Optimal. Leaf size=280 \[ -\frac{13267820528 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5182734375 \sqrt{33}}+\frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}}{2925}+\frac{3698 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{482625}+\frac{142391 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{7239375}-\frac{569519 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{28153125}-\frac{400516993 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2533781250}-\frac{13267820528 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{11402015625}-\frac{1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20730937500 \sqrt{33}} \]
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Rubi [A] time = 0.117828, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}}{2925}+\frac{3698 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{482625}+\frac{142391 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{7239375}-\frac{569519 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{28153125}-\frac{400516993 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2533781250}-\frac{13267820528 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{11402015625}-\frac{13267820528 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5182734375 \sqrt{33}}-\frac{1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20730937500 \sqrt{33}} \]
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)^{5/2} (3+5 x)^{3/2} \, dx &=\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{2}{75} \int \left (-\frac{115}{2}-\frac{155 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{4 \int \left (-3320-\frac{9245 x}{4}\right ) \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx}{14625}\\ &=\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{8 \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2} \left (-\frac{1423865}{8}+\frac{2135865 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{2413125}\\ &=\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{8 \int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{117464475}{16}+\frac{76885065 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{108590625}\\ &=-\frac{569519 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{28153125}+\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (-\frac{11836111305}{16}-\frac{18023264685 x}{16}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3800671875}\\ &=-\frac{400516993 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac{569519 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{28153125}+\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{1551878163945}{32}+74631490470 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{57010078125}\\ &=-\frac{13267820528 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{11402015625}-\frac{400516993 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac{569519 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{28153125}+\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{8 \int \frac{-\frac{50259437359155}{32}-\frac{79387348157685 x}{32}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{513090703125}\\ &=-\frac{13267820528 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{11402015625}-\frac{400516993 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac{569519 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{28153125}+\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac{6633910264 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5182734375}+\frac{1764163292393 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{228040312500}\\ &=-\frac{13267820528 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{11402015625}-\frac{400516993 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac{569519 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{28153125}+\frac{142391 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac{3698 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20730937500 \sqrt{33}}-\frac{13267820528 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5182734375 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.245946, size = 119, normalized size = 0.42 \[ \frac{\sqrt{2} \left (1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-888487137545 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+621672975000 x^5-336683182500 x^4-528977216250 x^3+48836706750 x^2+173484591165 x+12155574323\right )}{684120937500} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 170, normalized size = 0.6 \begin{align*}{\frac{1}{20523628125000\,{x}^{3}+15734781562500\,{x}^{2}-4788846562500\,x-4104725625000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 492567075000000\,{x}^{9}+937140435000000\,{x}^{8}+11007171000000\,{x}^{7}-937455630300000\,{x}^{6}+888487137545\,\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 ) -1764163292393\,\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 ) -362238910312500\,{x}^{5}+361521647968500\,{x}^{4}+215604575302050\,{x}^{3}-36835025076780\,{x}^{2}-33779897017530\,x-2188003378140 \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{5}{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 (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\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{5}{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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