Optimal. Leaf size=129 \[ \frac{124}{75} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{14 (1-2 x)^{3/2}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}-\frac{1496 \sqrt{3 x+2} \sqrt{1-2 x}}{15 \sqrt{5 x+3}}+\frac{4636}{75} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0405258, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}-\frac{1496 \sqrt{3 x+2} \sqrt{1-2 x}}{15 \sqrt{5 x+3}}+\frac{124}{75} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4636}{75} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{2}{3} \int \frac{\sqrt{1-2 x} (97+37 x)}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1496 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}+\frac{4}{15} \int \frac{-\frac{1459}{2}-1159 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1496 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}-\frac{682}{75} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{4636}{75} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{3 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{1496 \sqrt{1-2 x} \sqrt{2+3 x}}{15 \sqrt{3+5 x}}+\frac{4636}{75} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{124}{75} \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 = 0.132382, size = 131, normalized size = 1.02 \[ \frac{2590 \sqrt{2} \left (15 x^2+19 x+6\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-4636 \sqrt{2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (2314 x+1461)}{225 (3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 135, normalized size = 1.1 \begin{align*} -{\frac{2}{6750\,{x}^{3}+5175\,{x}^{2}-1575\,x-1350}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 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 ) -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 ) +69420\,{x}^{2}+9120\,x-21915 \right ) } \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{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{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, 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{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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