Optimal. Leaf size=81 \[ \frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{77 \sqrt{1-2 x}}+\frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{11 \sqrt{5 x+3}} \]
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Rubi [A] time = 0.0232588, 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 = {104, 21, 114, 113} \[ \frac{4 \sqrt{3 x+2} \sqrt{5 x+3}}{77 \sqrt{1-2 x}}+\frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{11 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 21
Rule 114
Rule 113
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{77 \sqrt{1-2 x}}-\frac{2}{77} \int \frac{-\frac{15}{2}+15 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{77 \sqrt{1-2 x}}+\frac{15}{77} \int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{77 \sqrt{1-2 x}}+\frac{\left (15 \sqrt{-3-5 x}\right ) \int \frac{\sqrt{\frac{3}{7}-\frac{6 x}{7}}}{\sqrt{-9-15 x} \sqrt{2+3 x}} \, dx}{11 \sqrt{7} \sqrt{3+5 x}}\\ &=\frac{4 \sqrt{2+3 x} \sqrt{3+5 x}}{77 \sqrt{1-2 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 )}{11 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [C] time = 0.0687077, size = 61, normalized size = 0.75 \[ \frac{2}{77} \left (\frac{2 \sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}-i \sqrt{33} E\left (i \sinh ^{-1}\left (\sqrt{15 x+9}\right )|-\frac{2}{33}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 135, normalized size = 1.7 \begin{align*} -{\frac{1}{2310\,{x}^{3}+1771\,{x}^{2}-539\,x-462}\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}+76\,x+24 \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{1}{\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}}{60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6}, 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}{\left (1 - 2 x\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}\, 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{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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