Optimal. Leaf size=191 \[ -\frac{7536 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{12005}+\frac{733812 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 \sqrt{3 x+2}}+\frac{10308 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{3/2}}+\frac{138 \sqrt{1-2 x} \sqrt{5 x+3}}{2695 (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{244604 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]
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Rubi [A] time = 0.066021, antiderivative size = 191, 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} \[ \frac{733812 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 \sqrt{3 x+2}}+\frac{10308 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{3/2}}+\frac{138 \sqrt{1-2 x} \sqrt{5 x+3}}{2695 (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} (3 x+2)^{5/2}}-\frac{7536 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{244604 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005} \]
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)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2}{77} \int \frac{-\frac{123}{2}-75 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{138 \sqrt{1-2 x} \sqrt{3+5 x}}{2695 (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{1887}{2}+\frac{1035 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{2695}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{138 \sqrt{1-2 x} \sqrt{3+5 x}}{2695 (2+3 x)^{5/2}}+\frac{10308 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{131913}{4}+\frac{38655 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{56595}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{138 \sqrt{1-2 x} \sqrt{3+5 x}}{2695 (2+3 x)^{5/2}}+\frac{10308 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{3/2}}+\frac{733812 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{1744335}{4}-\frac{2751795 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{396165}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{138 \sqrt{1-2 x} \sqrt{3+5 x}}{2695 (2+3 x)^{5/2}}+\frac{10308 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{3/2}}+\frac{733812 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 \sqrt{2+3 x}}+\frac{11304 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{12005}+\frac{733812 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{132055}\\ &=\frac{4 \sqrt{3+5 x}}{77 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{138 \sqrt{1-2 x} \sqrt{3+5 x}}{2695 (2+3 x)^{5/2}}+\frac{10308 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{3/2}}+\frac{733812 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 \sqrt{2+3 x}}-\frac{244604 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}-\frac{7536 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12005}\\ \end{align*}
Mathematica [A] time = 0.151163, size = 106, normalized size = 0.55 \[ \frac{4 \left (\sqrt{2} \left (61151 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-30065 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (-6604308 x^3-5720058 x^2+1424784 x+1546591\right )}{2 \sqrt{1-2 x} (3 x+2)^{5/2}}\right )}{132055} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 314, normalized size = 1.6 \begin{align*}{\frac{2}{1320550\,{x}^{2}+132055\,x-396165}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 541170\,\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}-1100718\,\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}+721560\,\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}-1467624\,\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}+240520\,\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 ) -489208\,\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 ) +33021540\,{x}^{4}+48413214\,{x}^{3}+10036254\,{x}^{2}-12007307\,x-4639773 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1620 \, x^{7} + 3672 \, x^{6} + 2025 \, x^{5} - 1077 \, x^{4} - 1312 \, x^{3} - 152 \, x^{2} + 176 \, x + 48}, 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{7}{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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