Optimal. Leaf size=107 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (110981 x+70429)}{222264 (3 x+2)^2}+\frac{328715 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.158053, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (110981 x+70429)}{222264 (3 x+2)^2}+\frac{328715 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 18.7503, size = 95, normalized size = 0.89 \[ - \frac{\sqrt{- 2 x + 1} \left (1664715 x + 1056435\right )}{666792 \left (3 x + 2\right )^{2}} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{378 \left (3 x + 2\right )^{3}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{12 \left (3 x + 2\right )^{4}} + \frac{328715 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2333772} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**5,x)
[Out]
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Mathematica [A] time = 0.123023, size = 63, normalized size = 0.59 \[ \frac{657430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} \left (9646695 x^3+18358575 x^2+11657098 x+2469626\right )}{(3 x+2)^4}}{4667544} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^5,x]
[Out]
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Maple [A] time = 0.016, size = 66, normalized size = 0.6 \[ -324\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{119095\, \left ( 1-2\,x \right ) ^{7/2}}{444528}}+{\frac{3126535\, \left ( 1-2\,x \right ) ^{5/2}}{1714608}}-{\frac{3040873\, \left ( 1-2\,x \right ) ^{3/2}}{734832}}+{\frac{328715\,\sqrt{1-2\,x}}{104976}} \right ) }+{\frac{328715\,\sqrt{21}}{2333772}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^5,x)
[Out]
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Maxima [A] time = 1.51911, size = 149, normalized size = 1.39 \[ -\frac{328715}{4667544} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{9646695 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 65657235 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 149002777 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 112749245 \, \sqrt{-2 \, x + 1}}{111132 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211019, size = 140, normalized size = 1.31 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (9646695 \, x^{3} + 18358575 \, x^{2} + 11657098 \, x + 2469626\right )} \sqrt{-2 \, x + 1} - 328715 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{4667544 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^5,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.226706, size = 135, normalized size = 1.26 \[ -\frac{328715}{4667544} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9646695 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 65657235 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 149002777 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 112749245 \, \sqrt{-2 \, x + 1}}{1778112 \,{\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^5,x, algorithm="giac")
[Out]