Optimal. Leaf size=253 \[ \frac{20549264 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{9261}+\frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]
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Rubi [A] time = 0.0963376, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{2}{27} \int \frac{(229-227 x) \sqrt{1-2 x}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}-\frac{4}{567} \int \frac{-\frac{50897}{2}+38346 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{5572105}{2}+\frac{7651875 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{19845}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{16 \int \frac{-\frac{842998695}{4}+\frac{480300975 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{416745}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{32 \int \frac{-8991074400+\frac{22250999925 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2917215}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}+\frac{64 \int \frac{-\frac{936620355825}{8}-\frac{739723463325 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{32089365}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}-\frac{113020952 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{9261}-\frac{683150096 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{9261}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}\\ \end{align*}
Mathematica [A] time = 0.275378, size = 115, normalized size = 0.45 \[ \frac{2 \left (-4 \sqrt{2} \left (85393762 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-43010905 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (138337894440 x^5+456548966244 x^4+602551975428 x^3+397527527442 x^2+131099014240 x+17289178827\right )}{(3 x+2)^{9/2} \sqrt{5 x+3}}\right )}{27783} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 504, normalized size = 2. \begin{align*}{\frac{2}{277830\,{x}^{2}+27783\,x-83349}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 27667578888\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-13935533220\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+73780210368\,\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}-37161421920\,\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}+73780210368\,\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}-37161421920\,\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}+32791204608\,\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}-16516187520\,\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}-830027366640\,{x}^{6}+5465200768\,\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 ) -2752697920\,\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 ) -2324280114144\,{x}^{5}-2245664953836\,{x}^{4}-577509238368\,{x}^{3}+405988496886\,{x}^{2}+289561969758\,x+51867536481 \right ) \left ( 2+3\,x \right ) ^{-{\frac{9}{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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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