\(\int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} (-a b c d+(-1+a b d+a c d+b c d) x-(a+b+c) d x^2+d x^3)} \, dx\) [736]

Optimal result

Integrand size = 81, antiderivative size = 57 \[ \int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(-1+a b d+a c d+b c d) x-(a+b+c) d x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {d} \sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}\right )}{\sqrt {d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(2*Defer[Int][1/Sqrt[x*(-a + x)*(-b + x)*(-c + x)], x])/d - 3*a*b*c*Defer[Int][1/(Sqrt[x*(-a + x)*(-b + x)*(-c
+ x)]*(a*b*c*d + (1 - b*c*d - a*(b + c)*d)*x + (a + b + c)*d*x^2 - d*x^3)), x] - (2*(1 - b*c*d - a*(b + c)*d)
*Defer[Int][x/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a*b*c*d + (1 - b*c*d - a*(b + c)*d)*x + (a + b + c)*d*x^2 -
d*x^3)), x])/d - (a + b + c)*Defer[Int][x^2/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a*b*c*d + (1 - b*c*d - a*(b
+ c)*d)*x + (a + b + c)*d*x^2 - d*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 d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )} \, dx \\ & = \int \left (\frac {2}{d \sqrt {x (-a+x) (-b+x) (-c+x)}}-\frac {3 a b c d+2 (1-b c d-a (b+c) d) x+(a+b+c) d x^2}{d \sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )}\right ) \, dx \\ & = -\frac {\int \frac {3 a b c d+2 (1-b c d-a (b+c) d) x+(a+b+c) d x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+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 {3 a b c d}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+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 b c d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )}+\frac {(a+b+c) d x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+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 {1}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )} \, dx\right )-(a+b+c) \int \frac {x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )} \, dx+\frac {2 \int \frac {1}{\sqrt {x (-a+x) (-b+x) (-c+x)}} \, 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 b c d+(1-b c d-a (b+c) d) x+(a+b+c) d x^2-d x^3\right )} \, dx}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.69 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.67 \[ \int \frac {a b c-(a+b+c) x^2+2 x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(-1+a b d+a c d+b c d) x-(a+b+c) d x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x (-a+x) (-b+x) (-c+x)}}{x}\right )}{\sqrt {d}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x}\right )}{\sqrt {d}}\)	\(34\)
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d}\, \sqrt {-x \left (a -x \right ) \left (b -x \right ) \left (c -x \right )}}{x}\right )}{\sqrt {d}}\)	\(34\)
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 )}{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 (-a d -d b -c d \right ) \textit {\_Z}^{2}+\left (a b d +c a d +b c d -1\right ) \textit {\_Z} -a b c d \right )}{\sum }\frac {\left (-\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 -3 a b c d -2 \underline {\hspace {1.25 ex}}\alpha \right ) \left (-a +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -\underline {\hspace {1.25 ex}}\alpha b d -d \underline {\hspace {1.25 ex}}\alpha c +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 )-\frac {\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 -d \underline {\hspace {1.25 ex}}\alpha c +a b d +c a d +b c d -1\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 a d -\underline {\hspace {1.25 ex}}\alpha b d -d \underline {\hspace {1.25 ex}}\alpha c +a b d +c a d -1}{b d \left (-c +a \right )}, \sqrt {\frac {\left (a -b \right ) c}{b \left (-c +a \right )}}\right )}{b c d}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d -2 \underline {\hspace {1.25 ex}}\alpha b d -2 d \underline {\hspace {1.25 ex}}\alpha c +a b d +c a d +b c d -1\right ) \left (c -a \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{d \,a^{2}}\)	\(564\)

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

[Out]

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

Fricas [F(-1)]

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

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

[Out]

Timed out

Sympy [F(-1)]

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

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

[Out]

Timed out

Maxima [F]

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

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

[Out]

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

Giac [F]

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

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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