Optimal. Leaf size=30 \[ -\frac{1}{3} \sqrt{19} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|\frac{55}{76}\right ) \]
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Rubi [A] time = 0.02086, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {718, 424} \[ -\frac{1}{3} \sqrt{19} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|\frac{55}{76}\right ) \]
Antiderivative was successfully verified.
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Rule 718
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{\sqrt{2+5 x-12 x^2}} \, dx &=-\left (\frac{1}{3} \sqrt{19} \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{55 x^2}{76}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{16-24 x}}{\sqrt{22}}\right )\right )\\ &=-\frac{1}{3} \sqrt{19} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|\frac{55}{76}\right )\\ \end{align*}
Mathematica [B] time = 0.0766456, size = 86, normalized size = 2.87 \[ \frac{\sqrt{19} \sqrt{-4 x-1} \sqrt{2-3 x} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right ),\frac{21}{76}\right )-E\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right )|\frac{21}{76}\right )\right )}{3 \sqrt{-12 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.119, size = 77, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{57}}{6840\,{x}^{2}-2850\,x-1140} \left ({\it EllipticF} \left ({\frac{1}{19}\sqrt{171+285\,x}},{\frac{2\,\sqrt{399}}{21}} \right ) -{\it EllipticE} \left ({\frac{1}{19}\sqrt{171+285\,x}},{\frac{2\,\sqrt{399}}{21}} \right ) \right ) \sqrt{190-285\,x}\sqrt{-140\,x-35}\sqrt{-12\,{x}^{2}+5\,x+2}} \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{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 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{-12 \, x^{2} + 5 \, x + 2} \sqrt{5 \, x + 3}}{12 \, x^{2} - 5 \, x - 2}, 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{5 x + 3}}{\sqrt{- \left (3 x - 2\right ) \left (4 x + 1\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{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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