Optimal. Leaf size=129 \[ -\frac{8}{147} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{62 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{3/2}}-\frac{62}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0383998, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {99, 152, 158, 113, 119} \[ \frac{62 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{3/2}}-\frac{8}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{62}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{3/2}}+\frac{2}{21} \int \frac{\frac{17}{2}+5 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{3/2}}+\frac{62 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}+\frac{4}{147} \int \frac{\frac{115}{2}+\frac{155 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{3/2}}+\frac{62 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}+\frac{44}{147} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{62}{147} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{3/2}}+\frac{62 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}-\frac{62}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{8}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.132373, size = 97, normalized size = 0.75 \[ \frac{2}{441} \left (35 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (93 x+55)}{(3 x+2)^{3/2}}+31 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 219, normalized size = 1.7 \begin{align*} -{\frac{2}{4410\,{x}^{2}+441\,x-1323} \left ( 105\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+93\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+70\,\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 ) +62\,\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 ) -2790\,{x}^{3}-1929\,{x}^{2}+672\,x+495 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{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}}{{\left (3 \, x + 2\right )}^{\frac{5}{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}}{54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8}, 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{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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