Optimal. Leaf size=191 \[ -\frac{21160 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1323}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac{703480 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{10124 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}+\frac{76 \sqrt{5 x+3} \sqrt{1-2 x}}{9 (3 x+2)^{5/2}}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
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Rubi [A] time = 0.0665178, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac{703480 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{10124 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}+\frac{76 \sqrt{5 x+3} \sqrt{1-2 x}}{9 (3 x+2)^{5/2}}-\frac{21160 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{(161-91 x) \sqrt{1-2 x}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{-\frac{16835}{2}+9520 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{730625}{2}+\frac{442925 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{6615}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{19484675}{4}-\frac{15388625 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{46305}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{116380 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1323}+\frac{703480 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1323}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{21160 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}\\ \end{align*}
Mathematica [A] time = 0.232558, size = 107, normalized size = 0.56 \[ \frac{4 \left (5 \sqrt{2} \left (35174 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17717 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (9496980 x^3+19312866 x^2+13103724 x+2967269\right )}{2 (3 x+2)^{7/2}}\right )}{3969} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{39690\,{x}^{2}+3969\,x-11907} \left ( 4783590\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9496980\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9567180\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-18993960\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6378120\,\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}-12662640\,\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}+1417360\,\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 ) -2813920\,\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 ) +284909400\,{x}^{5}+607876920\,{x}^{4}+365577498\,{x}^{3}-45486552\,{x}^{2}-109031709\,x-26705421 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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