Optimal. Leaf size=253 \[ -\frac{1199452 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{22235661}+\frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{9/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{5 x+3}}{22235661 \sqrt{3 x+2}}-\frac{392998 \sqrt{1-2 x} \sqrt{5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3}}{453789 (3 x+2)^{5/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{5 x+3}}{64827 (3 x+2)^{7/2}}+\frac{295 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 (3 x+2)^{9/2}}-\frac{6036028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661} \]
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Rubi [A] time = 0.0973988, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{9/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{5 x+3}}{22235661 \sqrt{3 x+2}}-\frac{392998 \sqrt{1-2 x} \sqrt{5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3}}{453789 (3 x+2)^{5/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{5 x+3}}{64827 (3 x+2)^{7/2}}+\frac{295 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 (3 x+2)^{9/2}}-\frac{1199452 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661}-\frac{6036028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661} \]
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{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{1}{7} \int \frac{\left (-\frac{555}{2}-490 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{2 \int \frac{-\frac{169585}{4}-\frac{144025 x}{2}}{\sqrt{1-2 x} (2+3 x)^{9/2} \sqrt{3+5 x}} \, dx}{1323}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{4 \int \frac{-245765-\frac{1683625 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{64827}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{3+5 x}}{453789 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{8 \int \frac{-\frac{7378905}{8}-\frac{3135525 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{2268945}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{3+5 x}}{453789 (2+3 x)^{5/2}}-\frac{392998 \sqrt{1-2 x} \sqrt{3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{16 \int \frac{-\frac{17369985}{8}-\frac{14737425 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{47647845}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{3+5 x}}{453789 (2+3 x)^{5/2}}-\frac{392998 \sqrt{1-2 x} \sqrt{3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{3+5 x}}{22235661 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{32 \int \frac{-\frac{185288025}{16}-\frac{113175525 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{333534915}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{3+5 x}}{453789 (2+3 x)^{5/2}}-\frac{392998 \sqrt{1-2 x} \sqrt{3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{3+5 x}}{22235661 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}+\frac{6036028 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{22235661}+\frac{6596986 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{22235661}\\ &=\frac{295 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 (2+3 x)^{9/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{3+5 x}}{64827 (2+3 x)^{7/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{3+5 x}}{453789 (2+3 x)^{5/2}}-\frac{392998 \sqrt{1-2 x} \sqrt{3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{3+5 x}}{22235661 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{9/2}}-\frac{6036028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661}-\frac{1199452 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661}\\ \end{align*}
Mathematica [A] time = 0.216084, size = 115, normalized size = 0.45 \[ \frac{8 \sqrt{2} \left (6877465 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3018014 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{24 \sqrt{5 x+3} \left (488918268 x^5+985046292 x^4+466728543 x^3-227945505 x^2-243200677 x-52688263\right )}{\sqrt{1-2 x} (3 x+2)^{9/2}}}{266827932} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 504, normalized size = 2. \begin{align*} -{\frac{2}{667069830\,{x}^{2}+66706983\,x-200120949}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 244459134\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+557074665\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+651891024\,\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}+1485532440\,\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}+651891024\,\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}+1485532440\,\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}+289729344\,\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}+660236640\,\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}-7333774020\,{x}^{6}+48288224\,\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 ) +110039440\,\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 ) -19175958792\,{x}^{5}-15866344773\,{x}^{4}-781374312\,{x}^{3}+5699519700\,{x}^{2}+2979130038\,x+474194367 \right ) \left ( 2+3\,x \right ) ^{-{\frac{9}{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 (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2916 \, x^{8} + 8748 \, x^{7} + 8505 \, x^{6} + 756 \, x^{5} - 3780 \, x^{4} - 2016 \, x^{3} + 112 \, x^{2} + 320 \, x + 64}, 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{11}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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