Optimal. Leaf size=62 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\sqrt{\frac{5}{7}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]
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Rubi [A] time = 0.0151155, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {99, 21, 113} \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\sqrt{\frac{5}{7}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 21
Rule 113
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{2}{7} \int \frac{5+\frac{15 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{5}{7} \int \frac{\sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}+\sqrt{\frac{5}{7}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right )\\ \end{align*}
Mathematica [A] time = 0.0532824, size = 63, normalized size = 1.02 \[ \frac{1}{7} \left (\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}-\sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 92, normalized size = 1.5 \begin{align*}{\frac{1}{210\,{x}^{3}+161\,{x}^{2}-49\,x-42}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ({\frac{1}{11}\sqrt{66+110\,x}},{\frac{i}{2}}\sqrt{66} \right ) -30\,{x}^{2}-38\,x-12 \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{5 \, x + 3}}{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\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{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{12 \, x^{3} - 4 \, 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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