\(\int \frac {x (-b+x) (a b-2 a x+x^2)}{\sqrt {x (-a+x) (-b+x)} (-a^2+2 a x+(-1+b^2 d) x^2-2 b d x^3+d x^4)} \, dx\) [1201]

Optimal result

Integrand size = 69, antiderivative size = 88 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{3/4}} \]

[Out]

-arctan(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/(a-x))/d^(3/4)+arctanh(d^(1/4)*(a*b*x+(-a-b)*x^2+x^3)^(1/2)/(a-x)
)/d^(3/4)

Rubi [F]

\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx \]

[In]

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

[Out]

(4*a*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^4*Sqrt[-b + x^2])/(Sqrt[-a + x^2]*(a^2 - 2*a
*x^2 + (1 - b^2*d)*x^4 + 2*b*d*x^6 - d*x^8)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*a*b*Sqrt[x]*Sqrt[-
a + x]*Sqrt[-b + x]*Defer[Subst][Defer[Int][(x^2*Sqrt[-b + x^2])/(Sqrt[-a + x^2]*(-a^2 + 2*a*x^2 - (1 - b^2*d)
*x^4 - 2*b*d*x^6 + d*x^8)), x], x, Sqrt[x]])/Sqrt[(a - x)*(b - x)*x] + (2*Sqrt[x]*Sqrt[-a + x]*Sqrt[-b + x]*De
fer[Subst][Defer[Int][(x^6*Sqrt[-b + x^2])/(Sqrt[-a + x^2]*(-a^2 + 2*a*x^2 - (1 - b^2*d)*x^4 - 2*b*d*x^6 + d*x
^8)), 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 {\sqrt {x} \sqrt {-b+x} \left (a b-2 a x+x^2\right )}{\sqrt {-a+x} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\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 {x^2 \sqrt {-b+x^2} \left (a b-2 a x^2+x^4\right )}{\sqrt {-a+x^2} \left (-a^2+2 a x^2+\left (-1+b^2 d\right ) x^4-2 b d x^6+d x^8\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 {2 a x^4 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (a^2-2 a x^2+\left (1-b^2 d\right ) x^4+2 b d x^6-d x^8\right )}+\frac {a b x^2 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\right )}+\frac {x^6 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\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^6 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (4 a \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (a^2-2 a x^2+\left (1-b^2 d\right ) x^4+2 b d x^6-d x^8\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 {x^2 \sqrt {-b+x^2}}{\sqrt {-a+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.75 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} x (-b+x)}{\sqrt {x (-a+x) (-b+x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt {x (-a+x) (-b+x)}}{a-x}\right )}{d^{3/4}} \]

[In]

Integrate[(x*(-b + x)*(a*b - 2*a*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-a^2 + 2*a*x + (-1 + b^2*d)*x^2 - 2*b*d
*x^3 + d*x^4)),x]

[Out]

(ArcTan[(d^(1/4)*x*(-b + x))/Sqrt[x*(-a + x)*(-b + x)]] + ArcTanh[(d^(1/4)*Sqrt[x*(-a + x)*(-b + x)])/(a - x)]
)/d^(3/4)

Maple [C] (warning: unable to verify)

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

Time = 1.43 (sec) , antiderivative size = 358, normalized size of antiderivative = 4.07

method	result	size
default	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-2 d b \,\textit {\_Z}^{3}+\left (d \,b^{2}-1\right ) \textit {\_Z}^{2}+2 \textit {\_Z} a -a^{2}\right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3} a d +\underline {\hspace {1.25 ex}}\alpha ^{3} b d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d -\underline {\hspace {1.25 ex}}\alpha a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha a +a^{2}\right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(358\)
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-2 d b \,\textit {\_Z}^{3}+\left (d \,b^{2}-1\right ) \textit {\_Z}^{2}+2 \textit {\_Z} a -a^{2}\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b d -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +\underline {\hspace {1.25 ex}}\alpha a \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha a -a^{2}\right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +\underline {\hspace {1.25 ex}}\alpha \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha +a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d}\)	\(360\)

[In]

int(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2+2*a*x+(b^2*d-1)*x^2-2*b*d*x^3+d*x^4),x,method=_RETU
RNVERBOSE)

[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))-1/d*b*sum((-2*_alpha^3*a*d+_alpha^3*b*d+3*_alpha^2*a*b*d-_alpha^2*b^2*d-_alpha*a*b^2*
d+_alpha^2-2*_alpha*a+a^2)/(-2*_alpha^3*d+3*_alpha^2*b*d-_alpha*b^2*d+_alpha-a)*(_alpha^3*d-_alpha^2*b*d-_alph
a+2*a-b)/(a^2-2*a*b+b^2)*(-(-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)*Ellip
ticPi((-(-b+x)/b)^(1/2),-(_alpha^3*d-_alpha^2*b*d-_alpha+2*a-b)*b/(a^2-2*a*b+b^2),(b/(-a+b))^(1/2)),_alpha=Roo
tOf(d*_Z^4-2*d*b*_Z^3+(b^2*d-1)*_Z^2+2*_Z*a-a^2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 677, normalized size of antiderivative = 7.69 \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=-\frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (b d^{3} x - d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (a d - d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (b d^{3} x - d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (a d - d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} - 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) + \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (i \, b d^{3} x - i \, d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (-i \, a d + i \, d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) - \frac {1}{4} i \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left ({\left (-i \, b d^{3} x + i \, d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}} + {\left (i \, a d - i \, d x\right )} \frac {1}{d^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right ) \]

[In]

integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2+2*a*x+(b^2*d-1)*x^2-2*b*d*x^3+d*x^4),x, algor
ithm="fricas")

[Out]

-1/4*(d^(-3))^(1/4)*log((2*b*d*x^3 - d*x^4 - (b^2*d + 1)*x^2 - a^2 + 2*a*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)
*((b*d^3*x - d^3*x^2)*(d^(-3))^(3/4) + (a*d - d*x)*(d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)
*sqrt(d^(-3)))/(2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x)) + 1/4*(d^(-3))^(1/4)*log((2*b*d*x^3 - d*x^
4 - (b^2*d + 1)*x^2 - a^2 + 2*a*x - 2*sqrt(a*b*x - (a + b)*x^2 + x^3)*((b*d^3*x - d^3*x^2)*(d^(-3))^(3/4) + (a
*d - d*x)*(d^(-3))^(1/4)) - 2*(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*d*x^3 - d*x^4 - (b^2*
d - 1)*x^2 + a^2 - 2*a*x)) + 1/4*I*(d^(-3))^(1/4)*log((2*b*d*x^3 - d*x^4 - (b^2*d + 1)*x^2 - a^2 + 2*a*x + 2*s
qrt(a*b*x - (a + b)*x^2 + x^3)*((I*b*d^3*x - I*d^3*x^2)*(d^(-3))^(3/4) + (-I*a*d + I*d*x)*(d^(-3))^(1/4)) + 2*
(a*b*d^2*x - (a + b)*d^2*x^2 + d^2*x^3)*sqrt(d^(-3)))/(2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x)) - 1
/4*I*(d^(-3))^(1/4)*log((2*b*d*x^3 - d*x^4 - (b^2*d + 1)*x^2 - a^2 + 2*a*x + 2*sqrt(a*b*x - (a + b)*x^2 + x^3)
*((-I*b*d^3*x + I*d^3*x^2)*(d^(-3))^(3/4) + (I*a*d - I*d*x)*(d^(-3))^(1/4)) + 2*(a*b*d^2*x - (a + b)*d^2*x^2 +
d^2*x^3)*sqrt(d^(-3)))/(2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x))

Sympy [F(-1)]

Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(-b+x)*(a*b-2*a*x+x**2)/(x*(-a+x)*(-b+x))**(1/2)/(-a**2+2*a*x+(b**2*d-1)*x**2-2*b*d*x**3+d*x**4),x
)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]

[In]

integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2+2*a*x+(b^2*d-1)*x^2-2*b*d*x^3+d*x^4),x, algor
ithm="maxima")

[Out]

integrate((a*b - 2*a*x + x^2)*(b - x)*x/((2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x)*sqrt((a - x)*(b -
x)*x)), x)

Giac [F]

\[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\int { \frac {{\left (a b - 2 \, a x + x^{2}\right )} {\left (b - x\right )} x}{{\left (2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}} \,d x } \]

[In]

integrate(x*(-b+x)*(a*b-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/2)/(-a^2+2*a*x+(b^2*d-1)*x^2-2*b*d*x^3+d*x^4),x, algor
ithm="giac")

[Out]

integrate((a*b - 2*a*x + x^2)*(b - x)*x/((2*b*d*x^3 - d*x^4 - (b^2*d - 1)*x^2 + a^2 - 2*a*x)*sqrt((a - x)*(b -
x)*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x (-b+x) \left (a b-2 a x+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx=\int -\frac {x\,\left (b-x\right )\,\left (x^2-2\,a\,x+a\,b\right )}{\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (-a^2+2\,a\,x+d\,x^4-2\,b\,d\,x^3+\left (b^2\,d-1\right )\,x^2\right )} \,d x \]

[In]

int(-(x*(b - x)*(a*b - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/2)*(x^2*(b^2*d - 1) + 2*a*x + d*x^4 - a^2 - 2*b*d
*x^3)),x)

[Out]

int(-(x*(b - x)*(a*b - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/2)*(x^2*(b^2*d - 1) + 2*a*x + d*x^4 - a^2 - 2*b*d
*x^3)), x)