Optimal. Leaf size=128 \[ \frac{(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac{43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac{43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac{43 \sqrt{1-2 x}}{7938 (3 x+2)}-\frac{43 \sqrt{1-2 x}}{1134 (3 x+2)^2}+\frac{43 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.13383, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{7/2}}{105 (3 x+2)^5}-\frac{43 (1-2 x)^{5/2}}{315 (3 x+2)^4}+\frac{43 (1-2 x)^{3/2}}{567 (3 x+2)^3}+\frac{43 \sqrt{1-2 x}}{7938 (3 x+2)}-\frac{43 \sqrt{1-2 x}}{1134 (3 x+2)^2}+\frac{43 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 13.4362, size = 112, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{105 \left (3 x + 2\right )^{5}} - \frac{43 \left (- 2 x + 1\right )^{\frac{5}{2}}}{315 \left (3 x + 2\right )^{4}} + \frac{43 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567 \left (3 x + 2\right )^{3}} + \frac{43 \sqrt{- 2 x + 1}}{7938 \left (3 x + 2\right )} - \frac{43 \sqrt{- 2 x + 1}}{1134 \left (3 x + 2\right )^{2}} + \frac{43 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{83349} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**6,x)
[Out]
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Mathematica [A] time = 0.117651, size = 68, normalized size = 0.53 \[ \frac{\frac{21 \sqrt{1-2 x} \left (17415 x^4-116415 x^3-53772 x^2+3322 x-7018\right )}{(3 x+2)^5}+430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{833490} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^6,x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.6 \[ 7776\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{43\, \left ( 1-2\,x \right ) ^{9/2}}{381024}}-{\frac{37\, \left ( 1-2\,x \right ) ^{7/2}}{34992}}+{\frac{172\, \left ( 1-2\,x \right ) ^{5/2}}{32805}}-{\frac{2107\, \left ( 1-2\,x \right ) ^{3/2}}{314928}}+{\frac{2107\,\sqrt{1-2\,x}}{629856}} \right ) }+{\frac{43\,\sqrt{21}}{83349}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^6,x)
[Out]
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Maxima [A] time = 1.53027, size = 173, normalized size = 1.35 \[ -\frac{43}{166698} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17415 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 163170 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 809088 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1032430 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 516215 \, \sqrt{-2 \, x + 1}}{19845 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21532, size = 161, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (17415 \, x^{4} - 116415 \, x^{3} - 53772 \, x^{2} + 3322 \, x - 7018\right )} \sqrt{-2 \, x + 1} + 215 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{833490 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,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+3*x)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.214494, size = 157, normalized size = 1.23 \[ -\frac{43}{166698} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{17415 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 163170 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 809088 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 1032430 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 516215 \, \sqrt{-2 \, x + 1}}{635040 \,{\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="giac")
[Out]