Optimal. Leaf size=222 \[ \frac{1400888 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{132055}-\frac{46585232 \sqrt{1-2 x} \sqrt{3 x+2}}{290521 \sqrt{5 x+3}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14928 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{46585232 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055} \]
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Rubi [A] time = 0.0818852, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ -\frac{46585232 \sqrt{1-2 x} \sqrt{3 x+2}}{290521 \sqrt{5 x+3}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14928 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1400888 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055}+\frac{46585232 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{2}{77} \int \frac{-\frac{163}{2}-105 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{4 \int \frac{-1291+\frac{1725 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{2695}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{14928 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{301413}{4}+83970 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{56595}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{14928 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{16 \int \frac{-\frac{12741465}{4}+\frac{7879995 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{396165}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{14928 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{46585232 \sqrt{1-2 x} \sqrt{2+3 x}}{290521 \sqrt{3+5 x}}+\frac{32 \int \frac{-\frac{331786305}{8}-\frac{131020965 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4357815}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{14928 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{46585232 \sqrt{1-2 x} \sqrt{2+3 x}}{290521 \sqrt{3+5 x}}-\frac{2101332 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{132055}-\frac{139755696 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1452605}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{14928 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2101332 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{46585232 \sqrt{1-2 x} \sqrt{2+3 x}}{290521 \sqrt{3+5 x}}+\frac{46585232 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055}+\frac{1400888 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{132055}\\ \end{align*}
Mathematica [A] time = 0.165434, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{6289006320 x^4+9225477612 x^3+1919527182 x^2-2283681406 x-884250959}{\sqrt{1-2 x} (3 x+2)^{5/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (11646308 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5867645 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )\right )}{1452605} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 314, normalized size = 1.4 \begin{align*} -{\frac{2}{14526050\,{x}^{2}+1452605\,x-4357815}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 105617610\,\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}-209633544\,\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}+140823480\,\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}-279511392\,\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}+46941160\,\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 ) -93170464\,\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 ) +6289006320\,{x}^{4}+9225477612\,{x}^{3}+1919527182\,{x}^{2}-2283681406\,x-884250959 \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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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}}{8100 \, x^{8} + 23220 \, x^{7} + 21141 \, x^{6} + 690 \, x^{5} - 9791 \, x^{4} - 4696 \, x^{3} + 424 \, x^{2} + 768 \, 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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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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