\(\int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} (-a^3 d+(b+3 a^2 d) x-(1+3 a d) x^2+d x^3)} \, dx\) [570]

Optimal result

Integrand size = 84, antiderivative size = 44 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]

[Out]

-2*arctanh((a*b*x+(-a-b)*x^2+x^3)^(1/2)/d^(1/2)/(a-x)^2)/d^(1/2)

Rubi [F]

\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx \]

[In]

Int[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*x - (1
+ 3*a*d)*x^2 + d*x^3)),x]

[Out]

(2*a*b*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][Sqrt[-a + x^2]/(Sqrt[-b + x^2]*(a^3*d - b*(1
+ (3*a^2*d)/b)*x^2 + (1 + 3*a*d)*x^4 - d*x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] - (4*(a - b)*Sqrt[x]*
Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[-a + x^2])/(Sqrt[-b + x^2]*(a^3*d - b*(1 + (3*a^2*
d)/b)*x^2 + (1 + 3*a*d)*x^4 - d*x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*Sqrt[x]*Sqrt[-a + x]*Sqrt
[-b + x]*Defer[Subst][Defer[Int][(x^4*Sqrt[-a + x^2])/(Sqrt[-b + x^2]*(-(a^3*d) + b*(1 + (3*a^2*d)/b)*x^2 - (1
+ 3*a*d)*x^4 + d*x^6)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \left (-a b+(2 a-2 b) x+x^2\right )}{\sqrt {x} \sqrt {-b+x} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {\sqrt {-a+x^2} \left (-a b+(2 a-2 b) x^2+x^4\right )}{\sqrt {-b+x^2} \left (-a^3 d+\left (b+3 a^2 d\right ) x^2-(1+3 a d) x^4+d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \left (\frac {a b \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )}+\frac {2 (-a+b) x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )}+\frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^2-(1+3 a d) x^4+d x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^3 d+b \left (1+\frac {3 a^2 d}{b}\right ) x^2-(1+3 a d) x^4+d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}-\frac {\left (4 (a-b) \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {\sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^3 d-b \left (1+\frac {3 a^2 d}{b}\right ) x^2+(1+3 a d) x^4-d x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.80 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x)}}{\sqrt {d} (a-x)^2}\right )}{\sqrt {d}} \]

[In]

Integrate[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*
x - (1 + 3*a*d)*x^2 + d*x^3)),x]

[Out]

(-2*ArcTanh[Sqrt[x*(-a + x)*(-b + x)]/(Sqrt[d]*(a - x)^2)])/Sqrt[d]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 9.07

method	result	size
default	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -d \,a^{3}\right )}{\sum }\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d +\underline {\hspace {1.25 ex}}\alpha a b d +d \,a^{3}+a^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +6 \underline {\hspace {1.25 ex}}\alpha a d -3 a^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\)	\(399\)
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -1\right ) \textit {\_Z}^{2}+\left (3 a^{2} d +b \right ) \textit {\_Z} -d \,a^{3}\right )}{\sum }\frac {\left (-4 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +5 \underline {\hspace {1.25 ex}}\alpha \,a^{2} d -\underline {\hspace {1.25 ex}}\alpha a b d -d \,a^{3}-a^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha a d +\underline {\hspace {1.25 ex}}\alpha b d +3 a^{2} d -3 a b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha \right ) b}{d \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right )}, \sqrt {\frac {b}{-a +b}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -6 \underline {\hspace {1.25 ex}}\alpha a d +3 a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +b \right ) \left (a^{3}-3 b \,a^{2}+3 a \,b^{2}-b^{3}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2}}\)	\(401\)

[In]

int((b*a^2-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-d*a^3+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x^3),x,
method=_RETURNVERBOSE)

[Out]

-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))-2/d^2*b*sum((4*_alpha^2*a*d-2*_alpha^2*b*d-5*_alpha*a^2*d+_alpha*a*b*d+a^3*d+a^2*b*d+
_alpha^2-_alpha*b)/(-3*_alpha^2*d+6*_alpha*a*d-3*a^2*d+2*_alpha-b)*(_alpha^2*d-3*_alpha*a*d+_alpha*b*d+3*a^2*d
-3*a*b*d+b^2*d-_alpha)/(a^3-3*a^2*b+3*a*b^2-b^3)*(-(-b+x)/b)^(1/2)*((-a+x)/(-a+b))^(1/2)*(x/b)^(1/2)/(x*(a*b-a
*x-b*x+x^2))^(1/2)*EllipticPi((-(-b+x)/b)^(1/2),-(_alpha^2*d-3*_alpha*a*d+_alpha*b*d+3*a^2*d-3*a*b*d+b^2*d-_al
pha)*b/d/(a^3-3*a^2*b+3*a*b^2-b^3),(b/(-a+b))^(1/2)),_alpha=RootOf(d*_Z^3+(-3*a*d-1)*_Z^2+(3*a^2*d+b)*_Z-d*a^3
))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (35) = 70\).

Time = 0.70 (sec) , antiderivative size = 441, normalized size of antiderivative = 10.02 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\left [\frac {\log \left (\frac {a^{6} d^{2} + d^{2} x^{6} - 6 \, {\left (a d^{2} - d\right )} x^{5} + {\left (15 \, a^{2} d^{2} - 6 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} - 9 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} - 6 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 4 \, {\left (a^{4} d + d x^{4} - {\left (4 \, a d - 1\right )} x^{3} + {\left (6 \, a^{2} d - a - b\right )} x^{2} - {\left (4 \, a^{3} d - a b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 6 \, {\left (a^{5} d^{2} - a^{3} b d\right )} x}{a^{6} d^{2} + d^{2} x^{6} - 2 \, {\left (3 \, a d^{2} + d\right )} x^{5} + {\left (15 \, a^{2} d^{2} + 2 \, {\left (3 \, a + b\right )} d + 1\right )} x^{4} - 2 \, {\left (10 \, a^{3} d^{2} + 3 \, {\left (a^{2} + a b\right )} d + b\right )} x^{3} + {\left (15 \, a^{4} d^{2} + b^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} d\right )} x^{2} - 2 \, {\left (3 \, a^{5} d^{2} + a^{3} b d\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (a^{3} d - d x^{3} + {\left (3 \, a d - 1\right )} x^{2} - {\left (3 \, a^{2} d - b\right )} x\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a^{2} b d x + {\left (2 \, a + b\right )} d x^{3} - d x^{4} - {\left (a^{2} + 2 \, a b\right )} d x^{2}\right )}}\right )}{d}\right ] \]

[In]

integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="fricas")

[Out]

[1/2*log((a^6*d^2 + d^2*x^6 - 6*(a*d^2 - d)*x^5 + (15*a^2*d^2 - 6*(3*a + b)*d + 1)*x^4 - 2*(10*a^3*d^2 - 9*(a^
2 + a*b)*d + b)*x^3 + (15*a^4*d^2 + b^2 - 6*(a^3 + 3*a^2*b)*d)*x^2 - 4*(a^4*d + d*x^4 - (4*a*d - 1)*x^3 + (6*a
^2*d - a - b)*x^2 - (4*a^3*d - a*b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 6*(a^5*d^2 - a^3*b*d)*x)/(a^6
*d^2 + d^2*x^6 - 2*(3*a*d^2 + d)*x^5 + (15*a^2*d^2 + 2*(3*a + b)*d + 1)*x^4 - 2*(10*a^3*d^2 + 3*(a^2 + a*b)*d
+ b)*x^3 + (15*a^4*d^2 + b^2 + 2*(a^3 + 3*a^2*b)*d)*x^2 - 2*(3*a^5*d^2 + a^3*b*d)*x))/sqrt(d), sqrt(-d)*arctan
(1/2*(a^3*d - d*x^3 + (3*a*d - 1)*x^2 - (3*a^2*d - b)*x)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d)/(a^2*b*d*x +
(2*a + b)*d*x^3 - d*x^4 - (a^2 + 2*a*b)*d*x^2))/d]

Sympy [F(-1)]

Timed out. \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((a**2*b-a*(2*a-b)*x-(-a+2*b)*x**2+x**3)/(x*(-a+x)*(-b+x))**(1/2)/(-a**3*d+(3*a**2*d+b)*x-(3*a*d+1)*x
**2+d*x**3),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]

[In]

integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="maxima")

[Out]

-integrate((a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*
sqrt((a - x)*(b - x)*x)), x)

Giac [F]

\[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\int { -\frac {a^{2} b - {\left (2 \, a - b\right )} a x + {\left (a - 2 \, b\right )} x^{2} + x^{3}}{{\left (a^{3} d - d x^{3} + {\left (3 \, a d + 1\right )} x^{2} - {\left (3 \, a^{2} d + b\right )} x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]

[In]

integrate((a^2*b-a*(2*a-b)*x-(-a+2*b)*x^2+x^3)/(x*(-a+x)*(-b+x))^(1/2)/(-a^3*d+(3*a^2*d+b)*x-(3*a*d+1)*x^2+d*x
^3),x, algorithm="giac")

[Out]

integrate(-(a^2*b - (2*a - b)*a*x + (a - 2*b)*x^2 + x^3)/((a^3*d - d*x^3 + (3*a*d + 1)*x^2 - (3*a^2*d + b)*x)*
sqrt((a - x)*(b - x)*x)), x)

Mupad [B] (verification not implemented)

Time = 11.50 (sec) , antiderivative size = 368, normalized size of antiderivative = 8.36 \[ \int \frac {a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^3 d+\left (b+3 a^2 d\right ) x-(1+3 a d) x^2+d x^3\right )} \, dx=\frac {\ln \left (\frac {\left (a-b+x+a^2\,d-2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+d\,x^2-2\,a\,d\,x\right )\,\left (a\,x^2-a^4\,d-2\,b\,x^2+b^2\,x-2\,d\,x^4+x^3-a^5\,d^2+d^2\,x^5+2\,a^2\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a^2\,d\,x^2-5\,a\,d^2\,x^4+5\,a^4\,d^2\,x-a\,b\,x+10\,a^2\,d^2\,x^3-10\,a^3\,d^2\,x^2+a^3\,b\,d+4\,a\,d\,x^3+2\,a^3\,d\,x+2\,b\,d\,x^3-2\,a\,b\,\sqrt {d}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-3\,a\,b\,d\,x^2\right )}{\left (-d\,a^3+3\,d\,a^2\,x-3\,d\,a\,x^2+d\,x^3-x^2+b\,x\right )\,\left (a^4\,d^2-4\,a^3\,d^2\,x+2\,a^3\,d-2\,a^2\,b\,d+6\,a^2\,d^2\,x^2-2\,a^2\,d\,x+a^2-2\,a\,b-4\,a\,d^2\,x^3+2\,a\,d\,x^2+2\,a\,x+b^2+2\,b\,d\,x^2-2\,b\,x+d^2\,x^4-2\,d\,x^3+x^2\right )}\right )}{\sqrt {d}} \]

[In]

int((a^2*b + x^2*(a - 2*b) + x^3 - a*x*(2*a - b))/((x*(a - x)*(b - x))^(1/2)*(x*(b + 3*a^2*d) - a^3*d + d*x^3
- x^2*(3*a*d + 1))),x)

[Out]

log(((a - b + x + a^2*d - 2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) + d*x^2 - 2*a*d*x)*(a*x^2 - a^4*d - 2*b*x^2 + b^
2*x - 2*d*x^4 + x^3 - a^5*d^2 + d^2*x^5 + 2*a^2*d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a^2*d*x^2 - 5*a*d^2*x^4
+ 5*a^4*d^2*x - a*b*x + 10*a^2*d^2*x^3 - 10*a^3*d^2*x^2 + a^3*b*d + 4*a*d*x^3 + 2*a^3*d*x + 2*b*d*x^3 - 2*a*b*
d^(1/2)*(x*(a - x)*(b - x))^(1/2) - 3*a*b*d*x^2))/((b*x - a^3*d + d*x^3 - x^2 - 3*a*d*x^2 + 3*a^2*d*x)*(2*a*x
- 2*a*b - 2*b*x + 2*a^3*d - 2*d*x^3 + a^2 + b^2 + x^2 + a^4*d^2 + d^2*x^4 - 4*a*d^2*x^3 - 4*a^3*d^2*x + 6*a^2*
d^2*x^2 - 2*a^2*b*d + 2*a*d*x^2 - 2*a^2*d*x + 2*b*d*x^2)))/d^(1/2)