Optimal. Leaf size=46 \[ \log \left (\frac {-2 \left (\sqrt {(1-x) x \left (a^2-2 a x+x\right )}+x\right )-a^2+2 a x+x^2}{(a-x)^2}\right ) \]
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Rubi [C] time = 1.49, antiderivative size = 180, normalized size of antiderivative = 3.91, number of steps used = 7, number of rules used = 7, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {2056, 6733, 1710, 1104, 419, 1220, 537} \[ \frac {4 (1-a) \sqrt {1-x} \sqrt {x} \sqrt {\frac {(1-2 a) x}{a^2}+1} \Pi \left (\frac {1}{a};\sin ^{-1}\left (\sqrt {x}\right )|-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}}-\frac {2 (1-2 a) \sqrt {1-x} \sqrt {x} \sqrt {\frac {(1-2 a) x}{a^2}+1} F\left (\sin ^{-1}\left (\sqrt {x}\right )|-\frac {1-2 a}{a^2}\right )}{\sqrt {\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 537
Rule 1104
Rule 1220
Rule 1710
Rule 2056
Rule 6733
Rubi steps
\begin {align*} \int \frac {-a+(-1+2 a) x}{(-a+x) \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \int \frac {-a+(-1+2 a) x}{\sqrt {x} (-a+x) \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}} \, dx}{\sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname {Subst}\left (\int \frac {-a+(-1+2 a) x^2}{\left (-a+x^2\right ) \sqrt {a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac {\left (4 (1-a) a \sqrt {x} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \sqrt {a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}+\frac {\left (2 (-1+2 a) \sqrt {x} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+\left (1-2 a-a^2\right ) x^2+(-1+2 a) x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac {\left (4 (1-a) a \sqrt {1-x} \sqrt {x} \sqrt {1+\frac {(1-2 a) x}{a^2}} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \sqrt {1+\frac {2 (-1+2 a) x^2}{1-(-1+a)^2-2 a-a^2}} \sqrt {1+\frac {2 (-1+2 a) x^2}{1+(-1+a)^2-2 a-a^2}}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+\left (1-2 a-a^2\right ) x+(-1+2 a) x^2} \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}+\frac {\left (2 (-1+2 a) \sqrt {1-x} \sqrt {x} \sqrt {1+\frac {(1-2 a) x}{a^2}} \sqrt {a^2-\left (-1+2 a+a^2\right ) x+(-1+2 a) x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {2 (-1+2 a) x^2}{1-(-1+a)^2-2 a-a^2}} \sqrt {1+\frac {2 (-1+2 a) x^2}{1+(-1+a)^2-2 a-a^2}}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+\left (1-2 a-a^2\right ) x+(-1+2 a) x^2} \sqrt {a^2 x-\left (-1+2 a+a^2\right ) x^2+(-1+2 a) x^3}}\\ &=-\frac {2 (1-2 a) \sqrt {1-x} \sqrt {x} \sqrt {1+\frac {(1-2 a) x}{a^2}} F\left (\sin ^{-1}\left (\sqrt {x}\right )|-\frac {1-2 a}{a^2}\right )}{\sqrt {a^2 x+\left (1-2 a-a^2\right ) x^2-(1-2 a) x^3}}+\frac {4 (1-a) \sqrt {1-x} \sqrt {x} \sqrt {1+\frac {(1-2 a) x}{a^2}} \Pi \left (\frac {1}{a};\sin ^{-1}\left (\sqrt {x}\right )|-\frac {1-2 a}{a^2}\right )}{\sqrt {a^2 x+\left (1-2 a-a^2\right ) x^2-(1-2 a) x^3}}\\ \end {align*}
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Mathematica [C] time = 1.09, size = 133, normalized size = 2.89 \[ \frac {2 i (x-1)^{3/2} \sqrt {\frac {x}{x-1}} \sqrt {-\frac {a^2-2 a x+x}{(2 a-1) (x-1)}} \left (2 a \Pi \left (1-a;i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|-\frac {(a-1)^2}{2 a-1}\right )-\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right ),-\frac {(a-1)^2}{2 a-1}\right )\right )}{\sqrt {-\left ((x-1) x \left (a^2-2 a x+x\right )\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 63, normalized size = 1.37 \[ \log \left (-\frac {a^{2} - 2 \, {\left (a - 1\right )} x - x^{2} + 2 \, \sqrt {{\left (2 \, a - 1\right )} x^{3} + a^{2} x - {\left (a^{2} + 2 \, a - 1\right )} x^{2}}}{a^{2} - 2 \, a x + x^{2}}\right ) \]
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 \[ \int -\frac {{\left (2 \, a - 1\right )} x - a}{\sqrt {{\left (2 \, a - 1\right )} x^{3} + a^{2} x - {\left (a^{2} + 2 \, a - 1\right )} x^{2}} {\left (a - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 536, normalized size = 11.65 \[ -\frac {4 \sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}\, \sqrt {\frac {x -1}{\frac {a^{2}}{2 a -1}-1}}\, \sqrt {\frac {\left (2 a -1\right ) x}{a^{2}}}\, a^{3} \EllipticF \left (\sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}, \sqrt {\frac {a^{2}}{\left (2 a -1\right ) \left (\frac {a^{2}}{2 a -1}-1\right )}}\right )}{\left (2 a -1\right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}}-\frac {4 \left (a -1\right ) \sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}\, \sqrt {\frac {x -1}{\frac {a^{2}}{2 a -1}-1}}\, \sqrt {\frac {\left (2 a -1\right ) x}{a^{2}}}\, a^{3} \EllipticPi \left (\sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}, \frac {a^{2}}{\left (2 a -1\right ) \left (\frac {a^{2}}{2 a -1}-a \right )}, \sqrt {\frac {a^{2}}{\left (2 a -1\right ) \left (\frac {a^{2}}{2 a -1}-1\right )}}\right )}{\left (2 a -1\right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}\, \left (\frac {a^{2}}{2 a -1}-a \right )}+\frac {2 \sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}\, \sqrt {\frac {x -1}{\frac {a^{2}}{2 a -1}-1}}\, \sqrt {\frac {\left (2 a -1\right ) x}{a^{2}}}\, a^{2} \EllipticF \left (\sqrt {-\frac {\left (-\frac {a^{2}}{2 a -1}+x \right ) \left (2 a -1\right )}{a^{2}}}, \sqrt {\frac {a^{2}}{\left (2 a -1\right ) \left (\frac {a^{2}}{2 a -1}-1\right )}}\right )}{\left (2 a -1\right ) \sqrt {-a^{2} x^{2}+2 a \,x^{3}+a^{2} x -2 a \,x^{2}-x^{3}+x^{2}}} \]
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 \[ -\int \frac {{\left (2 \, a - 1\right )} x - a}{\sqrt {{\left (2 \, a - 1\right )} x^{3} + a^{2} x - {\left (a^{2} + 2 \, a - 1\right )} x^{2}} {\left (a - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.02 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 a x - a - x}{\sqrt {x \left (x - 1\right ) \left (- a^{2} + 2 a x - x\right )} \left (- a + x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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