Optimal. Leaf size=185 \[ -120 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0641452, antiderivative size = 185, 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} \[ \frac{5440 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{300 \sqrt{1-2 x} \sqrt{3 x+2}}{(5 x+3)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{2}{9} \int \frac{123-169 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{4}{63} \int \frac{\frac{18459}{2}-10605 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{300 \sqrt{1-2 x} \sqrt{2+3 x}}{(3+5 x)^{3/2}}-\frac{8 \int \frac{\frac{756063}{2}-\frac{467775 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2079}\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{300 \sqrt{1-2 x} \sqrt{2+3 x}}{(3+5 x)^{3/2}}+\frac{5440 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{19690209}{4}+7775460 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{22869}\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{300 \sqrt{1-2 x} \sqrt{2+3 x}}{(3+5 x)^{3/2}}+\frac{5440 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}+180 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+1088 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{404 \sqrt{1-2 x}}{9 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{300 \sqrt{1-2 x} \sqrt{2+3 x}}{(3+5 x)^{3/2}}+\frac{5440 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}-1088 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-120 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.224861, size = 104, normalized size = 0.56 \[ \frac{2}{3} \left (2 \sqrt{2} \left (272 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-137 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (122400 x^3+232590 x^2+147122 x+30977\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*}{\frac{2}{6\,x-3}\sqrt{1-2\,x} \left ( 4110\,\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}-8160\,\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}+5206\,\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}-10336\,\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}+1644\,\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 ) -3264\,\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 ) +244800\,{x}^{4}+342780\,{x}^{3}+61654\,{x}^{2}-85168\,x-30977 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{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{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{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}}}{3375 \, x^{6} + 12825 \, x^{5} + 20295 \, x^{4} + 17119 \, x^{3} + 8118 \, x^{2} + 2052 \, x + 216}, 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{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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