Optimal. Leaf size=160 \[ \frac{124}{539} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{23180 \sqrt{1-2 x} \sqrt{3 x+2}}{5929 \sqrt{5 x+3}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4636}{539} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0509455, 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 = {104, 152, 158, 113, 119} \[ -\frac{23180 \sqrt{1-2 x} \sqrt{3 x+2}}{5929 \sqrt{5 x+3}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{124}{539} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4636}{539} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{2}{77} \int \frac{-\frac{91}{2}-45 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{4}{539} \int \frac{-440+\frac{465 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{23180 \sqrt{1-2 x} \sqrt{2+3 x}}{5929 \sqrt{3+5 x}}+\frac{8 \int \frac{-\frac{21885}{4}-\frac{17385 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5929}\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{23180 \sqrt{1-2 x} \sqrt{2+3 x}}{5929 \sqrt{3+5 x}}-\frac{186}{539} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{13908 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5929}\\ &=\frac{4}{77 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{23180 \sqrt{1-2 x} \sqrt{2+3 x}}{5929 \sqrt{3+5 x}}+\frac{4636}{539} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{124}{539} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.153682, size = 98, normalized size = 0.61 \[ \frac{2 \left (\sqrt{2} \left (1295 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-2318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{69540 x^2+9544 x-22003}{\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}\right )}{5929} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 135, normalized size = 0.8 \begin{align*}{\frac{2}{177870\,{x}^{3}+136367\,{x}^{2}-41503\,x-35574} \left ( 2318\,\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 ) -1295\,\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 ) -69540\,{x}^{2}-9544\,x+22003 \right ) \sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}} \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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36}, 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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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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