Optimal. Leaf size=191 \[ \frac{496 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{7203}+\frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{5 x+3}}{7203 \sqrt{3 x+2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{3/2}}-\frac{22 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203} \]
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Rubi [A] time = 0.0646141, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{5 x+3}}{7203 \sqrt{3 x+2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{3/2}}-\frac{22 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203}+\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203} \]
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{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{1}{21} \int \frac{\sqrt{3+5 x} \left (\frac{51}{2}+15 x\right )}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx\\ &=-\frac{22 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{1}{147} \int \frac{\frac{1401}{2}+\frac{2445 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{22 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{2 \int \frac{\frac{4749}{4}+\frac{3435 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{3087}\\ &=-\frac{22 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{3+5 x}}{7203 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{4 \int \frac{\frac{9705}{4}+\frac{2535 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{21609}\\ &=-\frac{22 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{3+5 x}}{7203 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{169 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{7203}-\frac{2728 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7203}\\ &=-\frac{22 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{3/2}}-\frac{169 \sqrt{1-2 x} \sqrt{3+5 x}}{7203 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{169 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203}+\frac{496 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7203}\\ \end{align*}
Mathematica [A] time = 0.164424, size = 105, normalized size = 0.55 \[ \frac{-\sqrt{2} \left (8015 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+169 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{5 x+3} \left (1014 x^3-3544 x^2-9883 x-4675\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}}{21609} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.6 \begin{align*}{\frac{1}{21609\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 1014\,\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}+48090\,\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}+169\,\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}+8015\,\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}-338\,\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 ) -16030\,\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 ) -30420\,{x}^{4}+88068\,{x}^{3}+360282\,{x}^{2}+318144\,x+84150 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \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 (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\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{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{216 \, x^{6} + 108 \, x^{5} - 198 \, x^{4} - 71 \, x^{3} + 66 \, x^{2} + 12 \, x - 8}, 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 (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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