Optimal. Leaf size=249 \[ -\frac{923943703 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{691031250 \sqrt{33}}+\frac{2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac{8318 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac{25603 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1876875}-\frac{6794792 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{84459375}-\frac{923943703 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1520268750}-\frac{30660308017 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}} \]
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Rubi [A] time = 0.101815, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}}{3575}+\frac{8318 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{482625}+\frac{25603 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1876875}-\frac{6794792 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{84459375}-\frac{923943703 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1520268750}-\frac{923943703 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}}-\frac{30660308017 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx &=\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{2}{65} \int \left (-\frac{113}{2}-\frac{159 x}{2}\right ) (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx\\ &=\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{4 \int \left (-\frac{6063}{2}-\frac{12477 x}{4}\right ) \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx}{10725}\\ &=\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{8 \int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (-\frac{826005}{8}+\frac{691281 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{1447875}\\ &=\frac{25603 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1876875}+\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{8 \int \frac{(3+5 x)^{3/2} \left (\frac{83150493}{16}+7644141 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{50675625}\\ &=-\frac{6794792 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{84459375}+\frac{25603 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1876875}+\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (-\frac{5392906641}{16}-\frac{8315493327 x}{16}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{760134375}\\ &=-\frac{923943703 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1520268750}-\frac{6794792 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{84459375}+\frac{25603 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1876875}+\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{8 \int \frac{\frac{349425411903}{32}+\frac{275942772153 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6841209375}\\ &=-\frac{923943703 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1520268750}-\frac{6794792 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{84459375}+\frac{25603 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1876875}+\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac{923943703 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1382062500}+\frac{30660308017 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{7601343750}\\ &=-\frac{923943703 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1520268750}-\frac{6794792 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{84459375}+\frac{25603 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1876875}+\frac{8318 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{482625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}}{3575}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac{30660308017 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}}-\frac{923943703 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{691031250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.273851, size = 112, normalized size = 0.45 \[ \frac{-30830473835 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (14033250000 x^5+5400675000 x^4-13684072500 x^3-3707642250 x^2+5290733520 x+1020785999\right )+61320616034 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{22804031250 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{1368241875000\,{x}^{3}+1048985437500\,{x}^{2}-319256437500\,x-273648375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 12629925000000\,{x}^{8}+14543550000000\,{x}^{7}-11536182000000\,{x}^{6}+30830473835\,\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 ) -61320616034\,\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 ) -16439014800000\,{x}^{5}+4104920740500\,{x}^{4}+7811051450400\,{x}^{3}+260663905110\,{x}^{2}-1166697093390\,x-183741479820 \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{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{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 ({\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, 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{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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