\(\int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} (a d+(b c-d) x-(b+c) x^2+x^3)} \, dx\) [827]

Optimal result

Integrand size = 77, antiderivative size = 63 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]

[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)/(a-x))/d^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 10.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a d+(b c-d) x-(b+c) x^2+x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x) (-c+x)}}{\sqrt {d} (a-x)}\right )}{\sqrt {d}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 8.06 (sec) , antiderivative size = 509, normalized size of antiderivative = 8.08

method	result	size
default	\(-\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 (-b -c \right ) \textit {\_Z}^{2}+\left (b c -d \right ) \textit {\_Z} +a d \right )}{\sum }\frac {\left (-3 \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 -a b c +2 \underline {\hspace {1.25 ex}}\alpha d -2 a d \right ) \left (-a +x \right )^{2} \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^{2}-a b -a 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 b -\underline {\hspace {1.25 ex}}\alpha c +b c -d \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) c}{d \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{d}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -b c +d \right ) \left (c -a \right ) \left (a^{2}-a b -a c +b c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{a^{2}}\)	\(509\)
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 (-b -c \right ) \textit {\_Z}^{2}+\left (b c -d \right ) \textit {\_Z} +a d \right )}{\sum }\frac {\left (3 \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 +a b c -2 \underline {\hspace {1.25 ex}}\alpha d +2 a d \right ) \left (-a +x \right )^{2} \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^{2}-a b -a 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 b -\underline {\hspace {1.25 ex}}\alpha c +b c -d \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c +b c \right ) c}{d \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{d}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c -b c +d \right ) \left (c -a \right ) \left (a^{2}-a b -a c +b c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{a^{2}}\)	\(510\)

[In]

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

[Out]

-4*c*((c-a)*x/c/(-a+x))^(1/2)*(-a+x)^2*(a*(-b+x)/b/(-a+x))^(1/2)*(a*(-c+x)/c/(-a+x))^(1/2)/(c-a)/a/(x*(-a+x)*(
-b+x)*(-c+x))^(1/2)*EllipticF(((c-a)*x/c/(-a+x))^(1/2),((a-b)*c/b/(-c+a))^(1/2))+2*c/a^2*sum((-3*_alpha^2*a+_a
lpha^2*b+_alpha^2*c+2*_alpha*a*b+2*_alpha*a*c-2*_alpha*b*c-a*b*c+2*_alpha*d-2*a*d)/(-3*_alpha^2+2*_alpha*b+2*_
alpha*c-b*c+d)*(-a+x)^2/(c-a)*(_alpha^2+_alpha*a-_alpha*b-_alpha*c+a^2-a*b-a*c+b*c-d)/(a^2-a*b-a*c+b*c)*((c-a)
*x/c/(-a+x))^(1/2)*(a*(-b+x)/b/(-a+x))^(1/2)*(a*(-c+x)/c/(-a+x))^(1/2)/(x*(-a+x)*(-b+x)*(-c+x))^(1/2)*(Ellipti
cF(((c-a)*x/c/(-a+x))^(1/2),((a-b)*c/b/(-c+a))^(1/2))+(_alpha^2-_alpha*b-_alpha*c+b*c-d)/d*EllipticPi(((c-a)*x
/c/(-a+x))^(1/2),-(_alpha^2-_alpha*b-_alpha*c+b*c)*c/d/(-c+a),((a-b)*c/b/(-c+a))^(1/2))),_alpha=RootOf(_Z^3+(-
b-c)*_Z^2+(b*c-d)*_Z+a*d))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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