Optimal. Leaf size=160 \[ -\frac{1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac{187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac{2057 (1-2 x)^{5/2} \sqrt{5 x+3}}{1024}+\frac{22627 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480}+\frac{746691 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}+\frac{8213601 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]
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Rubi [A] time = 0.0509546, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac{17}{80} (1-2 x)^{5/2} (5 x+3)^{5/2}-\frac{187}{256} (1-2 x)^{5/2} (5 x+3)^{3/2}-\frac{2057 (1-2 x)^{5/2} \sqrt{5 x+3}}{1024}+\frac{22627 (1-2 x)^{3/2} \sqrt{5 x+3}}{20480}+\frac{746691 \sqrt{1-2 x} \sqrt{5 x+3}}{204800}+\frac{8213601 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{204800 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx &=-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{17}{8} \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{187}{32} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{6171}{512} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{2057 (1-2 x)^{5/2} \sqrt{3+5 x}}{1024}-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{22627 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{2048}\\ &=\frac{22627 (1-2 x)^{3/2} \sqrt{3+5 x}}{20480}-\frac{2057 (1-2 x)^{5/2} \sqrt{3+5 x}}{1024}-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{746691 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{40960}\\ &=\frac{746691 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{22627 (1-2 x)^{3/2} \sqrt{3+5 x}}{20480}-\frac{2057 (1-2 x)^{5/2} \sqrt{3+5 x}}{1024}-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{8213601 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{409600}\\ &=\frac{746691 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{22627 (1-2 x)^{3/2} \sqrt{3+5 x}}{20480}-\frac{2057 (1-2 x)^{5/2} \sqrt{3+5 x}}{1024}-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{8213601 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{204800 \sqrt{5}}\\ &=\frac{746691 \sqrt{1-2 x} \sqrt{3+5 x}}{204800}+\frac{22627 (1-2 x)^{3/2} \sqrt{3+5 x}}{20480}-\frac{2057 (1-2 x)^{5/2} \sqrt{3+5 x}}{1024}-\frac{187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac{8213601 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{204800 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0696038, size = 75, normalized size = 0.47 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (5120000 x^5+8448000 x^4+1456000 x^3-4238560 x^2-2224900 x+555399\right )-8213601 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2048000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{4096000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -102400000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-168960000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-29120000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+84771200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8213601\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +44498000\,x\sqrt{-10\,{x}^{2}-x+3}-11107980\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59841, size = 134, normalized size = 0.84 \begin{align*} -\frac{1}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{29}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{187}{128} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{187}{2560} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{67881}{10240} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{8213601}{4096000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{67881}{204800} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51784, size = 301, normalized size = 1.88 \begin{align*} -\frac{1}{204800} \,{\left (5120000 \, x^{5} + 8448000 \, x^{4} + 1456000 \, x^{3} - 4238560 \, x^{2} - 2224900 \, x + 555399\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{8213601}{4096000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\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 [B] time = 2.34382, size = 427, normalized size = 2.67 \begin{align*} -\frac{1}{51200000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{41}{38400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{17}{960000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{17}{8000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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