Optimal. Leaf size=249 \[ -\frac{70536439 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{13820625 \sqrt{33}}+\frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]
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Rubi [A] time = 0.102512, 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} \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]
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)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx &=\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2}{65} \int \left (-\frac{69}{2}-\frac{93 x}{2}\right ) \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-870-\frac{201 x}{4}\right ) \sqrt{2+3 x} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{10725}\\ &=-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{4 \int \frac{(3+5 x)^{5/2} \left (\frac{639867}{8}+120063 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{482625}\\ &=-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{62883465}{8}-\frac{96001155 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{10135125}\\ &=-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{8251543035}{16}+\frac{3174139755 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{152026875}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{-\frac{267229618515}{16}-\frac{422105672655 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1368241875}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{70536439 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{27641250}+\frac{9380126059 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{608107500}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.292554, size = 115, normalized size = 0.46 \[ \frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (944944217 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+2364390000 x^4+496455750 x^3-1110242250 x^2-638983395 x+67302101\right )\right )}{912161250 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{54729675000\,{x}^{3}+41959417500\,{x}^{2}-12770257500\,x-10945935000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1262992500000\,{x}^{8}-3096245250000\,{x}^{7}-1783541025000\,{x}^{6}+4724721085\,\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 ) -9380126059\,\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 ) +1405783957500\,{x}^{5}+1870998115500\,{x}^{4}+236537814150\,{x}^{3}-380468567640\,{x}^{2}-100883569890\,x+12114378180 \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{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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 (-{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\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{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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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