Optimal. Leaf size=183 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]
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Rubi [A] time = 0.0550919, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]
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{(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac{195 x}{2}\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{1}{60} \int \frac{\left (-11805-\frac{74925 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}-\frac{\int \frac{(2+3 x) (3+5 x)^{3/2} \left (\frac{7494225}{4}+\frac{23531625 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{3000}\\ &=\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{295101237 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{102400}\\ &=\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{9738340821 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{819200}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3276800}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1638400 \sqrt{5}}\\ &=\frac{9738340821 \sqrt{1-2 x} \sqrt{3+5 x}}{1638400}+\frac{295101237 \sqrt{1-2 x} (3+5 x)^{3/2}}{409600}+\frac{999}{160} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac{13}{8} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1638400 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.055032, size = 84, normalized size = 0.46 \[ \frac{107121749031 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (276480000 x^6+1479168000 x^5+3687379200 x^4+5945485120 x^3+7755469800 x^2+11734056318 x-16267424049\right )}{16384000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 174, normalized size = 1. \begin{align*} -{\frac{1}{65536000\,x-32768000} \left ( -5529600000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-29583360000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-73747584000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-118909702400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+214243498062\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-155109396000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-107121749031\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -234681126360\,x\sqrt{-10\,{x}^{2}-x+3}+325348480980\,\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 = 3.8259, size = 193, normalized size = 1.05 \begin{align*} -\frac{3375 \, x^{7}}{4 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{80325 \, x^{6}}{16 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3574125 \, x^{5}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{25493477 \, x^{4}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1415345109 \, x^{3}}{40960 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{8193669099 \, x^{2}}{163840 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{107121749031}{32768000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{46134951291 \, x}{1638400 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{48802272147}{1638400 \, \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 = 1.77867, size = 382, normalized size = 2.09 \begin{align*} \frac{107121749031 \, \sqrt{10}{\left (2 \, 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 (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{32768000 \,{\left (2 \, 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 = 2.32109, size = 166, normalized size = 0.91 \begin{align*} -\frac{107121749031}{16384000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 35 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4299 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3832457 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 295101237 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16230568035 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 535608745155 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{204800000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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