Optimal. Leaf size=222 \[ -\frac{23012 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{588245}+\frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{588245 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{5/2}}-\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{7/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{189368 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245} \]
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Rubi [A] time = 0.0826907, antiderivative size = 222, 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 = {98, 152, 158, 113, 119} \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{588245 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{5/2}}-\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{7/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{23012 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245}-\frac{189368 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{1}{7} \int \frac{-\frac{577}{2}-490 x}{\sqrt{1-2 x} (2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2}{343} \int \frac{-\frac{3347}{2}-\frac{5725 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{25461}{4}-\frac{21135 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{12005}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 (2+3 x)^{3/2}}-\frac{8 \int \frac{-18633-\frac{40785 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{252105}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{588245 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{1042005}{8}-\frac{355065 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1764735}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{588245 \sqrt{2+3 x}}+\frac{126566 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{588245}+\frac{189368 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{588245}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{343 (2+3 x)^{7/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{3+5 x}}{12005 (2+3 x)^{5/2}}-\frac{5438 \sqrt{1-2 x} \sqrt{3+5 x}}{84035 (2+3 x)^{3/2}}+\frac{189368 \sqrt{1-2 x} \sqrt{3+5 x}}{588245 \sqrt{2+3 x}}-\frac{189368 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245}-\frac{23012 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245}\\ \end{align*}
Mathematica [A] time = 0.185379, size = 109, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (95165 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+94684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (5112936 x^4+7326810 x^3+1004571 x^2-2279324 x-809083\right )}{\sqrt{1-2 x} (3 x+2)^{7/2}}\right )}{1764735} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 409, normalized size = 1.8 \begin{align*} -{\frac{2}{17647350\,{x}^{2}+1764735\,x-5294205}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2556468\,\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}+2569455\,\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}+5112936\,\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}+5138910\,\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}+3408624\,\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}+3425940\,\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}+757472\,\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 ) +761320\,\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 ) -76694040\,{x}^{5}-155918574\,{x}^{4}-81009855\,{x}^{3}+25148721\,{x}^{2}+32650161\,x+7281747 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{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 (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{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 (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32}, 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 (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{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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