Optimal. Leaf size=222 \[ -\frac{234856 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{83349}-\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac{7810384 \sqrt{5 x+3} \sqrt{1-2 x}}{83349 \sqrt{3 x+2}}+\frac{112436 \sqrt{5 x+3} \sqrt{1-2 x}}{11907 (3 x+2)^{3/2}}+\frac{832 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}-\frac{7810384 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349} \]
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Rubi [A] time = 0.0818294, antiderivative size = 222, 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, 152, 158, 113, 119} \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac{7810384 \sqrt{5 x+3} \sqrt{1-2 x}}{83349 \sqrt{3 x+2}}+\frac{112436 \sqrt{5 x+3} \sqrt{1-2 x}}{11907 (3 x+2)^{3/2}}+\frac{832 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}-\frac{234856 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349}-\frac{7810384 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}-\frac{4}{567} \int \frac{\sqrt{1-2 x} \left (-435+\frac{255 x}{2}\right )}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}+\frac{832 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{8 \int \frac{\frac{79845}{4}-\frac{45525 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{8505}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}+\frac{832 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{112436 \sqrt{1-2 x} \sqrt{3+5 x}}{11907 (2+3 x)^{3/2}}+\frac{16 \int \frac{869010-\frac{2108175 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{178605}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}+\frac{832 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{112436 \sqrt{1-2 x} \sqrt{3+5 x}}{11907 (2+3 x)^{3/2}}+\frac{7810384 \sqrt{1-2 x} \sqrt{3+5 x}}{83349 \sqrt{2+3 x}}+\frac{32 \int \frac{\frac{92710725}{8}+\frac{36611175 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1250235}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}+\frac{832 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{112436 \sqrt{1-2 x} \sqrt{3+5 x}}{11907 (2+3 x)^{3/2}}+\frac{7810384 \sqrt{1-2 x} \sqrt{3+5 x}}{83349 \sqrt{2+3 x}}+\frac{1291708 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{83349}+\frac{7810384 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{83349}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{27 (2+3 x)^{9/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{63 (2+3 x)^{7/2}}+\frac{832 \sqrt{1-2 x} \sqrt{3+5 x}}{567 (2+3 x)^{5/2}}+\frac{112436 \sqrt{1-2 x} \sqrt{3+5 x}}{11907 (2+3 x)^{3/2}}+\frac{7810384 \sqrt{1-2 x} \sqrt{3+5 x}}{83349 \sqrt{2+3 x}}-\frac{7810384 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349}-\frac{234856 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349}\\ \end{align*}
Mathematica [A] time = 0.24582, size = 111, normalized size = 0.5 \[ \frac{4 \left (\sqrt{2} \left (1952596 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-983815 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (316320552 x^4+854146674 x^3+865270206 x^2+389804925 x+65886031\right )}{2 (3 x+2)^{9/2}}\right )}{250047} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 504, normalized size = 2.3 \begin{align*}{\frac{2}{2500470\,{x}^{2}+250047\,x-750141} \left ( 159378030\,\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}-316320552\,\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}+425008080\,\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}-843521472\,\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}+425008080\,\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}-843521472\,\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}+188892480\,\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}-374898432\,\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}+9489616560\,{x}^{6}+31482080\,\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 ) -62483072\,\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 ) +26573361876\,{x}^{5}+25673661234\,{x}^{4}+6602638302\,{x}^{3}-4641436149\,{x}^{2}-3310586232\,x-592974279 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{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}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, 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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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