Optimal. Leaf size=79 \[ -\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{3 \sqrt{x^2-3 x+1}}-\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1} \]
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Rubi [A] time = 0.0391474, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {692, 691, 689, 221} \[ -\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Rule 692
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{(3-2 x)^{3/2}}{\sqrt{1-3 x+x^2}} \, dx &=-\frac{4}{3} \sqrt{3-2 x} \sqrt{1-3 x+x^2}+\frac{5}{3} \int \frac{1}{\sqrt{3-2 x} \sqrt{1-3 x+x^2}} \, dx\\ &=-\frac{4}{3} \sqrt{3-2 x} \sqrt{1-3 x+x^2}+\frac{\left (\sqrt{5} \sqrt{-1+3 x-x^2}\right ) \int \frac{1}{\sqrt{3-2 x} \sqrt{-\frac{1}{5}+\frac{3 x}{5}-\frac{x^2}{5}}} \, dx}{3 \sqrt{1-3 x+x^2}}\\ &=-\frac{4}{3} \sqrt{3-2 x} \sqrt{1-3 x+x^2}-\frac{\left (2 \sqrt{5} \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 )}{3 \sqrt{1-3 x+x^2}}\\ &=-\frac{4}{3} \sqrt{3-2 x} \sqrt{1-3 x+x^2}-\frac{2\ 5^{3/4} \sqrt{-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt{1-3 x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0276804, size = 76, normalized size = 0.96 \[ -\frac{2 \sqrt{3-2 x} \left (\sqrt{5} \sqrt{-x^2+3 x-1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{5} (3-2 x)^2\right )+2 x^2-6 x+2\right )}{3 \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 118, normalized size = 1.5 \begin{align*}{\frac{1}{6\,{x}^{3}-27\,{x}^{2}+33\,x-9}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1} \left ( \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 ) -8\,{x}^{3}+36\,{x}^{2}-44\,x+12 \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{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\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{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\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 = 10.5448, size = 41, normalized size = 0.52 \begin{align*} \frac{\sqrt{5} i \left (3 - 2 x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{\left (3 - 2 x\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{9}{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{{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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