Optimal. Leaf size=222 \[ -\frac{299863 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{2953125}-\frac{8}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{1972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}+\frac{196499 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{1509007 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125} \]
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Rubi [A] time = 0.0839128, 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 = {97, 154, 158, 113, 119} \[ -\frac{8}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{1972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}+\frac{196499 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{299863 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{1509007 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (-\frac{5}{2}-30 x\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{4}{675} \int \frac{\left (\frac{465}{4}-\frac{7395 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{684795}{8}-\frac{627105 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{70875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{167228 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{8 \int \frac{\sqrt{2+3 x} \left (-\frac{721125}{2}+\frac{8842455 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1771875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{196499 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{167228 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{8 \int \frac{\frac{111172815}{16}+\frac{67905315 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{26578125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{196499 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{167228 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{1509007 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2953125}+\frac{3298493 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5906250}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt{3+5 x}}+\frac{196499 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{590625}+\frac{167228 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{118125}-\frac{1972 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{4725}-\frac{8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{1509007 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{299863 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}\\ \end{align*}
Mathematica [A] time = 0.303946, size = 112, normalized size = 0.5 \[ \frac{6877465 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (945000 x^4-382500 x^3-844650 x^2+650155 x+443337\right )}{\sqrt{5 x+3}}+3018014 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17718750} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 155, normalized size = 0.7 \begin{align*} -{\frac{1}{531562500\,{x}^{3}+407531250\,{x}^{2}-124031250\,x-106312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -170100000\,{x}^{6}+6877465\,\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 ) +3018014\,\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 ) +40500000\,{x}^{5}+220212000\,{x}^{4}-114638400\,{x}^{3}-149984310\,{x}^{2}+25709190\,x+26600220 \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 (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\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 (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, 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 (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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