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

Optimal. Leaf size=57 \[ -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {d} \sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}\right )}{\sqrt {d}} \]

[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)

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Rubi [F]
time = 3.33, 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 d+(-1+a b d+a c d+b c d) x-(a+b+c) d x^2+d 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*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*} \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-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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 42.70, size = 8060, normalized size = 141.40 \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*d) + (-1 + a*b*d + a*
c*d + b*c*d)*x - (a + b + c)*d*x^2 + d*x^3)),x]

[Out]

Result too large to show

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.39, size = 565, normalized size = 9.91

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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-a d -b d -c d \right ) \textit {\_Z}^{2}+\left (a b d +a c 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 (-c +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b +a b d -1\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} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c d +a b d +a c d +b c d -1\right ) \EllipticPi \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c d +a c d +b c d -1}{b d \left (a -c \right )}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{a b d}\right )}{\left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2} d +2 d \underline {\hspace {1.25 ex}}\alpha a +2 d \underline {\hspace {1.25 ex}}\alpha b +2 \underline {\hspace {1.25 ex}}\alpha c d -a b d -a c d -b c d +1\right ) \left (a -c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{d \,c^{2}}\)	\(565\)
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 )}{d \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 (d \,\textit {\_Z}^{3}+\left (-a d -b d -c d \right ) \textit {\_Z}^{2}+\left (a b d +a c 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 (-c +x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b +a b d -1\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} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c d +a b d +a c d +b c d -1\right ) \EllipticPi \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} d -d \underline {\hspace {1.25 ex}}\alpha a -d \underline {\hspace {1.25 ex}}\alpha b -\underline {\hspace {1.25 ex}}\alpha c d +a c d +b c d -1}{b d \left (a -c \right )}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{a b d}\right )}{\left (3 \underline {\hspace {1.25 ex}}\alpha ^{2} d -2 d \underline {\hspace {1.25 ex}}\alpha a -2 d \underline {\hspace {1.25 ex}}\alpha b -2 \underline {\hspace {1.25 ex}}\alpha c d +a b d +a c d +b c d -1\right ) \left (a -c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{d \,c^{2}}\)	\(565\)

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*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]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*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)

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Fricas [F(-1)] Timed out
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*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

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Sympy [F(-1)] Timed out
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*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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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*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)

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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 (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 \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)*(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)

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