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

3.8.53.1 Optimal result

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

output

-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)
 

3.8.53.2 Mathematica [A] (verified)

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

input

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]
 

output

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

3.8.53.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {2 x^3-(a+b+c) x^2+a b c}{\left (-x^3+(a+b+c) x^2-(b c+a (b+c)-d) x+a b c\right ) \sqrt {x^3-(a+b+c) x^2+(b c+a (b+c)) x-a b c}}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {2 x^3-(a+b+c) x^2+a b c}{\sqrt {-((a-x) (x-b) (x-c))} \left (-x^3+(a+b+c) x^2-(b c+a (b+c)-d) x+a b c\right )}d\sqrt {x}}{\sqrt {-(x (a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7269

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \frac {2 x^3-(a+b+c) x^2+a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-x^3+(a+b+c) x^2-(b c+a (b+c)-d) x+a b c\right )}d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(a+b+c) x^2-2 (b c+a (b+c)-d) x+3 a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-x^3+(a+b+c) x^2-(b c+a (b+c)-d) x+a b c\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(a+b+c) x^2-2 (b c+a (b+c)-d) x+3 a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-x^3+a \left (\frac {b+c}{a}+1\right ) x^2-a b \left (\frac {a c+b c-d}{a b}+1\right ) x+a b c\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \sqrt {-x^2 (a+b+c)+x (a (b+c)+b c)-a b c+x^3} \int \left (\frac {(a+b+c) x^2-2 (b c+a (b+c)-d) x+3 a b c}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c} \left (-x^3+a \left (\frac {b+c}{a}+1\right ) x^2-a b \left (\frac {a c+b c-d}{a b}+1\right ) x+a b c\right )}-\frac {2}{\sqrt {a-x} \sqrt {x-b} \sqrt {x-c}}\right )d\sqrt {x}}{\sqrt {-((a-x) (b-x) (c-x))} \sqrt {-(x (a-x) (b-x) (c-x))}}\)

input

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]
 

output

$Aborted
 

3.8.53.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7269

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ 
a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ 
p])*z^(q*FracPart[p])))   Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a 
, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !F 
reeQ[z, x]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.8.53.4 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}\)	\(524\)

input

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,method=_RETURNVERBOSE)
 

output

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

3.8.53.5 Fricas [F(-1)]

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

input

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")
 

output

Timed out
 

3.8.53.6 Sympy [F(-1)]

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

input

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)
 

output

Timed out
 

3.8.53.7 Maxima [F]

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

input

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")
 

output

-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)
 

3.8.53.8 Giac [F]

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

input

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")
 

output

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)
 

3.8.53.9 Mupad [F(-1)]

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

input

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)
 

output

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)