Optimal. Leaf size=51 \[ -\frac{2 \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{\sqrt [4]{5} \sqrt{x^2-3 x+1}} \]
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Rubi [A] time = 0.0240225, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {691, 689, 221} \[ -\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt [4]{5} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-2 x} \sqrt{1-3 x+x^2}} \, dx &=\frac{\sqrt{-1+3 x-x^2} \int \frac{1}{\sqrt{3-2 x} \sqrt{-\frac{1}{5}+\frac{3 x}{5}-\frac{x^2}{5}}} \, dx}{\sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{\left (2 \sqrt{-1+3 x-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{5}}} \, dx,x,\sqrt{3-2 x}\right )}{\sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{2 \sqrt{-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt [4]{5} \sqrt{1-3 x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0138915, size = 63, normalized size = 1.24 \[ -\frac{2 \sqrt{3-2 x} \sqrt{-x^2+3 x-1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{5} (3-2 x)^2\right )}{\sqrt{5} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 102, normalized size = 2. \begin{align*}{\frac{1}{10\,{x}^{3}-45\,{x}^{2}+55\,x-15}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}{2 \, x^{3} - 9 \, x^{2} + 11 \, x - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 - 2 x} \sqrt{x^{2} - 3 x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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