Optimal. Leaf size=59 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {x^3 (-a-b-c)+x^2 (a b+a c+b c)-a b c x+x^4}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]
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Rubi [F] time = 6.44, 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 {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right )} \, dx &=\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-(a b+a c+b c) x+(a+b+c) 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-(b c+a (b+c)) x+(a+b+c) 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)-\left (a^2 (b+c)+b c (b+c)+a \left (b^2+c^2+3 b c d\right )\right ) x+\left (a^2+b^2+c^2+2 b c d+2 a (b+c) d\right ) x^2}{(-1+d) \sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) 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)-\left (a^2 (b+c)+b c (b+c)+a \left (b^2+c^2+3 b c d\right )\right ) x+\left (a^2+b^2+c^2+2 b c d+2 a (b+c) d\right ) x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) 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)}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) x^2-(1-d) x^3\right )}+\frac {\left (-a^2 (b+c)-b c (b+c)-a \left (b^2+c^2+3 b c d\right )\right ) x}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) x^2-(1-d) x^3\right )}+\frac {\left (a^2+b^2+c^2+2 b c d+2 a (b+c) d\right ) x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) 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)) \int \frac {1}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) x^2-(1-d) x^3\right )} \, dx}{1-d}+\frac {\left (a^2+b^2+c^2+2 b c d+2 a (b+c) d\right ) \int \frac {x^2}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) x^2-(1-d) x^3\right )} \, dx}{1-d}-\frac {\left (a^2 (b+c)+b c (b+c)+a \left (b^2+c^2+3 b c d\right )\right ) \int \frac {x}{\sqrt {x (-a+x) (-b+x) (-c+x)} \left (a b c-(b c+a (b+c)) x+(a+b+c) x^2-(1-d) x^3\right )} \, dx}{1-d}\\ \end {align*}
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Mathematica [C] time = 12.64, size = 28348, normalized size = 480.47 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.75, size = 59, normalized size = 1.00 \begin {gather*} \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^2}\right )}{\sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [A] time = 4.22, size = 62, normalized size = 1.05 \begin {gather*} -\frac {2 \, \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 {-d}}\right )}{\sqrt {-d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 628, normalized size = 10.64 \begin {gather*} -\frac {2 \left (a +b +c \right ) 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 (-1+d \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 (\left (-1+d \right ) \textit {\_Z}^{3}+\left (a +b +c \right ) \textit {\_Z}^{2}+\left (-a b -a c -b c \right ) \textit {\_Z} +a b c \right )}{\sum }\frac {\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} a c d -2 \underline {\hspace {1.25 ex}}\alpha ^{2} b c d +3 \underline {\hspace {1.25 ex}}\alpha a b c d -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}+\underline {\hspace {1.25 ex}}\alpha \,a^{2} b +\underline {\hspace {1.25 ex}}\alpha \,a^{2} c +\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^{2} c +\underline {\hspace {1.25 ex}}\alpha b \,c^{2}-a^{2} b c -a \,b^{2} c -a b \,c^{2}\right ) \left (-c +x \right )^{2} \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha c d -c^{2} 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 ) \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 (-d \,\underline {\hspace {1.25 ex}}\alpha ^{2}+\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 \right ) \EllipticPi \left (\sqrt {\frac {\left (a -c \right ) x}{a \left (-c +x \right )}}, -\frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{2}+\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}{b \left (a -c \right )}, \sqrt {\frac {\left (-b +c \right ) a}{b \left (c -a \right )}}\right )}{a b}\right )}{\left (-3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2}+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 \right ) \left (a -c \right ) \sqrt {x \left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )}}\right )}{\left (-1+d \right ) c^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, a b c x + {\left (a + b + c\right )} x^{3} - 2 \, {\left (a b + a c + b c\right )} x^{2}}{\sqrt {-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )} x} {\left ({\left (d - 1\right )} x^{3} + a b c + {\left (a + b + c\right )} x^{2} - {\left (a b + a c + b c\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3\,\left (a+b+c\right )-2\,x^2\,\left (a\,b+a\,c+b\,c\right )+3\,a\,b\,c\,x}{\left (\left (d-1\right )\,x^3+\left (a+b+c\right )\,x^2+\left (-a\,b-a\,c-b\,c\right )\,x+a\,b\,c\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.
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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.
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