Optimal. Leaf size=142 \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{299 \sqrt{5 x+3} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{697}{88} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (7306140 x+17606479)}{70400}+\frac{13246251 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
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Rubi [A] time = 0.0445481, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 150, 153, 147, 54, 216} \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{299 \sqrt{5 x+3} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{697}{88} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (7306140 x+17606479)}{70400}+\frac{13246251 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 153
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^3 \left (41+\frac{135 x}{2}\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{299 (2+3 x)^3 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-4779-\frac{31365 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{697}{88} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{299 (2+3 x)^3 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}+\frac{1}{990} \int \frac{(2+3 x) \left (\frac{1680165}{4}+\frac{5479605 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{697}{88} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{299 (2+3 x)^3 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (17606479+7306140 x)}{70400}+\frac{13246251 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12800}\\ &=-\frac{697}{88} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{299 (2+3 x)^3 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (17606479+7306140 x)}{70400}+\frac{13246251 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6400 \sqrt{5}}\\ &=-\frac{697}{88} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{299 (2+3 x)^3 \sqrt{3+5 x}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (17606479+7306140 x)}{70400}+\frac{13246251 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6400 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0744262, size = 79, normalized size = 0.56 \[ \frac{437126283 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2851200 x^4+15040080 x^3+52700868 x^2-183672928 x+66038637\right )}{2112000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 154, normalized size = 1.1 \begin{align*}{\frac{1}{4224000\, \left ( 2\,x-1 \right ) ^{2}} \left ( -57024000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1748505132\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-300801600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1748505132\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1054017360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+437126283\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3673458560\,x\sqrt{-10\,{x}^{2}-x+3}-1320772740\,\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 = 1.51629, size = 339, normalized size = 2.39 \begin{align*} -\frac{437126283 \, \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 (2851200 \, x^{4} + 15040080 \, x^{3} + 52700868 \, x^{2} - 183672928 \, x + 66038637\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4224000 \,{\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 = 2.04273, size = 131, normalized size = 0.92 \begin{align*} \frac{13246251}{64000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (891 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 115 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8919 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 291417650 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4808389113 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{26400000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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