - 2*b *d*x**3 + b*x**2 + d*x**4 - x**3),x)*a*b*d*i + 5*int((sqrt(x)*sqrt(b - x)* sqrt( - a + x)*x)/(a**2*b - a**2*x - a*b**2*d*x + 2*a*b*d*x**2 - 2*a*b*x - a*d*x**3 + 2*a*x**2 + b**2*d*x**2 - 2*b*d*x**3 + b*x**2 + d*x**4 - x**3), x)*a*i + 2*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x)/(a**2*b - a**2*x - a *b**2*d*x + 2*a*b*d*x**2 - 2*a*b*x - a*d*x**3 + 2*a*x**2 + b**2*d*x**2 - 2 *b*d*x**3 + b*x**2 + d*x**4 - x**3),x)*b**2*d*i + 2*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x)*x)/(a**2*b - a**2*x - a*b**2*d*x + 2*a*b*d*x**2 - 2*a*b* x - a*d*x**3 + 2*a*x**2 + b**2*d*x**2 - 2*b*d*x**3 + b*x**2 + d*x**4 - x** 3),x)*b*i + 3*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x))/(a**2*b*x - a**2*x* *2 - a*b**2*d*x**2 + 2*a*b*d*x**3 - 2*a*b*x**2 - a*d*x**4 + 2*a*x**3 + b** 2*d*x**3 - 2*b*d*x**4 + b*x**3 + d*x**5 - x**4),x)*a**2*b**2*d*i + 5*int(( sqrt(x)*sqrt(b - x)*sqrt( - a + x))/(a**2*b*x - a**2*x**2 - a*b**2*d*x**2 + 2*a*b*d*x**3 - 2*a*b*x**2 - a*d*x**4 + 2*a*x**3 + b**2*d*x**3 - 2*b*d*x* *4 + b*x**3 + d*x**5 - x**4),x)*a**2*b*i + 2*int((sqrt(x)*sqrt(b - x)*sqrt ( - a + x))/(a**2*b*x - a**2*x**2 - a*b**2*d*x**2 + 2*a*b*d*x**3 - 2*a*b*x **2 - a*d*x**4 + 2*a*x**3 + b**2*d*x**3 - 2*b*d*x**4 + b*x**3 + d*x**5 - x **4),x)*a*b**3*d*i + 2*int((sqrt(x)*sqrt(b - x)*sqrt( - a + x))/(a**2*b...