Optimal. Leaf size=157 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]
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Rubi [A] time = 0.0464469, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{(-1-33 x) (1-2 x)^{3/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{125} \int \frac{(1-2 x)^{3/2} (2+3 x) \left (-131+\frac{69 x}{2}\right )}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{3+5 x}}{200000}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{66997 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{80000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{66997 (1-2 x)^{3/2} \sqrt{3+5 x}}{800000}-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{3+5 x}}{200000}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{2210901 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1600000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2210901 \sqrt{1-2 x} \sqrt{3+5 x}}{8000000}+\frac{66997 (1-2 x)^{3/2} \sqrt{3+5 x}}{800000}-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{3+5 x}}{200000}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{24319911 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{16000000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2210901 \sqrt{1-2 x} \sqrt{3+5 x}}{8000000}+\frac{66997 (1-2 x)^{3/2} \sqrt{3+5 x}}{800000}-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{3+5 x}}{200000}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{24319911 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8000000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2210901 \sqrt{1-2 x} \sqrt{3+5 x}}{8000000}+\frac{66997 (1-2 x)^{3/2} \sqrt{3+5 x}}{800000}-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{3+5 x}}{200000}+\frac{33}{125} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8000000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0437636, size = 93, normalized size = 0.59 \[ \frac{-10 \left (69120000 x^6-9504000 x^5-91502400 x^4+31284920 x^3+44775890 x^2-8158469 x-6089453\right )-24319911 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{80000000 \sqrt{1-2 x} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 150, normalized size = 1. \begin{align*}{\frac{1}{160000000} \left ( 691200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+250560000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-789744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+121599555\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-82022800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+72959733\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +406747500\,x\sqrt{-10\,{x}^{2}-x+3}+121789060\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.92002, size = 170, normalized size = 1.08 \begin{align*} -\frac{216 \, x^{6}}{25 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{297 \, x^{5}}{250 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{57189 \, x^{4}}{5000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{782123 \, x^{3}}{200000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4477589 \, x^{2}}{800000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{24319911}{160000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{8158469 \, x}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6089453}{8000000 \, \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 = 1.83697, size = 336, normalized size = 2.14 \begin{align*} -\frac{24319911 \, \sqrt{10}{\left (5 \, x + 3\right )} \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 ) - 20 \,{\left (34560000 \, x^{5} + 12528000 \, x^{4} - 39487200 \, x^{3} - 4101140 \, x^{2} + 20337375 \, x + 6089453\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{160000000 \,{\left (5 \, x + 3\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 [A] time = 2.52191, size = 203, normalized size = 1.29 \begin{align*} \frac{1}{200000000} \,{\left (4 \,{\left (24 \,{\left (36 \,{\left (16 \, \sqrt{5}{\left (5 \, x + 3\right )} - 211 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 22859 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 969335 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 5816745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{24319911}{80000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{156250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{78125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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