3.6.19 \(\int \frac {3 a b x-2 (a+b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} (-a b+(a+b) x-x^2+d x^3)} \, dx\) [519]

3.6.19.1 Optimal result

Integrand size = 56, antiderivative size = 40 \[ \int \frac {3 a b x-2 (a+b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]

output

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

3.6.19.2 Mathematica [A] (verified)

Time = 12.47 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {3 a b x-2 (a+b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x (-a+x) (-b+x)}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]

input

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

output

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

3.6.19.3 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.

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {-x (a+b)+a b+x^2} \left (\frac {a b \int \frac {1}{\sqrt {x^2-(a+b) x+a b} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt {x}}{d}-\frac {(-3 a b d+a+b) \int \frac {x}{\sqrt {x^2-(a+b) x+a b} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt {x}}{d}+\frac {(-2 a d-2 b d+1) \int \frac {x^2}{\sqrt {x^2-(a+b) x+a b} \left (-d x^3+x^2-a \left (\frac {b}{a}+1\right ) x+a b\right )}d\sqrt {x}}{d}-\frac {\left (\sqrt {a} \sqrt {b}+x\right ) \sqrt {\frac {-x (a+b)+a b+x^2}{\left (\sqrt {a} \sqrt {b}+x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{a} \sqrt [4]{b}}\right ),\frac {1}{4} \left (\frac {a+b}{\sqrt {a} \sqrt {b}}+2\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt {-x (a+b)+a b+x^2}}\right )}{\sqrt {x (a-x) (b-x)}}\)

input

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

output

$Aborted
 

3.6.19.3.1 Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

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 7293

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

3.6.19.4 Maple [C] (warning: unable to verify)

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

Time = 0.88 (sec) , antiderivative size = 293, normalized size of antiderivative = 7.32

method	result	size
elliptic	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d +2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d -3 \underline {\hspace {1.25 ex}}\alpha a b d -\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha a +\underline {\hspace {1.25 ex}}\alpha b -a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha +a \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -2 \underline {\hspace {1.25 ex}}\alpha +a +b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2} b^{2}}\)	\(293\)
default	\(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-\textit {\_Z}^{2}+\left (a +b \right ) \textit {\_Z} -a b \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a b d +\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha a -\underline {\hspace {1.25 ex}}\alpha b +a b \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha +a \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {-b +x}{b}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d +\underline {\hspace {1.25 ex}}\alpha b d +d \,b^{2}-\underline {\hspace {1.25 ex}}\alpha +a}{b^{2} d}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 \underline {\hspace {1.25 ex}}\alpha -a -b \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )}{d^{2} b^{2}}\)	\(296\)

input

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

output

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

3.6.19.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (31) = 62\).

Time = 0.99 (sec) , antiderivative size = 285, normalized size of antiderivative = 7.12 \[ \int \frac {3 a b x-2 (a+b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\left [\frac {\log \left (\frac {d^{2} x^{6} + 6 \, d x^{5} - {\left (6 \, {\left (a + b\right )} d - 1\right )} x^{4} + a^{2} b^{2} + 2 \, {\left (3 \, a b d - a - b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 4 \, {\left (d x^{4} + a b x - {\left (a + b\right )} x^{2} + x^{3}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {d} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}{d^{2} x^{6} - 2 \, d x^{5} + {\left (2 \, {\left (a + b\right )} d + 1\right )} x^{4} + a^{2} b^{2} - 2 \, {\left (a b d + a + b\right )} x^{3} + {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2} - 2 \, {\left (a^{2} b + a b^{2}\right )} x}\right )}{2 \, \sqrt {d}}, \frac {\sqrt {-d} \arctan \left (\frac {{\left (d x^{3} + a b - {\left (a + b\right )} x + x^{2}\right )} \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} \sqrt {-d}}{2 \, {\left (a b d x^{2} - {\left (a + b\right )} d x^{3} + d x^{4}\right )}}\right )}{d}\right ] \]

input

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

output

[1/2*log((d^2*x^6 + 6*d*x^5 - (6*(a + b)*d - 1)*x^4 + a^2*b^2 + 2*(3*a*b*d 
 - a - b)*x^3 + (a^2 + 4*a*b + b^2)*x^2 - 4*(d*x^4 + a*b*x - (a + b)*x^2 + 
 x^3)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(d) - 2*(a^2*b + a*b^2)*x)/(d^2* 
x^6 - 2*d*x^5 + (2*(a + b)*d + 1)*x^4 + a^2*b^2 - 2*(a*b*d + a + b)*x^3 + 
(a^2 + 4*a*b + b^2)*x^2 - 2*(a^2*b + a*b^2)*x))/sqrt(d), sqrt(-d)*arctan(1 
/2*(d*x^3 + a*b - (a + b)*x + x^2)*sqrt(a*b*x - (a + b)*x^2 + x^3)*sqrt(-d 
)/(a*b*d*x^2 - (a + b)*d*x^3 + d*x^4))/d]
 

3.6.19.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.19.7 Maxima [F]

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

input

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

output

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

3.6.19.8 Giac [F]

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

input

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

output

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

3.6.19.9 Mupad [B] (verification not implemented)

Time = 8.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.72 \[ \int \frac {3 a b x-2 (a+b) x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^3\right )} \, dx=\frac {\ln \left (\frac {a\,b-a\,x-b\,x+d\,x^3+x^2-2\,\sqrt {d}\,x\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}}{a\,x-a\,b+b\,x+d\,x^3-x^2}\right )}{\sqrt {d}} \]

input

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

output

log((a*b - a*x - b*x + d*x^3 + x^2 - 2*d^(1/2)*x*(x*(a - x)*(b - x))^(1/2) 
)/(a*x - a*b + b*x + d*x^3 - x^2))/d^(1/2)