Optimal. Leaf size=125 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]
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Rubi [A] time = 0.205132, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 20.1066, size = 104, normalized size = 0.83 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (500850 x + 358146\right )}{71442 \left (3 x + 2\right )} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{9 \left (3 x + 2\right )^{3}} + \frac{36038 \sqrt{- 2 x + 1}}{3969} - \frac{36038 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{11907} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**4,x)
[Out]
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Mathematica [A] time = 0.128696, size = 68, normalized size = 0.54 \[ \frac{\sqrt{1-2 x} \left (-31500 x^4+81900 x^3+259614 x^2+199243 x+47939\right )}{567 (3 x+2)^3}-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.6 \[{\frac{250}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2050}{243}\sqrt{1-2\,x}}+{\frac{2}{9\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3938}{21} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{23306}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{26824}{27}\sqrt{1-2\,x}} \right ) }-{\frac{36038\,\sqrt{21}}{11907}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^4,x)
[Out]
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Maxima [A] time = 1.50661, size = 149, normalized size = 1.19 \[ \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{4 \,{\left (17721 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}\right )}}{1701 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*(-2*x + 1)^(3/2)/(3*x + 2)^4,x, algorithm="maxima")
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Fricas [A] time = 0.212197, size = 134, normalized size = 1.07 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (31500 \, x^{4} - 81900 \, x^{3} - 259614 \, x^{2} - 199243 \, x - 47939\right )} \sqrt{-2 \, x + 1} - 18019 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{11907 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*(-2*x + 1)^(3/2)/(3*x + 2)^4,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((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.221543, size = 138, normalized size = 1.1 \[ \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{17721 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}}{3402 \,{\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*(-2*x + 1)^(3/2)/(3*x + 2)^4,x, algorithm="giac")
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