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

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

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Rubi [F]  time = 7.24, 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 {x \left (3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right )}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 12.27, size = 32877, normalized size = 557.24 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.98, size = 59, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {x^3 (-a-b-c)+x^2 (a b+a c+b c)-a b c x+x^4}}{x^2}\right )}{\sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[(Sqrt[d]*Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4])/x^2])/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(x*(3*a*b*c-2*(a*b+a*c+b*c)*x+(a+b+c)*x^2)/(x*(-a+x)*(-b+x)*(-c+x))^(1/2)/(-a*b*c*d+(a*b+a*c+b*c)*d*x
-(a+b+c)*d*x^2+(-1+d)*x^3),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 4.03, size = 69, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left | d \right |} \arctan \left (\frac {\sqrt {-\frac {a b c}{x^{3}} + \frac {a b}{x^{2}} + \frac {a c}{x^{2}} + \frac {b c}{x^{2}} - \frac {a}{x} - \frac {b}{x} - \frac {c}{x} + 1}}{\sqrt {-\frac {1}{d}}}\right )}{\sqrt {-d} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.12, size = 674, normalized size = 11.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (3 \, a b c + {\left (a + b + c\right )} x^{2} - 2 \, {\left (a b + a c + b c\right )} x\right )} x}{{\left (a b c d + {\left (a + b + c\right )} d x^{2} - {\left (d - 1\right )} x^{3} - {\left (a b + a c + b c\right )} d x\right )} \sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,\left (x^2\,\left (a+b+c\right )-2\,x\,\left (a\,b+a\,c+b\,c\right )+3\,a\,b\,c\right )}{\left (\left (d-1\right )\,x^3-d\,\left (a+b+c\right )\,x^2+d\,\left (a\,b+a\,c+b\,c\right )\,x-a\,b\,c\,d\right )\,\sqrt {-x\,\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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