Optimal. Leaf size=249 \[ -\frac{486785077 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{248771250 \sqrt{33}}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}}{10725}+\frac{2314 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{111375}-\frac{121031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{30405375}-\frac{3872003 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{30405375}-\frac{486785077 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{547296750}-\frac{8120161139 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{124385625 \sqrt{33}} \]
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Rubi [A] time = 0.100282, 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} \[ \frac{2}{65} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{7/2}+\frac{326 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{7/2}}{10725}+\frac{2314 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{111375}-\frac{121031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{30405375}-\frac{3872003 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{30405375}-\frac{486785077 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{547296750}-\frac{486785077 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{248771250 \sqrt{33}}-\frac{8120161139 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{124385625 \sqrt{33}} \]
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} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx &=\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{2}{65} \int \frac{\left (-\frac{111}{2}-\frac{163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{5653}{2}-\frac{15041 x}{4}\right ) \sqrt{1-2 x} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx}{10725}\\ &=\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{8 \int \frac{\left (-\frac{518563}{8}-\frac{121031 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1447875}\\ &=-\frac{121031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{30405375}+\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{8 \int \frac{(3+5 x)^{3/2} \left (\frac{71027395}{16}+\frac{58080045 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{30405375}\\ &=-\frac{3872003 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{30405375}-\frac{121031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{30405375}+\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{8 \int \frac{\left (-\frac{2382196995}{8}-\frac{7301776155 x}{16}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{456080625}\\ &=-\frac{486785077 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{547296750}-\frac{3872003 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{30405375}-\frac{121031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{30405375}+\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{8 \int \frac{\frac{308389708545}{32}+\frac{121802417085 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4104725625}\\ &=-\frac{486785077 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{547296750}-\frac{3872003 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{30405375}-\frac{121031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{30405375}+\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}+\frac{486785077 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{497542500}+\frac{8120161139 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1368241875}\\ &=-\frac{486785077 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{547296750}-\frac{3872003 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{30405375}-\frac{121031 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{30405375}+\frac{2314 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{111375}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{7/2}}{10725}+\frac{2}{65} (1-2 x)^{5/2} \sqrt{2+3 x} (3+5 x)^{7/2}-\frac{8120161139 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{124385625 \sqrt{33}}-\frac{486785077 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{248771250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.286436, size = 112, normalized size = 0.45 \[ \frac{-16416737015 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (8419950000 x^5+2577015000 x^4-7942630500 x^3-1730459250 x^2+2923422930 x+495379991\right )+32480644556 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{8209451250 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.009, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{492567075000\,{x}^{3}+377634757500\,{x}^{2}-114932317500\,x-98513415000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 7577955000000\,{x}^{8}+8129079000000\,{x}^{7}-7138416600000\,{x}^{6}+16416737015\,\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 ) -32480644556\,\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 ) -9094592520000\,{x}^{5}+2641153459500\,{x}^{4}+4256073746100\,{x}^{3}+39376043490\,{x}^{2}-630245925510\,x-89168398380 \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}} \sqrt{3 \, x + 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 (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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}} \sqrt{3 \, x + 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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