Optimal. Leaf size=103 \[ \frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{\sqrt{x^2-3 x+1}}-\frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]
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Rubi [A] time = 0.0523656, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {691, 690, 307, 221, 1181, 21, 424} \[ \frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}}-\frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Rule 691
Rule 690
Rule 307
Rule 221
Rule 1181
Rule 21
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{3-2 x}}{\sqrt{1-3 x+x^2}} \, dx &=\frac{\sqrt{-1+3 x-x^2} \int \frac{\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{x^2}{\sqrt{1-\frac{x^4}{5}}} \, dx,x,\sqrt{3-2 x}\right )}{\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{1-3 x+x^2}}-\frac{\left (2 \sqrt{-1+3 x-x^2}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{5}}}{\sqrt{1-\frac{x^4}{5}}} \, dx,x,\sqrt{3-2 x}\right )}{\sqrt{1-3 x+x^2}}\\ &=\frac{2 \sqrt [4]{5} \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{1-3 x+x^2}}-\frac{\left (2 \sqrt{-1+3 x-x^2}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{5}}}{\sqrt{\frac{1}{\sqrt{5}}-\frac{x^2}{5}} \sqrt{\frac{1}{\sqrt{5}}+\frac{x^2}{5}}} \, dx,x,\sqrt{3-2 x}\right )}{\sqrt{5} \sqrt{1-3 x+x^2}}\\ &=\frac{2 \sqrt [4]{5} \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{1-3 x+x^2}}-\frac{\left (2 \sqrt{-1+3 x-x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{1}{\sqrt{5}}+\frac{x^2}{5}}}{\sqrt{\frac{1}{\sqrt{5}}-\frac{x^2}{5}}} \, dx,x,\sqrt{3-2 x}\right )}{\sqrt{1-3 x+x^2}}\\ &=-\frac{2 \sqrt [4]{5} \sqrt{-1+3 x-x^2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{1-3 x+x^2}}+\frac{2 \sqrt [4]{5} \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{1-3 x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0133973, size = 65, normalized size = 0.63 \[ -\frac{2 (3-2 x)^{3/2} \sqrt{-x^2+3 x-1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{1}{5} (3-2 x)^2\right )}{3 \sqrt{5} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 105, normalized size = 1. \begin{align*} -{\frac{\sqrt{5}}{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 EllipticE} \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{\sqrt{-2 \, x + 3}}{\sqrt{x^{2} - 3 \, x + 1}}\,{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{-2 \, x + 3}}{\sqrt{x^{2} - 3 \, x + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.5848, size = 41, normalized size = 0.4 \begin{align*} \frac{\sqrt{5} i \left (3 - 2 x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{\left (3 - 2 x\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{7}{4}\right )} \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{\sqrt{-2 \, x + 3}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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