Optimal. Leaf size=128 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]
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Rubi [A] time = 0.036617, antiderivative size = 128, 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 = {100, 147, 50, 54, 216} \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 100
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x} \, dx &=-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{1}{50} \int \left (-245-\frac{777 x}{2}\right ) \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x} \, dx\\ &=-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{323491 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{32000}\\ &=-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{3558401 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{256000}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2560000}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1280000 \sqrt{5}}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1280000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.152811, size = 79, normalized size = 0.62 \[ -\frac{10 \sqrt{5 x+3} \left (13824000 x^5+27820800 x^4+12527040 x^3-8941640 x^2-11567238 x+4282349\right )+39142411 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12800000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 121, normalized size = 1. \begin{align*}{\frac{1}{25600000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+347328000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+298934400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+39142411\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +60050800\,x\sqrt{-10\,{x}^{2}-x+3}-85646980\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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 = 1.93748, size = 117, normalized size = 0.91 \begin{align*} -\frac{27}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{5211}{4000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{19191}{16000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{323491}{64000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{39142411}{25600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{323491}{1280000} \, \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 = 1.93203, size = 289, normalized size = 2.26 \begin{align*} \frac{1}{1280000} \,{\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{39142411}{25600000} \, \sqrt{10} \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 ) \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 [B] time = 2.3003, size = 317, normalized size = 2.48 \begin{align*} \frac{9}{64000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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