\(\int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3)} \, dx\) [756]

Optimal result

Integrand size = 87, antiderivative size = 59 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 10.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x) (-c+x)}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x^{2} \sqrt {d}}\right )}{\sqrt {d}}\)	\(34\)
default	\(-\frac {2 \left (a +b +c \right ) c \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \left (-a +x \right )^{2} \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{\left (d -1\right ) \left (c -a \right ) a \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}+\frac {2 c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\left (d -1\right ) \textit {\_Z}^{3}+\left (a +b +c \right ) \textit {\_Z}^{2}+\left (-a b -a c -b c \right ) \textit {\_Z} +a b c \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} a c d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b c d +3 \underline {\hspace {1.25 ex}}\alpha a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}+\underline {\hspace {1.25 ex}}\alpha \,a^{2} b +\underline {\hspace {1.25 ex}}\alpha \,a^{2} c +\underline {\hspace {1.25 ex}}\alpha a \,b^{2}+\underline {\hspace {1.25 ex}}\alpha a \,c^{2}+\underline {\hspace {1.25 ex}}\alpha \,b^{2} c +\underline {\hspace {1.25 ex}}\alpha b \,c^{2}-a^{2} b c -a \,b^{2} c -a b \,c^{2}\right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )-\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c}{b \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{b c}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +3 \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha a -2 \underline {\hspace {1.25 ex}}\alpha b -2 \underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{\left (d -1\right ) a^{4} d}\)	\(625\)
elliptic	\(-\frac {2 \left (a +b +c \right ) c \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \left (-a +x \right )^{2} \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{\left (d -1\right ) \left (c -a \right ) a \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}+\frac {2 c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\left (d -1\right ) \textit {\_Z}^{3}+\left (a +b +c \right ) \textit {\_Z}^{2}+\left (-a b -a c -b c \right ) \textit {\_Z} +a b c \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} a c d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b c d -3 \underline {\hspace {1.25 ex}}\alpha a b c d +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}-\underline {\hspace {1.25 ex}}\alpha \,a^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} c -\underline {\hspace {1.25 ex}}\alpha a \,b^{2}-\underline {\hspace {1.25 ex}}\alpha a \,c^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} c -\underline {\hspace {1.25 ex}}\alpha b \,c^{2}+a^{2} b c +a \,b^{2} c +a b \,c^{2}\right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d +a^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha b +\underline {\hspace {1.25 ex}}\alpha c -b c \right ) \sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}\, \sqrt {\frac {a \left (-b +x \right )}{b \left (-a +x \right )}}\, \sqrt {\frac {a \left (-c +x \right )}{c \left (-a +x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )-\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c +b c \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \frac {-\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +a b +a c}{b \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{b c}\right )}{\left (d -1\right ) \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha a +2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -a b -a c -b c \right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{a^{4} d}\)	\(628\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05 \[ \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {-\frac {a b c}{x^{3}} + \frac {a b}{x^{2}} + \frac {a c}{x^{2}} + \frac {b c}{x^{2}} - \frac {a}{x} - \frac {b}{x} - \frac {c}{x} + 1}}{\sqrt {-d}}\right )}{\sqrt {-d}} \]

[In]

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

[Out]

-2*arctan(sqrt(-a*b*c/x^3 + a*b/x^2 + a*c/x^2 + b*c/x^2 - a/x - b/x - c/x + 1)/sqrt(-d))/sqrt(-d)

Mupad [F(-1)]

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

[In]

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

[Out]

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