Optimal. Leaf size=79 \[ -\frac{2 \sqrt{-x^2+3 x-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{15 \sqrt [4]{5} \sqrt{x^2-3 x+1}}-\frac{4 \sqrt{x^2-3 x+1}}{15 (3-2 x)^{3/2}} \]
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Rubi [A] time = 0.0344653, 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 = {693, 691, 689, 221} \[ -\frac{4 \sqrt{x^2-3 x+1}}{15 (3-2 x)^{3/2}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{15 \sqrt [4]{5} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(3-2 x)^{5/2} \sqrt{1-3 x+x^2}} \, dx &=-\frac{4 \sqrt{1-3 x+x^2}}{15 (3-2 x)^{3/2}}+\frac{1}{15} \int \frac{1}{\sqrt{3-2 x} \sqrt{1-3 x+x^2}} \, dx\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{15 (3-2 x)^{3/2}}+\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}{15 \sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{15 (3-2 x)^{3/2}}-\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 )}{15 \sqrt{5} \sqrt{1-3 x+x^2}}\\ &=-\frac{4 \sqrt{1-3 x+x^2}}{15 (3-2 x)^{3/2}}-\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 )}{15 \sqrt [4]{5} \sqrt{1-3 x+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0147454, size = 65, normalized size = 0.82 \[ \frac{2 \sqrt{-x^2+3 x-1} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{1}{5} (3-2 x)^2\right )}{3 \sqrt{5} (3-2 x)^{3/2} \sqrt{x^2-3 x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.206, size = 172, normalized size = 2.2 \begin{align*}{\frac{1}{75\, \left ( -3+2\,x \right ) ^{2}} \left ( 2\,\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 ( 1/10\,\sqrt{2}\sqrt{5}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}},\sqrt{2} \right ) x-3\,\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 ( 1/10\,\sqrt{2}\sqrt{5}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}},\sqrt{2} \right ) -20\,{x}^{2}+60\,x-20 \right ) \sqrt{3-2\,x}{\frac{1}{\sqrt{{x}^{2}-3\,x+1}}}} \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{5}{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}}{8 \, x^{5} - 60 \, x^{4} + 170 \, x^{3} - 225 \, x^{2} + 135 \, x - 27}, 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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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