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

Optimal. Leaf size=59 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^3 (-a-b-c)+x^2 (a b+a c+b c)-a b c x+x^4}}{\sqrt {d} x}\right )}{\sqrt {d}} \]

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Rubi [F]  time = 4.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 13.10, size = 6921, normalized size = 117.31 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.62, size = 59, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]

[Out]

(-2*ArcTanh[Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4]/(Sqrt[d]*x)])/Sqrt[d]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)

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maple [C]  time = 0.14, size = 522, normalized size = 8.85

method	result	size
default	\(-\frac {4 a \sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}\, \left (-c +x \right )^{2} \sqrt {\frac {c \left (-b +x \right )}{b \left (-c +x \right )}}\, \sqrt {\frac {c \left (-a +x \right )}{a \left (-c +x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{\left (a -c \right ) c \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}-\frac {2 a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\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 (-c +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 +a b -d \right ) \sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}\, \sqrt {\frac {c \left (-b +x \right )}{b \left (-c +x \right )}}\, \sqrt {\frac {c \left (-a +x \right )}{a \left (-c +x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, \sqrt {\frac {\left (-b +c \right ) a}{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 ) \EllipticPi \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +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 c +b c -d}{b \left (a -c \right )}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{a b}\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 (a -c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{c^{2} d}\)	\(522\)
elliptic	\(-\frac {4 a \sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}\, \left (-c +x \right )^{2} \sqrt {\frac {c \left (-b +x \right )}{b \left (-c +x \right )}}\, \sqrt {\frac {c \left (-a +x \right )}{a \left (-c +x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{\left (a -c \right ) c \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}+\frac {2 a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\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 (-c +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 +a b -d \right ) \sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}\, \sqrt {\frac {c \left (-b +x \right )}{b \left (-c +x \right )}}\, \sqrt {\frac {c \left (-a +x \right )}{a \left (-c +x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, \sqrt {\frac {\left (-b +c \right ) a}{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 ) \EllipticPi \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +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 c +b c -d}{b \left (a -c \right )}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{a b}\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 (a -c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{c^{2} d}\)	\(525\)

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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