Optimal. Leaf size=68 \[ -\frac{67 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{\sqrt{1-2 x}}{42 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.0691723, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{67 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{\sqrt{1-2 x}}{42 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 6.89843, size = 56, normalized size = 0.82 \[ - \frac{67 \sqrt{- 2 x + 1}}{294 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1}}{42 \left (3 x + 2\right )^{2}} - \frac{67 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3087} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)/(2+3*x)**3/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0951428, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{1-2 x} (201 x+127)}{294 (3 x+2)^2}-\frac{67 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
[Out]
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Maple [A] time = 0.015, size = 48, normalized size = 0.7 \[ -36\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{67\, \left ( 1-2\,x \right ) ^{3/2}}{1764}}+{\frac{65\,\sqrt{1-2\,x}}{756}} \right ) }-{\frac{67\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)/(2+3*x)^3/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50663, size = 100, normalized size = 1.47 \[ \frac{67}{6174} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{201 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 455 \, \sqrt{-2 \, x + 1}}{147 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23157, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (201 \, x + 127\right )} \sqrt{-2 \, x + 1} - 67 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{6174 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)/(2+3*x)**3/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220194, size = 92, normalized size = 1.35 \[ \frac{67}{6174} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{201 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 455 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]