Integrand size = 82, antiderivative size = 77 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \] Output:
-arctan(d^(1/4)*x/(a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)-arctanh(d^(1/4)*x/ (a*b*x+(-a-b)*x^2+x^3)^(1/2))/d^(1/4)
Time = 9.44 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt {x (-a+x) (-b+x)}}\right )}{\sqrt [4]{d}} \] Input:
Integrate[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a ^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x ^4)),x]
Output:
-((ArcTan[(d^(1/4)*x)/Sqrt[x*(-a + x)*(-b + x)]] + ArcTanh[(d^(1/4)*x)/Sqr t[x*(-a + x)*(-b + x)]])/d^(1/4))
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.
| \(\displaystyle \int \frac {(x-a) (x-b) \left (x^2-a b\right )}{\sqrt {x (x-a) (x-b)} \left (x^2 \left (a^2+4 a b+b^2-d\right )+a^2 b^2-2 x^3 (a+b)-2 a b x (a+b)+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int -\frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x} \sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \frac {(a-x) (b-x) \left (a b-x^2\right )}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \int \left (\frac {-\left ((a+b) x^3\right )+\left (a^2+4 b a+b^2-d\right ) x^2-3 a b (a+b) x+2 a^2 b^2}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}-\frac {1}{\sqrt {x^2-(a+b) x+a b}}\right )d\sqrt {x}}{\sqrt {x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (2 a^2 b^2 \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-3 a b (a+b) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}+\left (a^2+4 a b+b^2-d\right ) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-(a+b) \int \frac {x^3}{\sqrt {x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt {x}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\) |
Input:
Int[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x ]
Output:
$Aborted
Time = 0.91 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {-2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )+\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )}{2 d^{\frac {1}{4}}}\) | \(76\) |
| pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {x \left (a -x \right ) \left (b -x \right )}}{x \,d^{\frac {1}{4}}}\right )-\ln \left (\frac {d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}{-d^{\frac {1}{4}} x +\sqrt {x \left (a -x \right ) \left (b -x \right )}}\right )}{2 d^{\frac {1}{4}}}\) | \(78\) |
| elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (-2 a -2 b \right ) \textit {\_Z}^{3}+\left (a^{2}+4 a b +b^{2}-d \right ) \textit {\_Z}^{2}+\left (-2 a^{2} b -2 a \,b^{2}\right ) \textit {\_Z} +a^{2} b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a -\underline {\hspace {1.25 ex}}\alpha ^{3} b +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+2 a^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} d \right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +b d \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2} b +\underline {\hspace {1.25 ex}}\alpha d +b d}{b d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2}-4 \underline {\hspace {1.25 ex}}\alpha a b -\underline {\hspace {1.25 ex}}\alpha \,b^{2}+a^{2} b +a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha d \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b d}\) | \(427\) |
Input:
int((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b*(a+b)* x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x,method=_RETURNVERBOSE)
Output:
-1/2*(-2*arctan((x*(a-x)*(b-x))^(1/2)/x/d^(1/4))+ln((d^(1/4)*x+(x*(a-x)*(b -x))^(1/2))/(-d^(1/4)*x+(x*(a-x)*(b-x))^(1/2))))/d^(1/4)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 731, normalized size of antiderivative = 9.49 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="fricas")
Output:
-1/4*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2* (a^2*b + a*b^2)*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt (d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2 *b + a*b^2)*x))/d^(1/4) + 1/4*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(d^(3/4)*x + (a*b*d - (a + b)*d*x + d*x^2)/d^(3/4)) + 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a* b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4) + 1/4*I*log((a^2*b^2 - 2* (a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2* sqrt(a*b*x - (a + b)*x^2 + x^3)*(I*d^(3/4)*x + (-I*a*b*d + I*(a + b)*d*x - I*d*x^2)/d^(3/4)) - 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x ))/d^(1/4) - 1/4*I*log((a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 + d)*x^2 - 2*(a^2*b + a*b^2)*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*(-I*d^ (3/4)*x + (I*a*b*d - I*(a + b)*d*x + I*d*x^2)/d^(3/4)) - 2*(a*b*d*x - (a + b)*d*x^2 + d*x^3)/sqrt(d))/(a^2*b^2 - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a^2*b + a*b^2)*x))/d^(1/4)
Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((-a+x)*(-b+x)*(-a*b+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(a**2*b**2-2* a*b*(a+b)*x+(a**2+4*a*b+b**2-d)*x**2-2*(a+b)*x**3+x**4),x)
Output:
Timed out
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \] Input:
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="maxima")
Output:
-integrate((a*b - x^2)*(a - x)*(b - x)/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \] Input:
integrate((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b* (a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="giac")
Output:
integrate(-(a*b - x^2)*(a - x)*(b - x)/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*sqrt((a - x)*(b - x)*x)), x)
Time = 9.05 (sec) , antiderivative size = 1150, normalized size of antiderivative = 14.94 \[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx =\text {Too large to display} \] Input:
int(-((a - x)*(b - x)*(a*b - x^2))/((x*(a - x)*(b - x))^(1/2)*(x^4 - 2*x^3 *(a + b) + a^2*b^2 + x^2*(4*a*b - d + a^2 + b^2) - 2*a*b*x*(a + b))),x)
Output:
symsum(-(b*(x/b)^(1/2)*((b - x)/b)^(1/2)*((a - x)/(a - b))^(1/2)*ellipticP i(-b/(root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z *(a + b) + a^2*b^2, z, k) - b), asin(((b - x)/b)^(1/2)), -b/(a - b))*(a*ro ot(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^3 + b*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^ 2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^3 + d*root(z^4 - z^3*(2*a + 2* b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 - a^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*( a + b) + a^2*b^2, z, k)^2 - b^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4* a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 - 2*a^2*b^2 - 4*a*b* root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 + 3*a*b^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4* a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k) + 3*b*a^2*root(z^4 - z ^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2 , z, k)))/((root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2 *a*b*z*(a + b) + a^2*b^2, z, k) - b)*(x*(a - x)*(b - x))^(1/2)*(3*a*root(z ^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a ^2*b^2, z, k)^2 - a^2*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k) + 3*b*root(z^4 - z^3*(2*a + 2*b) + z^2*(- d + 4*a*b + a^2 + b^2) - 2*a*b*z*(a + b) + a^2*b^2, z, k)^2 -...
\[ \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx =\text {Too large to display} \] Input:
int((-a+x)*(-b+x)*(-a*b+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(a^2*b^2-2*a*b*(a+b)* x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x)
Output:
(2*sqrt(x)*sqrt(b - x)*sqrt( - a + x)*i - 2*sqrt(x)*sqrt(b - x)*sqrt(a - x ) - 5*int((sqrt(b - x)*sqrt( - a + x))/(sqrt(x)*a**2*b**2 - 2*sqrt(x)*a**2 *b*x + sqrt(x)*a**2*x**2 - 2*sqrt(x)*a*b**2*x + 4*sqrt(x)*a*b*x**2 - 2*sqr t(x)*a*x**3 + sqrt(x)*b**2*x**2 - 2*sqrt(x)*b*x**3 - sqrt(x)*d*x**2 + sqrt (x)*x**4),x)*a**3*b*i - 15*int((sqrt(b - x)*sqrt( - a + x))/(sqrt(x)*a**2* b**2 - 2*sqrt(x)*a**2*b*x + sqrt(x)*a**2*x**2 - 2*sqrt(x)*a*b**2*x + 4*sqr t(x)*a*b*x**2 - 2*sqrt(x)*a*x**3 + sqrt(x)*b**2*x**2 - 2*sqrt(x)*b*x**3 - sqrt(x)*d*x**2 + sqrt(x)*x**4),x)*a**2*b**2*i - 5*int((sqrt(b - x)*sqrt( - a + x))/(sqrt(x)*a**2*b**2 - 2*sqrt(x)*a**2*b*x + sqrt(x)*a**2*x**2 - 2*s qrt(x)*a*b**2*x + 4*sqrt(x)*a*b*x**2 - 2*sqrt(x)*a*x**3 + sqrt(x)*b**2*x** 2 - 2*sqrt(x)*b*x**3 - sqrt(x)*d*x**2 + sqrt(x)*x**4),x)*a*b**3*i + int((s qrt(b - x)*sqrt( - a + x))/(sqrt(x)*a**2*b**2 - 2*sqrt(x)*a**2*b*x + sqrt( x)*a**2*x**2 - 2*sqrt(x)*a*b**2*x + 4*sqrt(x)*a*b*x**2 - 2*sqrt(x)*a*x**3 + sqrt(x)*b**2*x**2 - 2*sqrt(x)*b*x**3 - sqrt(x)*d*x**2 + sqrt(x)*x**4),x) *a*b*d*i + 5*int((sqrt(x)*sqrt(b - x)*sqrt(a - x)*x)/(a**2*b**2 - 2*a**2*b *x + a**2*x**2 - 2*a*b**2*x + 4*a*b*x**2 - 2*a*x**3 + b**2*x**2 - 2*b*x**3 - d*x**2 + x**4),x)*a**2 + 15*int((sqrt(x)*sqrt(b - x)*sqrt(a - x)*x)/(a* *2*b**2 - 2*a**2*b*x + a**2*x**2 - 2*a*b**2*x + 4*a*b*x**2 - 2*a*x**3 + b* *2*x**2 - 2*b*x**3 - d*x**2 + x**4),x)*a*b + 5*int((sqrt(x)*sqrt(b - x)*sq rt(a - x)*x)/(a**2*b**2 - 2*a**2*b*x + a**2*x**2 - 2*a*b**2*x + 4*a*b*x...