Optimal. Leaf size=113 \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
[Out]
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Rubi [A] time = 0.243521, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]
[Out]
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Rubi in Sympy [A] time = 28.1951, size = 100, normalized size = 0.88 \[ \frac{1138 \sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )} + \frac{49 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{2}} + \frac{7 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{3}} + \frac{78506 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} - 110 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x),x)
[Out]
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Mathematica [A] time = 0.182168, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (10242 x^2+13999 x+4797\right )}{21 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.7 \[ -54\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1138\, \left ( 1-2\,x \right ) ^{5/2}}{63}}-{\frac{6926\, \left ( 1-2\,x \right ) ^{3/2}}{81}}+{\frac{8204\,\sqrt{1-2\,x}}{81}} \right ) }+{\frac{78506\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-110\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x)
[Out]
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Maxima [A] time = 1.50485, size = 173, normalized size = 1.53 \[ 55 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39253}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (5121 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}\right )}}{21 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235447, size = 185, normalized size = 1.64 \[ \frac{\sqrt{21}{\left (1155 \, \sqrt{55} \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (10242 \, x^{2} + 13999 \, x + 4797\right )} \sqrt{-2 \, x + 1} + 39253 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{441 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)*(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)/(2+3*x)**4/(3+5*x),x)
[Out]
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GIAC/XCAS [A] time = 0.214848, size = 166, normalized size = 1.47 \[ 55 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{39253}{441} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{5121 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}}{42 \,{\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^4),x, algorithm="giac")
[Out]