\(\int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} (-a b c+(a b+a c+b c-d) x-(a+b+c) x^2+x^3)} \, dx\) [753]

Optimal result

Integrand size = 76, antiderivative size = 59 \[ \int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c+(a b+a c+b c-d) x-(a+b+c) x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} x}\right )}{\sqrt {d}} \]

[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)/d^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 10.49 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c+(a b+a c+b c-d) x-(a+b+c) x^2+x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x) (-c+x)}}{\sqrt {d} x}\right )}{\sqrt {d}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58

method	result	size
default	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x \sqrt {d}}\right )}{\sqrt {d}}\)	\(34\)
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x \sqrt {d}}\right )}{\sqrt {d}}\)	\(34\)
elliptic	\(-\frac {4 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 (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 (\textit {\_Z}^{3}+\left (-a -b -c \right ) \textit {\_Z}^{2}+\left (a b +a c +b c -d \right ) \textit {\_Z} -a b c \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a -\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha ^{2} c +2 \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 c -3 a b c -2 \underline {\hspace {1.25 ex}}\alpha d \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c -d \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}-\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 -d \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, \frac {\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 -d}{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}+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 +d \right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{a^{2} d}\)	\(524\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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