Optimal. Leaf size=220 \[ -\frac{2092 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{84035}+\frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{924385 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{5/2}}+\frac{458 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{189368 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035 \sqrt{33}} \]
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Rubi [A] time = 0.0781987, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{924385 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{132055 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{18865 (3 x+2)^{5/2}}+\frac{458 \sqrt{5 x+3}}{1617 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{2092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}-\frac{189368 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035 \sqrt{33}} \]
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{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac{2}{21} \int \frac{-31-\frac{105 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}+\frac{4 \int \frac{\frac{10041}{4}+\frac{17175 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{5/2}}+\frac{8 \int \frac{\frac{76383}{8}+\frac{63405 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{56595}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 (2+3 x)^{3/2}}+\frac{16 \int \frac{\frac{55899}{2}+\frac{122355 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1188495}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{924385 \sqrt{2+3 x}}+\frac{32 \int \frac{\frac{3126015}{16}+\frac{1065195 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8319465}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{924385 \sqrt{2+3 x}}+\frac{11506 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{84035}+\frac{189368 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{924385}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac{458 \sqrt{3+5 x}}{1617 \sqrt{1-2 x} (2+3 x)^{5/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{18865 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{132055 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{924385 \sqrt{2+3 x}}-\frac{189368 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035 \sqrt{33}}-\frac{2092 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}\\ \end{align*}
Mathematica [A] time = 0.175554, size = 108, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (95165 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+94684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{\sqrt{5 x+3} \left (10225872 x^4+2723436 x^3-7133292 x^2-807691 x+1339677\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}\right )}{2773155} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 406, normalized size = 1.9 \begin{align*} -{\frac{2}{2773155\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 1704312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1712970\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1420260\,\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}+1427475\,\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}-378736\,\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}-380660\,\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}-378736\,\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 ) -380660\,\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 ) -51129360\,{x}^{5}-44294796\,{x}^{4}+27496152\,{x}^{3}+25438331\,{x}^{2}-4275312\,x-4019031 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{7}{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}}{648 \, x^{7} + 756 \, x^{6} - 378 \, x^{5} - 609 \, x^{4} + 56 \, x^{3} + 168 \, x^{2} - 16}, 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{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{7}{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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