Optimal. Leaf size=218 \[ -\frac{31288 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]
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Rubi [A] time = 0.0827641, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (\frac{23}{2}-30 x\right ) (2+3 x)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}+\frac{4}{825} \int \frac{\left (\frac{4875}{4}-\frac{7185 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}+\frac{958 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{1925}-\frac{4 \int \frac{(2+3 x)^{3/2} \left (-\frac{32295}{4}+\frac{77295 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{28875}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}+\frac{5153 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{48125}+\frac{958 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{1925}+\frac{4 \int \frac{\sqrt{2+3 x} \left (\frac{1297125}{8}+\frac{567045 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{721875}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}-\frac{12601 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{240625}+\frac{5153 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{48125}+\frac{958 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{1925}-\frac{4 \int \frac{-\frac{42883065}{8}-\frac{66309255 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10828125}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}-\frac{12601 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{240625}+\frac{5153 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{48125}+\frac{958 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{1925}+\frac{15644 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{109375}+\frac{1473539 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2406250}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (2+3 x)^{7/2}}{165 \sqrt{3+5 x}}-\frac{12601 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{240625}+\frac{5153 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{48125}+\frac{958 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{1925}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.315991, size = 112, normalized size = 0.51 \[ \frac{-441035 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (3341250 x^4+8575875 x^3+6882975 x^2+1854575 x+54083\right )}{(5 x+3)^{3/2}}+1473539 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{7218750} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.031, size = 234, normalized size = 1.1 \begin{align*}{\frac{1}{43312500\,{x}^{2}+7218750\,x-14437500} \left ( 2205175\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7367695\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+200475000\,{x}^{6}+1323105\,\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 ) -4420617\,\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 ) +547965000\,{x}^{5}+431912250\,{x}^{4}+8586750\,{x}^{3}-115868770\,{x}^{2}-36550670\,x-1081660 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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 (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\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 (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, 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 (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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