Optimal. Leaf size=128 \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.125811, 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 \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 13.1683, size = 109, normalized size = 0.85 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{105 \left (3 x + 2\right )^{5}} + \frac{\sqrt{- 2 x + 1}}{1029 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1}}{441 \left (3 x + 2\right )^{2}} + \frac{2 \sqrt{- 2 x + 1}}{315 \left (3 x + 2\right )^{3}} - \frac{2 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{4}} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)*(1-2*x)**(1/2)/(2+3*x)**6,x)
[Out]
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Mathematica [A] time = 0.100768, size = 68, normalized size = 0.53 \[ \frac{\frac{21 \sqrt{1-2 x} \left (405 x^4+1395 x^3+2004 x^2-864 x-1019\right )}{(3 x+2)^5}+10 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{108045} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^6,x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.6 \[ 7776\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{9/2}}{49392}}+{\frac{ \left ( 1-2\,x \right ) ^{7/2}}{4536}}-{\frac{8\, \left ( 1-2\,x \right ) ^{5/2}}{8505}}+{\frac{13\, \left ( 1-2\,x \right ) ^{3/2}}{13608}}+{\frac{7\,\sqrt{1-2\,x}}{11664}} \right ) }+{\frac{2\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)*(1-2*x)^(1/2)/(2+3*x)^6,x)
[Out]
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Maxima [A] time = 1.47698, size = 173, normalized size = 1.35 \[ -\frac{1}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (405 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4410 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 18816 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}\right )}}{5145 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213214, size = 161, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (405 \, x^{4} + 1395 \, x^{3} + 2004 \, x^{2} - 864 \, x - 1019\right )} \sqrt{-2 \, x + 1} + 5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{108045 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*sqrt(-2*x + 1)/(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((3+5*x)*(1-2*x)**(1/2)/(2+3*x)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.219379, size = 157, normalized size = 1.23 \[ -\frac{1}{21609} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{405 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 4410 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 18816 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}}{82320 \,{\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="giac")
[Out]