Optimal. Leaf size=249 \[ -\frac{2295970088 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]
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Rubi [A] time = 0.100438, 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} \[ -\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{1}{13} \int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac{257}{2}+205 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{1}{715} \int \frac{\left (-\frac{45285}{2}-\frac{71515 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{12799775}{4}+\frac{9975285 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{32175}\\ &=-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{\int \frac{\left (-\frac{1364822645}{4}-\frac{2082147465 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1126125}\\ &=-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{179052019605}{8}+34439551320 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{16891875}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{\int \frac{-\frac{5798711966295}{8}-\frac{9159406524465 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{152026875}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{1147985044 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4606875}+\frac{610627101631 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{405405000}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.28098, size = 115, normalized size = 0.46 \[ \frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (61511810003 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2104987500 x^5+9351247500 x^4+18620894250 x^3+22592085750 x^2+19961825445 x+16001700059\right )\right )}{608107500 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{36486450000\,{x}^{3}+27972945000\,{x}^{2}-8513505000\,x-7297290000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1894488750000\,{x}^{8}-9868564125000\,{x}^{7}-22769118225000\,{x}^{6}+307559050015\,\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 ) -610627101631\,\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 ) -30838634482500\,{x}^{5}-27960569725500\,{x}^{4}-20079090637650\,{x}^{3}-2782614262260\,{x}^{2}+6953485592490\,x+2880306010620 \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 \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{\sqrt{-2 \, x + 1}}\,{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 (-\frac{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{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 \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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