Optimal. Leaf size=59 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt {d} x^2}\right )}{\sqrt {d}} \]
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Rubi [F]
time = 4.44, 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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Rubi steps
\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] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 56.70, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.27, size = 628, normalized size = 10.64 Too large to display
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Giac [A]
time = 0.81, 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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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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