Optimal. Leaf size=51 \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]
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Rubi [A] time = 0.0261794, 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, 112, 110} \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 714
Rule 12
Rule 112
Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{2 x-3 x^2}} \, dx &=\int \frac{\sqrt{d+e x}}{\sqrt{2} \sqrt{1-\frac{3 x}{2}} \sqrt{x}} \, dx\\ &=\frac{\int \frac{\sqrt{d+e x}}{\sqrt{1-\frac{3 x}{2}} \sqrt{x}} \, dx}{\sqrt{2}}\\ &=\frac{\sqrt{d+e x} \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{1-\frac{3 x}{2}} \sqrt{x}} \, dx}{\sqrt{2} \sqrt{1+\frac{e x}{d}}}\\ &=\frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{1+\frac{e x}{d}}}\\ \end{align*}
Mathematica [B] time = 0.314161, size = 117, normalized size = 2.29 \[ \frac{2 (3 x-2) \sqrt{-\frac{d}{e}} (d+e x)-2 d \sqrt{9-\frac{6}{x}} x^{3/2} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|-\frac{2 e}{3 d}\right )}{3 \sqrt{-x (3 x-2)} \sqrt{-\frac{d}{e}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.2, size = 215, normalized size = 4.2 \begin{align*} -{\frac{2\,d}{3\,ex \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }\sqrt{ex+d}\sqrt{-x \left ( -2+3\,x \right ) }\sqrt{{\frac{ex+d}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{-{\frac{ex}{d}}} \left ( 3\,d{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) +2\,{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e-3\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) d-2\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e \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{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 \, x^{2} - 2 \, 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{\sqrt{d + e x}}{\sqrt{- x \left (3 x - 2\right )}}\, 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{\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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