\(\int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} (a+(-1+b c d) x-(b+c) d x^2+d x^3)} \, dx\) [830]

Optimal result

Integrand size = 77, antiderivative size = 63 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{a-x}\right )}{\sqrt {d}} \]

[Out]

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

Rubi [F]

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

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

[Out]

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

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

Mathematica [A] (verified)

Time = 10.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.67 \[ \int \frac {-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a+(-1+b c d) x-(b+c) d x^2+d x^3\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x) (-c+x)}}{a-x}\right )}{\sqrt {d}} \]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 8.15 (sec) , antiderivative size = 531, normalized size of antiderivative = 8.43

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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-d b -c d \right ) \textit {\_Z}^{2}+\left (b c d -1\right ) \textit {\_Z} +a \right )}{\sum }\frac {\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +\underline {\hspace {1.25 ex}}\alpha ^{2} b d +\underline {\hspace {1.25 ex}}\alpha ^{2} c d +2 \underline {\hspace {1.25 ex}}\alpha a b d +2 \underline {\hspace {1.25 ex}}\alpha a c d -2 \underline {\hspace {1.25 ex}}\alpha b c d -a b c d +2 \underline {\hspace {1.25 ex}}\alpha -2 a \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{2} d -a b d -c a d +b c d -1\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 )+\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +b c d -1\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {d \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}{-c +a}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha b d +2 \underline {\hspace {1.25 ex}}\alpha c d -b c d +1\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 )}{d^{2} a^{2}}\)	\(531\)
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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-d b -c d \right ) \textit {\_Z}^{2}+\left (b c d -1\right ) \textit {\_Z} +a \right )}{\sum }\frac {\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -\underline {\hspace {1.25 ex}}\alpha ^{2} b d -\underline {\hspace {1.25 ex}}\alpha ^{2} c d -2 \underline {\hspace {1.25 ex}}\alpha a b d -2 \underline {\hspace {1.25 ex}}\alpha a c d +2 \underline {\hspace {1.25 ex}}\alpha b c d +a b c d -2 \underline {\hspace {1.25 ex}}\alpha +2 a \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha a d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +a^{2} d -a b d -c a d +b c d -1\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 )+\left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -\underline {\hspace {1.25 ex}}\alpha b d -\underline {\hspace {1.25 ex}}\alpha c d +b c d -1\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (c -a \right ) x}{c \left (-a +x \right )}}, -\frac {d \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}{-c +a}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha b d -2 \underline {\hspace {1.25 ex}}\alpha c d +b c d -1\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 )}{d^{2} a^{2}}\)	\(531\)

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

[Out]

-4/d*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/d^2*c/a^2*sum((-3*_alpha^
2*a*d+_alpha^2*b*d+_alpha^2*c*d+2*_alpha*a*b*d+2*_alpha*a*c*d-2*_alpha*b*c*d-a*b*c*d+2*_alpha-2*a)/(-3*_alpha^
2*d+2*_alpha*b*d+2*_alpha*c*d-b*c*d+1)*(-a+x)^2/(c-a)*(_alpha^2*d+_alpha*a*d-_alpha*b*d-_alpha*c*d+a^2*d-a*b*d
-a*c*d+b*c*d-1)/(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)*(EllipticF(((c-a)*x/c/(-a+x))^(1/2),((a-b)*c/b/(-c+a))^(1/2))+(_alpha^2*d-_alp
ha*b*d-_alpha*c*d+b*c*d-1)*EllipticPi(((c-a)*x/c/(-a+x))^(1/2),-d*(_alpha^2-_alpha*b-_alpha*c+b*c)*c/(-c+a),((
a-b)*c/b/(-c+a))^(1/2))),_alpha=RootOf(d*_Z^3+(-b*d-c*d)*_Z^2+(b*c*d-1)*_Z+a))

Fricas [F(-1)]

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

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

[Out]

Timed out

Sympy [F(-1)]

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

[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+(b*c*d-1)*x-(b+c)*d*x*
*2+d*x**3),x)

[Out]

Timed out

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

[In]

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

[Out]

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