Optimal. Leaf size=165 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]
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Rubi [A] time = 0.0511839, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x} \, dx &=-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}-\frac{1}{60} \int \left (-186-\frac{579 x}{2}\right ) (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{7189 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{4000}\\ &=-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{79079 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{64000}\\ &=\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{869869 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{384000}\\ &=\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{9568559 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{2560000}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{25600000}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{12800000 \sqrt{5}}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{12800000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0498672, size = 75, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (230400000 x^5+94464000 x^4-237187200 x^3-61262560 x^2+102523580 x+9303927\right )-315762447 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{384000000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 138, normalized size = 0.8 \begin{align*}{\frac{1}{768000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4608000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+1889280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4743744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1225251200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+315762447\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2050471600\,x\sqrt{-10\,{x}^{2}-x+3}+186078540\,\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 = 2.64744, size = 140, normalized size = 0.85 \begin{align*} -\frac{3}{5} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{93}{500} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{18251}{40000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27893}{480000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{869869}{640000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{105254149}{256000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{869869}{12800000} \, \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 = 1.56109, size = 320, normalized size = 1.94 \begin{align*} \frac{1}{38400000} \,{\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{105254149}{256000000} \, \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 [A] time = 143.037, size = 695, normalized size = 4.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.64639, size = 427, normalized size = 2.59 \begin{align*} \frac{3}{640000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{16000000} \, \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{23}{1920000} \, \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{1}{6000} \, \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}{100} \, \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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