Optimal. Leaf size=51 \[ \frac{2 \sqrt{\frac{e x}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right ),-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]
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Rubi [A] time = 0.0253953, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {714, 12, 117, 115} \[ \frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 714
Rule 12
Rule 117
Rule 115
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \sqrt{2 x-3 x^2}} \, dx &=\int \frac{1}{\sqrt{2} \sqrt{1-\frac{3 x}{2}} \sqrt{x} \sqrt{d+e x}} \, dx\\ &=\frac{\int \frac{1}{\sqrt{1-\frac{3 x}{2}} \sqrt{x} \sqrt{d+e x}} \, dx}{\sqrt{2}}\\ &=\frac{\sqrt{1+\frac{e x}{d}} \int \frac{1}{\sqrt{1-\frac{3 x}{2}} \sqrt{x} \sqrt{1+\frac{e x}{d}}} \, dx}{\sqrt{2} \sqrt{d+e x}}\\ &=\frac{2 \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.120767, size = 76, normalized size = 1.49 \[ -\frac{\sqrt{6-\frac{4}{x}} x^{3/2} \sqrt{\frac{d}{e x}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),-\frac{3 d}{2 e}\right )}{\sqrt{-x (3 x-2)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 115, normalized size = 2.3 \begin{align*} -2\,{\frac{d\sqrt{ex+d}\sqrt{-x \left ( -2+3\,x \right ) }}{ex \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) \sqrt{-{\frac{ex}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{{\frac{ex+d}{d}}}} \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{e x + d} \sqrt{-3 \, x^{2} + 2 \, x}}\,{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{e x + d} \sqrt{-3 \, x^{2} + 2 \, x}}{3 \, e x^{3} +{\left (3 \, d - 2 \, e\right )} x^{2} - 2 \, d x}, 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{- x \left (3 x - 2\right )} \sqrt{d + e x}}\, 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{e x + d} \sqrt{-3 \, x^{2} + 2 \, x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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