Optimal. Leaf size=187 \[ -\frac{9124 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{26411 \sqrt{33}}+\frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{290521 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 (3 x+2)^{3/2}}+\frac{1072 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{184636 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}} \]
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Rubi [A] time = 0.0660767, 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 = {104, 152, 158, 113, 119} \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{290521 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{41503 (3 x+2)^{3/2}}+\frac{1072 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}}-\frac{9124 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}}-\frac{184636 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{193}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{1072 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{4 \int \frac{\frac{17541}{4}+6030 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{17787}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{1072 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{41503 (2+3 x)^{3/2}}+\frac{8 \int \frac{\frac{123867}{4}-\frac{21915 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{373527}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{1072 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{41503 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{290521 \sqrt{2+3 x}}+\frac{16 \int \frac{\frac{2718405}{8}+\frac{2077155 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2614689}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{1072 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{41503 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{290521 \sqrt{2+3 x}}+\frac{4562 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{26411}+\frac{184636 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{290521}\\ &=\frac{4 \sqrt{3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac{1072 \sqrt{3+5 x}}{17787 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{41503 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{290521 \sqrt{2+3 x}}-\frac{184636 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}}-\frac{9124 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26411 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.181132, size = 103, normalized size = 0.55 \[ \frac{2 \left (\sqrt{2} \left (92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17045 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (3323448 x^3-1066908 x^2-1478206 x+597945\right )}{(1-2 x)^{3/2} (3 x+2)^{3/2}}\right )}{871563} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 311, normalized size = 1.7 \begin{align*}{\frac{2}{871563\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 102270\,\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}-553908\,\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}+17045\,\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}-92318\,\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}-34090\,\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 ) +184636\,\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 ) +16617240\,{x}^{4}+4635804\,{x}^{3}-10591754\,{x}^{2}-1444893\,x+1793835 \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{1}{\sqrt{5 \, x + 3}{\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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1080 \, x^{7} + 1188 \, x^{6} - 666 \, x^{5} - 949 \, x^{4} + 117 \, x^{3} + 258 \, x^{2} - 4 \, x - 24}, 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{1}{\sqrt{5 \, x + 3}{\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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