Optimal. Leaf size=160 \[ \frac{16}{3} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{9 \sqrt{5 x+3}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{532}{3} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.052417, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{2660 \sqrt{1-2 x} \sqrt{3 x+2}}{9 \sqrt{5 x+3}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{16}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{532}{3} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2}{9} \int \frac{88-99 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{4}{63} \int \frac{\frac{7469}{2}-2310 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{9 \sqrt{3+5 x}}-\frac{8}{693} \int \frac{\frac{97251}{2}+\frac{153615 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{9 \sqrt{3+5 x}}-\frac{88}{3} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{532}{3} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{88 \sqrt{1-2 x}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2660 \sqrt{1-2 x} \sqrt{2+3 x}}{9 \sqrt{3+5 x}}+\frac{532}{3} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{16}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.123611, size = 100, normalized size = 0.62 \[ -\frac{4}{9} \sqrt{2} \left (133 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-67 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{2 \sqrt{1-2 x} \left (3990 x^2+5188 x+1683\right )}{3 (3 x+2)^{3/2} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 219, normalized size = 1.4 \begin{align*} -{\frac{2}{90\,{x}^{2}+9\,x-27}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 402\,\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}-798\,\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}+268\,\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 ) -532\,\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 ) +23940\,{x}^{3}+19158\,{x}^{2}-5466\,x-5049 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{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{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}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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