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 = 2.93293, 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 \[ \frac{4 (1-a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} \Pi \left (\frac{1}{a};\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}}-\frac{2 (1-2 a) \sqrt{1-x} \sqrt{x} \sqrt{\frac{(1-2 a) x}{a^2}+1} F\left (\sin ^{-1}\left (\sqrt{x}\right )|-\frac{1-2 a}{a^2}\right )}{\sqrt{\left (-a^2-2 a+1\right ) x^2+a^2 x-(1-2 a) x^3}} \]
Antiderivative was successfully verified.
[In] Int[(-a + (-1 + 2*a)*x)/((-a + x)*Sqrt[a^2*x - (-1 + 2*a + a^2)*x^2 + (-1 + 2*a)*x^3]),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-a+(-1+2*a)*x)/(-a+x)/(a**2*x-(a**2+2*a-1)*x**2+(-1+2*a)*x**3)**(1/2),x)
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Mathematica [C] time = 0.590397, size = 133, normalized size = 2.89 \[ \frac{2 i (x-1)^{3/2} \sqrt{\frac{x}{x-1}} \sqrt{-\frac{a^2-2 a x+x}{(2 a-1) (x-1)}} \left (2 a \Pi \left (1-a;i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|-\frac{(a-1)^2}{2 a-1}\right )-F\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )|-\frac{(a-1)^2}{2 a-1}\right )\right )}{\sqrt{-(x-1) x \left (a^2-2 a x+x\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(-a + (-1 + 2*a)*x)/((-a + x)*Sqrt[a^2*x - (-1 + 2*a + a^2)*x^2 + (-1 + 2*a)*x^3]),x]
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Maple [C] time = 0.079, size = 536, normalized size = 11.7 \[ 2\,{\frac{{a}^{2}}{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3}}{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticF} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) }-4\,{\frac{{a}^{3} \left ( a-1 \right ) }{ \left ( -1+2\,a \right ) \sqrt{-{a}^{2}{x}^{2}+2\,a{x}^{3}+{a}^{2}x-2\,a{x}^{2}-{x}^{3}+{x}^{2}}}\sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }}\sqrt{{(-1+x) \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}}\sqrt{{\frac{ \left ( -1+2\,a \right ) x}{{a}^{2}}}}{\it EllipticPi} \left ( \sqrt{-{\frac{-1+2\,a}{{a}^{2}} \left ( x-{\frac{{a}^{2}}{-1+2\,a}} \right ) }},{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}},\sqrt{{\frac{{a}^{2}}{-1+2\,a} \left ({\frac{{a}^{2}}{-1+2\,a}}-1 \right ) ^{-1}}} \right ) \left ({\frac{{a}^{2}}{-1+2\,a}}-a \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-a+(-1+2*a)*x)/(-a+x)/(a^2*x-(a^2+2*a-1)*x^2+(-1+2*a)*x^3)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (2 \, a - 1\right )} x - a}{\sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}{\left (a - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="maxima")
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Fricas [A] time = 0.25922, size = 85, normalized size = 1.85 \[ \log \left (-\frac{a^{2} - 2 \,{\left (a - 1\right )} x - x^{2} + 2 \, \sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}}{a^{2} - 2 \, a x + x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-a+(-1+2*a)*x)/(-a+x)/(a**2*x-(a**2+2*a-1)*x**2+(-1+2*a)*x**3)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (2 \, a - 1\right )} x - a}{\sqrt{{\left (2 \, a - 1\right )} x^{3} + a^{2} x -{\left (a^{2} + 2 \, a - 1\right )} x^{2}}{\left (a - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((2*a - 1)*x - a)/(sqrt((2*a - 1)*x^3 + a^2*x - (a^2 + 2*a - 1)*x^2)*(a - x)),x, algorithm="giac")
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