Optimal. Leaf size=143 \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.1656, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x],x]
[Out]
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Rubi in Sympy [A] time = 14.1562, size = 129, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (9 x + 6\right )}{50} - \frac{567 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4000} + \frac{4123 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{24000} - \frac{45353 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{64000} + \frac{498883 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{640000} + \frac{5487713 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6400000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**2*(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.110383, size = 70, normalized size = 0.49 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+7286400 x^3-3141280 x^2-4872460 x+382101\right )-16463139 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{19200000} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x],x]
[Out]
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Maple [A] time = 0.013, size = 121, normalized size = 0.9 \[{\frac{1}{38400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-145728000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+62825600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+16463139\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +97449200\,x\sqrt{-10\,{x}^{2}-x+3}-7642020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.49271, size = 117, normalized size = 0.82 \[ \frac{9}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{687}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2159}{24000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{45353}{32000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{5487713}{12800000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{45353}{640000} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224981, size = 97, normalized size = 0.68 \[ -\frac{1}{38400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 16463139 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 30.8409, size = 490, normalized size = 3.43 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(3/2)*(2+3*x)**2*(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.253068, size = 317, normalized size = 2.22 \[ -\frac{3}{32000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{128000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2),x, algorithm="giac")
[Out]