Optimal. Leaf size=81 \[ \frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}} \]
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Rubi [A] time = 0.0222998, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {99, 21, 114, 113} \[ \frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 21
Rule 114
Rule 113
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{7 \sqrt{2+3 x}}+\frac{2}{7} \int \frac{\frac{5}{2}-5 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{7 \sqrt{2+3 x}}+\frac{5}{7} \int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{7 \sqrt{2+3 x}}+\frac{\left (5 \sqrt{-3-5 x}\right ) \int \frac{\sqrt{\frac{3}{7}-\frac{6 x}{7}}}{\sqrt{-9-15 x} \sqrt{2+3 x}} \, dx}{\sqrt{7} \sqrt{3+5 x}}\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{7 \sqrt{2+3 x}}+\frac{2 \sqrt{\frac{5}{7}} \sqrt{-3-5 x} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{2+3 x}\right )|\frac{2}{35}\right )}{3 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [C] time = 0.0835631, size = 70, normalized size = 0.86 \[ \frac{-6 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-2 i \sqrt{33} (3 x+2) E\left (i \sinh ^{-1}\left (\sqrt{15 x+9}\right )|-\frac{2}{33}\right )}{63 x+42} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 135, normalized size = 1.7 \begin{align*} -{\frac{1}{630\,{x}^{3}+483\,{x}^{2}-147\,x-126}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -2\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +60\,{x}^{2}+6\,x-18 \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}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{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}}{18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4}, 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}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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