Optimal. Leaf size=191 \[ -\frac{1048 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1323}-\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{5/2}}+\frac{36052 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{524 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{36052 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
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Rubi [A] time = 0.0657297, 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 = {97, 150, 152, 158, 113, 119} \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{5/2}}+\frac{36052 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{524 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{1048 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{36052 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
Antiderivative was successfully verified.
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Rule 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)^{9/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{7 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{\left (-\frac{705}{2}-\frac{75 x}{2}\right ) \sqrt{1-2 x}}{(2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{7 (2+3 x)^{5/2}}+\frac{524 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{8 \int \frac{\frac{31665}{4}-5025 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{2835}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{7 (2+3 x)^{5/2}}+\frac{524 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{36052 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{16 \int \frac{106800+\frac{675975 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{19845}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{7 (2+3 x)^{5/2}}+\frac{524 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{36052 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{5764 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1323}+\frac{36052 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1323}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{7 (2+3 x)^{5/2}}+\frac{524 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{36052 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}-\frac{36052 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{1048 \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 = 0.24708, size = 106, normalized size = 0.55 \[ \frac{4 \left (\sqrt{2} \left (9013 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4690 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (486702 x^3+988524 x^2+671007 x+151859\right )}{2 (3 x+2)^{7/2}}\right )}{3969} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{39690\,{x}^{2}+3969\,x-11907} \left ( 253260\,\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}-486702\,\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}+506520\,\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}-973404\,\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}+337680\,\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}-648936\,\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}+75040\,\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 ) -144208\,\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 ) +14601060\,{x}^{5}+31115826\,{x}^{4}+18715464\,{x}^{3}-2327925\,{x}^{2}-5583486\,x-1366731 \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{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\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}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, 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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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