Optimal. Leaf size=121 \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]
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Rubi [A] time = 0.143712, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 13.8666, size = 105, normalized size = 0.87 \[ \frac{31 \left (- 2 x + 1\right )^{\frac{7}{2}}}{588 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{252 \left (3 x + 2\right )^{4}} - \frac{4993 \left (- 2 x + 1\right )^{\frac{5}{2}}}{10584 \left (3 x + 2\right )^{2}} + \frac{24965 \left (- 2 x + 1\right )^{\frac{3}{2}}}{31752 \left (3 x + 2\right )} + \frac{24965 \sqrt{- 2 x + 1}}{15876} - \frac{24965 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{47628} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**5,x)
[Out]
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Mathematica [A] time = 0.131967, size = 68, normalized size = 0.56 \[ \frac{\frac{21 \sqrt{1-2 x} \left (302400 x^4+1231065 x^3+1526937 x^2+762598 x+134558\right )}{(3 x+2)^4}-49930 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{95256} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^5,x]
[Out]
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Maple [A] time = 0.019, size = 75, normalized size = 0.6 \[{\frac{200}{243}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{47185}{672} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{129289}{288} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{824705}{864} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1749055}{2592}\sqrt{1-2\,x}} \right ) }-{\frac{24965\,\sqrt{21}}{47628}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^5,x)
[Out]
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Maxima [A] time = 1.50568, size = 161, normalized size = 1.33 \[ \frac{24965}{95256} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{200}{243} \, \sqrt{-2 \, x + 1} - \frac{1273995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 8145207 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12243385 \, \sqrt{-2 \, x + 1}}{6804 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^5,x, algorithm="maxima")
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Fricas [A] time = 0.215477, size = 147, normalized size = 1.21 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (302400 \, x^{4} + 1231065 \, x^{3} + 1526937 \, x^{2} + 762598 \, x + 134558\right )} \sqrt{-2 \, x + 1} + 24965 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{95256 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(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((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.215345, size = 147, normalized size = 1.21 \[ \frac{24965}{95256} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{200}{243} \, \sqrt{-2 \, x + 1} + \frac{1273995 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 8145207 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12243385 \, \sqrt{-2 \, x + 1}}{108864 \,{\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^5,x, algorithm="giac")
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