Optimal. Leaf size=156 \[ -\frac{412 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3025 \sqrt{33}}+\frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 (5 x+3)^{3/2}}+\frac{4157 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}} \]
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Rubi [A] time = 0.0527488, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{7 (3 x+2)^{3/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 \sqrt{5 x+3}}-\frac{107 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 (5 x+3)^{3/2}}-\frac{412 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}+\frac{4157 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}} \]
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{(2+3 x)^{5/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{3/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{1}{11} \int \frac{\left (-\frac{25}{2}-3 x\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{2+3 x}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{3/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2 \int \frac{-\frac{1573}{4}-309 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{1815}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{2+3 x}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{3/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}+\frac{4 \int \frac{-\frac{6123}{4}-\frac{12471 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{2+3 x}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{3/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}+\frac{206 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3025}-\frac{4157 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{33275}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{2+3 x}}{1815 (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{3/2}}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{4157 \sqrt{1-2 x} \sqrt{2+3 x}}{19965 \sqrt{3+5 x}}+\frac{4157 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}-\frac{412 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3025 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.228527, size = 97, normalized size = 0.62 \[ \frac{\sqrt{2} \left (10955 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{5 \sqrt{6 x+4} \left (20785 x^2+22313 x+5881\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}-4157 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{99825} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 219, normalized size = 1.4 \begin{align*} -{\frac{1}{598950\,{x}^{2}+99825\,x-199650}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 54775\,\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}-20785\,\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}+32865\,\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 ) -12471\,\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 ) +623550\,{x}^{3}+1085090\,{x}^{2}+622690\,x+117620 \right ) \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 (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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 (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, 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 (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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