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.
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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.
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IntegrateAlgebraic [A] time = 0.63, 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}\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 [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.
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maple [C] time = 0.09, size = 525, normalized size = 8.90 \begin {gather*} -\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} \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 {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.
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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.
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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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