Optimal. Leaf size=128 \[ -\frac{2 \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}}-\frac{4 \sqrt{x^2-3 x+1}}{5 \sqrt{3-2 x}}+\frac{2 \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}} \]
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Rubi [A] time = 0.0681879, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {693, 691, 690, 307, 221, 1181, 21, 424} \[ -\frac{4 \sqrt{x^2-3 x+1}}{5 \sqrt{3-2 x}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}}+\frac{2 \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1181
Rule 21
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(3-2 x)^{3/2} \sqrt{1-3 x+x^2}} \, dx &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}-\frac{1}{5} \int \frac{\sqrt{3-2 x}}{\sqrt{1-3 x+x^2}} \, dx\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}-\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}{5 \sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}+\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 )}{5 \sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}-\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 )}{5 \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 )}{5 \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}-\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 )}{5^{3/4} \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 )}{5 \sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}-\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 )}{5^{3/4} \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 )}{5 \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{5 \sqrt{3-2 x}}+\frac{2 \sqrt{-1+3 x-x^2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \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 )}{5^{3/4} \sqrt{1-3 x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0129933, size = 63, normalized size = 0.49 \[ \frac{2 \sqrt{-x^2+3 x-1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{1}{5} (3-2 x)^2\right )}{\sqrt{5} \sqrt{3-2 x} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 116, normalized size = 0.9 \begin{align*}{\frac{1}{50\,{x}^{3}-225\,{x}^{2}+275\,x-75}\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 EllipticE} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) \sqrt{5}+20\,{x}^{2}-60\,x+20 \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}{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}\,{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}}{4 \, x^{4} - 24 \, x^{3} + 49 \, x^{2} - 39 \, x + 9}, 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}{\left (3 - 2 x\right )^{\frac{3}{2}} \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}{\left (-2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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