Optimal. Leaf size=191 \[ \frac{1112}{35} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{36968 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{44 \sqrt{1-2 x}}{5 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{36968}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0690799, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac{36968 \sqrt{1-2 x} \sqrt{3 x+2}}{21 \sqrt{5 x+3}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{44 \sqrt{1-2 x}}{5 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{1112}{35} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{36968}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2}{15} \int \frac{121-165 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{44 \sqrt{1-2 x}}{5 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{4}{315} \int \frac{\frac{18249}{2}-10395 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{44 \sqrt{1-2 x}}{5 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{8 \int \frac{389235-\frac{481635 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2205}\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{44 \sqrt{1-2 x}}{5 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{36968 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}-\frac{16 \int \frac{\frac{20273715}{4}+\frac{16011765 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{24255}\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{44 \sqrt{1-2 x}}{5 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{36968 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}-\frac{6116}{35} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{36968}{35} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{44 \sqrt{1-2 x}}{5 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{6116 \sqrt{1-2 x}}{35 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{36968 \sqrt{1-2 x} \sqrt{2+3 x}}{21 \sqrt{3+5 x}}+\frac{36968}{35} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1112}{35} \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 = 0.148149, size = 105, normalized size = 0.55 \[ \frac{2}{105} \left (-2 \sqrt{2} \left (9242 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4655 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (831780 x^3+1636038 x^2+1071882 x+233897\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 314, normalized size = 1.6 \begin{align*}{\frac{2}{1050\,{x}^{2}+105\,x-315}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 166356\,\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}-83790\,\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}+221808\,\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}-111720\,\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}+73936\,\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 ) -37240\,\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 ) -4990680\,{x}^{4}-7320888\,{x}^{3}-1523178\,{x}^{2}+1812264\,x+701691 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144}, 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{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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