Optimal. Leaf size=222 \[ \frac{58928}{147} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}-\frac{9795160 \sqrt{3 x+2} \sqrt{1-2 x}}{441 \sqrt{5 x+3}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2332 \sqrt{1-2 x}}{21 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{104 \sqrt{1-2 x}}{9 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1959032}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0804856, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}-\frac{9795160 \sqrt{3 x+2} \sqrt{1-2 x}}{441 \sqrt{5 x+3}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2332 \sqrt{1-2 x}}{21 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{104 \sqrt{1-2 x}}{9 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{58928}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1959032}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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} (3+5 x)^{3/2}} \, dx &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{2}{21} \int \frac{(196-161 x) \sqrt{1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{4}{315} \int \frac{-\frac{31955}{2}+21945 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2332 \sqrt{1-2 x}}{21 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{2417415}{2}+\frac{2754675 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{6615}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2332 \sqrt{1-2 x}}{21 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{16 \int \frac{-\frac{206265675}{4}+\frac{63807975 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{46305}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2332 \sqrt{1-2 x}}{21 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{9795160 \sqrt{1-2 x} \sqrt{2+3 x}}{441 \sqrt{3+5 x}}+\frac{32 \int \frac{-\frac{1342947375}{2}-\frac{4242528675 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{509355}\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2332 \sqrt{1-2 x}}{21 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{9795160 \sqrt{1-2 x} \sqrt{2+3 x}}{441 \sqrt{3+5 x}}-\frac{324104}{147} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{1959032}{147} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{104 \sqrt{1-2 x}}{9 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2332 \sqrt{1-2 x}}{21 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{324104 \sqrt{1-2 x}}{147 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{9795160 \sqrt{1-2 x} \sqrt{2+3 x}}{441 \sqrt{3+5 x}}+\frac{1959032}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{58928}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.252525, size = 110, normalized size = 0.5 \[ \frac{2}{441} \left (-4 \sqrt{2} \left (244879 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-123340 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (132234660 x^4+348250356 x^3+343801494 x^2+150788294 x+24789615\right )}{(3 x+2)^{7/2} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 409, normalized size = 1.8 \begin{align*} -{\frac{2}{4410\,{x}^{2}+441\,x-1323}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13320720\,\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}-26446932\,\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}+26641440\,\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}-52893864\,\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}+17760960\,\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}-35262576\,\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}+3946880\,\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 ) -7836128\,\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 ) +793407960\,{x}^{5}+1692798156\,{x}^{4}+1018057896\,{x}^{3}-126674718\,{x}^{2}-303627192\,x-74368845 \right ) \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}}}{{\left (5 \, x + 3\right )}^{\frac{3}{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}}{6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288}, 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}}}{{\left (5 \, x + 3\right )}^{\frac{3}{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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