Optimal. Leaf size=191 \[ -\frac{9124 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{252105}+\frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]
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Rubi [A] time = 0.0648077, 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 = {98, 152, 158, 113, 119} \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]
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}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{2}{147} \int \frac{-347-\frac{1175 x}{2}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{536 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{5/2}}-\frac{4 \int \frac{-\frac{5847}{4}-2010 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{5145}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{536 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{5/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{41289}{4}+\frac{7305 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{536 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{5/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{906135}{8}-\frac{692385 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{756315}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{536 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{5/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 \sqrt{2+3 x}}+\frac{50182 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{252105}+\frac{184636 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{252105}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{7/2}}-\frac{536 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{5/2}}+\frac{974 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{3/2}}+\frac{184636 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 \sqrt{2+3 x}}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}\\ \end{align*}
Mathematica [A] time = 0.156809, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17045 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2492586 x^3+5015853 x^2+3324960 x+727631\right )}{(3 x+2)^{7/2}}\right )}{756315} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{7563150\,{x}^{2}+756315\,x-2268945} \left ( 460215\,\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}-2492586\,\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}+920430\,\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}-4985172\,\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}+613620\,\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}-3323448\,\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}+136360\,\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 ) -738544\,\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 ) +74777580\,{x}^{5}+157953348\,{x}^{4}+92363085\,{x}^{3}-13338867\,{x}^{2}-27741747\,x-6548679 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \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}} \sqrt{-2 \, x + 1}}\,{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}}{486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, 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}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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