Optimal. Leaf size=187 \[ -\frac{988 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{9317 \sqrt{33}}-\frac{22090 \sqrt{1-2 x} \sqrt{3 x+2}}{307461 \sqrt{5 x+3}}-\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 (5 x+3)^{3/2}}+\frac{118 \sqrt{3 x+2}}{847 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{4418 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}} \]
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Rubi [A] time = 0.06873, antiderivative size = 187, 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 = {99, 152, 158, 113, 119} \[ -\frac{22090 \sqrt{1-2 x} \sqrt{3 x+2}}{307461 \sqrt{5 x+3}}-\frac{2470 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 (5 x+3)^{3/2}}+\frac{118 \sqrt{3 x+2}}{847 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2 \sqrt{3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{988 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}}+\frac{4418 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2}{33} \int \frac{-\frac{51}{2}-\frac{75 x}{2}}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{118 \sqrt{2+3 x}}{847 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{4 \int \frac{1380+\frac{7965 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{2541}\\ &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{118 \sqrt{2+3 x}}{847 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2470 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 (3+5 x)^{3/2}}-\frac{8 \int \frac{-\frac{19965}{8}-\frac{11115 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{83853}\\ &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{118 \sqrt{2+3 x}}{847 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2470 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 (3+5 x)^{3/2}}-\frac{22090 \sqrt{1-2 x} \sqrt{2+3 x}}{307461 \sqrt{3+5 x}}+\frac{16 \int \frac{-\frac{17595}{4}-\frac{99405 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{922383}\\ &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{118 \sqrt{2+3 x}}{847 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2470 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 (3+5 x)^{3/2}}-\frac{22090 \sqrt{1-2 x} \sqrt{2+3 x}}{307461 \sqrt{3+5 x}}-\frac{4418 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{102487}+\frac{494 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{9317}\\ &=\frac{2 \sqrt{2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{118 \sqrt{2+3 x}}{847 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{2470 \sqrt{1-2 x} \sqrt{2+3 x}}{27951 (3+5 x)^{3/2}}-\frac{22090 \sqrt{1-2 x} \sqrt{2+3 x}}{307461 \sqrt{3+5 x}}+\frac{4418 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}}-\frac{988 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9317 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.160808, size = 103, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (10360 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-2209 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{3 x+2} \left (-220900 x^3+34020 x^2+88821 x-15986\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}\right )}{307461} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*}{\frac{2}{307461\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 22090\,\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}-103600\,\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}+2209\,\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}-10360\,\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}-6627\,\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 ) +31080\,\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 ) -662700\,{x}^{4}-339740\,{x}^{3}+334503\,{x}^{2}+129684\,x-31972 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,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{\sqrt{3 \, x + 2}}{{\left (5 \, x + 3\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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, 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{\sqrt{3 \, x + 2}}{{\left (5 \, x + 3\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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