Optimal. Leaf size=222 \[ \frac{310208 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1715}-\frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1029 \sqrt{5 x+3}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{12276 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{176 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{10312712 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]
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Rubi [A] time = 0.0827338, 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} \[ -\frac{10312712 \sqrt{1-2 x} \sqrt{3 x+2}}{1029 \sqrt{5 x+3}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{12276 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{176 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{3 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{310208 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{10312712 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{2}{21} \int \frac{154-231 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{4}{735} \int \frac{\frac{33649}{2}-23100 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{12276 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{8 \int \frac{1272579-\frac{2900205 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{15435}\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{12276 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{217164255}{4}-33589710 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{108045}\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{12276 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1029 \sqrt{3+5 x}}-\frac{32 \int \frac{\frac{2827810755}{4}+\frac{4466693385 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1188495}\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{12276 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1029 \sqrt{3+5 x}}-\frac{1706144 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1715}-\frac{10312712 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1715}\\ &=\frac{2 \sqrt{1-2 x}}{3 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{176 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{12276 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1706144 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{10312712 \sqrt{1-2 x} \sqrt{2+3 x}}{1029 \sqrt{3+5 x}}+\frac{10312712 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}+\frac{310208 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1715}\\ \end{align*}
Mathematica [A] time = 0.168726, size = 110, normalized size = 0.5 \[ \frac{2 \left (-4 \sqrt{2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-649285 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (696108060 x^4+1833255216 x^3+1809835578 x^2+793777840 x+130497191\right )}{(3 x+2)^{7/2} \sqrt{5 x+3}}\right )}{5145} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 409, normalized size = 1.8 \begin{align*}{\frac{2}{51450\,{x}^{2}+5145\,x-15435}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 139221612\,\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}-70122780\,\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}+278443224\,\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}-140245560\,\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}+185628816\,\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}-93497040\,\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}+41250848\,\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 ) -20777120\,\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 ) -4176648360\,{x}^{5}-8911207116\,{x}^{4}-5359247820\,{x}^{3}+666839694\,{x}^{2}+1598350374\,x+391491573 \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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{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}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288}, 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{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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