Optimal. Leaf size=158 \[ -\frac{64}{245} \sqrt{33} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{6388 \sqrt{1-2 x} \sqrt{5 x+3}}{245 \sqrt{3 x+2}}+\frac{92 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{5 (3 x+2)^{5/2}}-\frac{6388}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0532224, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{6388 \sqrt{1-2 x} \sqrt{5 x+3}}{245 \sqrt{3 x+2}}+\frac{92 \sqrt{1-2 x} \sqrt{5 x+3}}{35 (3 x+2)^{3/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{5 (3 x+2)^{5/2}}-\frac{64}{245} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{6388}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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{1-2 x}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{5 (2+3 x)^{5/2}}-\frac{2}{5} \int \frac{-13+15 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{5 (2+3 x)^{5/2}}+\frac{92 \sqrt{1-2 x} \sqrt{3+5 x}}{35 (2+3 x)^{3/2}}-\frac{4}{105} \int \frac{-\frac{1137}{2}+345 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}}{5 (2+3 x)^{5/2}}+\frac{92 \sqrt{1-2 x} \sqrt{3+5 x}}{35 (2+3 x)^{3/2}}+\frac{6388 \sqrt{1-2 x} \sqrt{3+5 x}}{245 \sqrt{2+3 x}}-\frac{8}{735} \int \frac{-\frac{15165}{2}-\frac{23955 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}}{5 (2+3 x)^{5/2}}+\frac{92 \sqrt{1-2 x} \sqrt{3+5 x}}{35 (2+3 x)^{3/2}}+\frac{6388 \sqrt{1-2 x} \sqrt{3+5 x}}{245 \sqrt{2+3 x}}+\frac{1056}{245} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{6388}{245} \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}}{5 (2+3 x)^{5/2}}+\frac{92 \sqrt{1-2 x} \sqrt{3+5 x}}{35 (2+3 x)^{3/2}}+\frac{6388 \sqrt{1-2 x} \sqrt{3+5 x}}{245 \sqrt{2+3 x}}-\frac{6388}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{64}{245} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.219889, size = 101, normalized size = 0.64 \[ \frac{4}{735} \left (\sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (28746 x^2+39294 x+13469\right )}{2 (3 x+2)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 314, normalized size = 2. \begin{align*}{\frac{2}{7350\,{x}^{2}+735\,x-2205} \left ( 14490\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-28746\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+19320\,\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}-38328\,\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}+6440\,\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 ) -12776\,\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 ) +862380\,{x}^{4}+1265058\,{x}^{3}+263238\,{x}^{2}-313239\,x-121221 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{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{-2 \, x + 1}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{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}}{405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48}, 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{-2 \, x + 1}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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