Optimal. Leaf size=113 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{233 \sqrt{5 x+3} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (69780 x+168157)}{3520}+\frac{126513 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]
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Rubi [A] time = 0.0299392, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {97, 150, 147, 54, 216} \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{233 \sqrt{5 x+3} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (69780 x+168157)}{3520}+\frac{126513 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^3 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^2 \left (32+\frac{105 x}{2}\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{233 (2+3 x)^2 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-\frac{5349}{2}-\frac{17445 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{233 (2+3 x)^2 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (168157+69780 x)}{3520}+\frac{126513}{640} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{233 (2+3 x)^2 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (168157+69780 x)}{3520}+\frac{126513 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{320 \sqrt{5}}\\ &=-\frac{233 (2+3 x)^2 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (168157+69780 x)}{3520}+\frac{126513 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{320 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0600223, size = 74, normalized size = 0.65 \[ \frac{4174929 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (71280 x^3+431244 x^2-1786144 x+625431\right )}{105600 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 137, normalized size = 1.2 \begin{align*}{\frac{1}{211200\, \left ( 2\,x-1 \right ) ^{2}} \left ( 16699716\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1425600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-16699716\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-8624880\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4174929\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +35722880\,x\sqrt{-10\,{x}^{2}-x+3}-12508620\,\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 [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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Fricas [A] time = 1.53004, size = 304, normalized size = 2.69 \begin{align*} -\frac{4174929 \, \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 (71280 \, x^{3} + 431244 \, x^{2} - 1786144 \, x + 625431\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{211200 \,{\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 = 2.33807, size = 113, normalized size = 1. \begin{align*} \frac{126513}{3200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (891 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 85 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 2783318 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 45924219 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1320000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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