Optimal. Leaf size=189 \[ \frac{38536 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{245 \sqrt{33}}-\frac{116464 \sqrt{1-2 x} \sqrt{3 x+2}}{147 \sqrt{5 x+3}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{416 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{116464}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0657802, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{116464 \sqrt{1-2 x} \sqrt{3 x+2}}{147 \sqrt{5 x+3}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{416 \sqrt{1-2 x}}{105 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{5 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{38536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}+\frac{116464}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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} (3+5 x)^{3/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2}{5} \int \frac{-18+25 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{416 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{4}{105} \int \frac{-\frac{2737}{2}+1560 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{416 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{8}{735} \int \frac{-\frac{116785}{2}+\frac{72255 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{416 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{116464 \sqrt{1-2 x} \sqrt{2+3 x}}{147 \sqrt{3+5 x}}+\frac{16 \int \frac{-\frac{3041445}{4}-1201035 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8085}\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{416 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{116464 \sqrt{1-2 x} \sqrt{2+3 x}}{147 \sqrt{3+5 x}}-\frac{19268}{245} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{116464}{245} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{5 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{416 \sqrt{1-2 x}}{105 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{19268 \sqrt{1-2 x}}{245 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{116464 \sqrt{1-2 x} \sqrt{2+3 x}}{147 \sqrt{3+5 x}}+\frac{116464}{245} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{38536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{245 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.228598, size = 105, normalized size = 0.56 \[ \frac{2}{735} \left (-2 \sqrt{2} \left (29116 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-14665 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (2620440 x^3+5154174 x^2+3376856 x+736871\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 314, normalized size = 1.7 \begin{align*} -{\frac{2}{7350\,{x}^{2}+735\,x-2205}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 263970\,\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}-524088\,\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}+351960\,\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}-698784\,\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}+117320\,\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 ) -232928\,\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 ) +15722640\,{x}^{4}+23063724\,{x}^{3}+4798614\,{x}^{2}-5709342\,x-2210613 \right ) \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}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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}}{2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144}, 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}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\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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