Optimal. Leaf size=249 \[ -\frac{69808931 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{76781250 \sqrt{33}}+\frac{2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2145}+\frac{34 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2475}+\frac{32717 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1126125}-\frac{445024 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{9384375}-\frac{69808931 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{168918750}-\frac{1163388067 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{38390625 \sqrt{33}} \]
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Rubi [A] time = 0.0997942, 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} \[ \frac{2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2145}+\frac{34 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}}{2475}+\frac{32717 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1126125}-\frac{445024 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{9384375}-\frac{69808931 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{168918750}-\frac{69808931 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{76781250 \sqrt{33}}-\frac{1163388067 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{38390625 \sqrt{33}} \]
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} \sqrt{3+5 x} \, dx &=\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{2}{65} \int \left (-\frac{115}{2}-\frac{155 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{4 \int \left (-\frac{6465}{2}-\frac{9945 x}{4}\right ) \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx}{10725}\\ &=\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{8 \int \frac{(2+3 x)^{3/2} \sqrt{3+5 x} \left (-\frac{1119015}{8}+\frac{1472265 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{1447875}\\ &=\frac{32717 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{8 \int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{90410175}{16}+7509780 x\right )}{\sqrt{1-2 x}} \, dx}{50675625}\\ &=-\frac{445024 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{9384375}+\frac{32717 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{8 \int \frac{\left (-\frac{3071310615}{8}-\frac{9424205685 x}{16}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1266890625}\\ &=-\frac{69808931 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{168918750}-\frac{445024 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{9384375}+\frac{32717 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{8 \int \frac{\frac{397670986215}{32}+\frac{157057389045 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11402015625}\\ &=-\frac{69808931 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{168918750}-\frac{445024 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{9384375}+\frac{32717 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{69808931 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{153562500}+\frac{1163388067 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{422296875}\\ &=-\frac{69808931 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{168918750}-\frac{445024 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{9384375}+\frac{32717 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac{34 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac{2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{1163388067 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{38390625 \sqrt{33}}-\frac{69808931 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{76781250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.288223, size = 112, normalized size = 0.45 \[ \frac{-2349857545 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (935550000 x^5+433755000 x^4-936022500 x^3-309143250 x^2+380959290 x+84411073\right )+4653552268 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2533781250 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{152026875000\,{x}^{3}+116553937500\,{x}^{2}-35472937500\,x-30405375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 841995000000\,{x}^{8}+1035909000000\,{x}^{7}-739594800000\,{x}^{6}+2349857545\,\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 ) -4653552268\,\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 ) -1183572000000\,{x}^{5}+248043343500\,{x}^{4}+572236008300\,{x}^{3}+33887974470\,{x}^{2}-86298997530\,x-15193993140 \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 \sqrt{5 \, x + 3}{\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 (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 \sqrt{5 \, x + 3}{\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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