Optimal. Leaf size=116 \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]
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Rubi [A] time = 0.0320511, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 78, 50, 54, 216} \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{1}{66} \int \frac{(3+5 x)^{3/2} \left (\frac{1579}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{938 (3+5 x)^{5/2}}{363 \sqrt{1-2 x}}+\frac{40787 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{1452}\\ &=-\frac{40787 \sqrt{1-2 x} (3+5 x)^{3/2}}{5808}+\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{938 (3+5 x)^{5/2}}{363 \sqrt{1-2 x}}+\frac{40787}{352} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{40787}{704} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{40787 \sqrt{1-2 x} (3+5 x)^{3/2}}{5808}+\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{938 (3+5 x)^{5/2}}{363 \sqrt{1-2 x}}+\frac{40787}{128} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{40787}{704} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{40787 \sqrt{1-2 x} (3+5 x)^{3/2}}{5808}+\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{938 (3+5 x)^{5/2}}{363 \sqrt{1-2 x}}+\frac{40787 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{64 \sqrt{5}}\\ &=-\frac{40787}{704} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{40787 \sqrt{1-2 x} (3+5 x)^{3/2}}{5808}+\frac{49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{938 (3+5 x)^{5/2}}{363 \sqrt{1-2 x}}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{64 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0630137, size = 74, normalized size = 0.64 \[ \frac{122361 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2160 x^3+12780 x^2-52256 x+18351\right )}{1920 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 137, normalized size = 1.2 \begin{align*}{\frac{1}{3840\, \left ( 2\,x-1 \right ) ^{2}} \left ( 489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-43200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-255600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+122361\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1045120\,x\sqrt{-10\,{x}^{2}-x+3}-367020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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 = 3.72956, size = 208, normalized size = 1.79 \begin{align*} \frac{40787}{1280} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{297}{64} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{24 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (2 \, x - 1\right )}} + \frac{539 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5873 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50356, size = 293, normalized size = 2.53 \begin{align*} -\frac{122361 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, x + 1\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 (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3840 \,{\left (4 \, x^{2} - 4 \, x + 1\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 = 1.98279, size = 113, normalized size = 0.97 \begin{align*} \frac{40787}{640} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 247 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 81574 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1345971 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{24000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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