Optimal. Leaf size=218 \[ \frac{673072 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{290521 \sqrt{33}}-\frac{113693540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{694 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{1352}{17787 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{22738708 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}} \]
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Rubi [A] time = 0.0820513, antiderivative size = 218, 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 = {104, 152, 158, 113, 119} \[ -\frac{113693540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{694 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{1352}{17787 \sqrt{1-2 x} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{673072 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}}+\frac{22738708 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}} \]
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)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{2}{231} \int \frac{-\frac{233}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{4 \int \frac{\frac{34841}{4}+12675 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{694 \sqrt{1-2 x}}{41503 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{8 \int \frac{55293-\frac{46845 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{373527}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{694 \sqrt{1-2 x}}{41503 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{2+3 x} \sqrt{3+5 x}}+\frac{16 \int \frac{\frac{12510795}{8}-\frac{1893015 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2614689}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{694 \sqrt{1-2 x}}{41503 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{113693540 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}-\frac{32 \int \frac{\frac{161815545}{8}+\frac{255810465 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{28761579}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{694 \sqrt{1-2 x}}{41503 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{113693540 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}-\frac{336536 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{290521}-\frac{22738708 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3195731}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1352}{17787 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{694 \sqrt{1-2 x}}{41503 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{336536 \sqrt{1-2 x}}{290521 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{113693540 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}+\frac{22738708 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}}+\frac{673072 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{290521 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.201281, size = 109, normalized size = 0.5 \[ \frac{2 \left (\frac{-2046483720 x^4-615527112 x^3+1285584962 x^2+198573504 x-215753865}{(1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2908255 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )\right )}{9587193} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 311, normalized size = 1.4 \begin{align*} -{\frac{2}{9587193\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 34899060\,\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}-68216124\,\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}+5816510\,\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}-11369354\,\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}-11633020\,\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 ) +22738708\,\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 ) +2046483720\,{x}^{4}+615527112\,{x}^{3}-1285584962\,{x}^{2}-198573504\,x+215753865 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,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{5}{2}}{\left (-2 \, x + 1\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} \sqrt{-2 \, x + 1}}{5400 \, x^{8} + 9180 \, x^{7} + 234 \, x^{6} - 6743 \, x^{5} - 2262 \, x^{4} + 1641 \, x^{3} + 754 \, x^{2} - 132 \, 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{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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