Optimal. Leaf size=128 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
[Out]
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Rubi [A] time = 0.20574, antiderivative size = 128, 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 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 27.9593, size = 109, normalized size = 0.85 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}}{5 \left (5 x + 3\right )} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{75} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{875} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (1153845 x + 1788570\right )}{2953125} + \frac{258 \sqrt{- 2 x + 1}}{15625} - \frac{258 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.128429, size = 78, normalized size = 0.61 \[ -\frac{5 \sqrt{1-2 x} \left (787500 x^5+1395000 x^4+157275 x^3-924335 x^2-143235 x+161312\right )+1806 \sqrt{55} (5 x+3) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{546875 (5 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]
[Out]
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Maple [A] time = 0.016, size = 81, normalized size = 0.6 \[ -{\frac{9}{100} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{999}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{12393}{12500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{8}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{52}{3125}\sqrt{1-2\,x}}+{\frac{22}{78125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{258\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x)
[Out]
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Maxima [A] time = 1.49627, size = 132, normalized size = 1.03 \[ -\frac{9}{100} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{999}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{12393}{12500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21365, size = 123, normalized size = 0.96 \[ \frac{\sqrt{5}{\left (903 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt{-2 \, x + 1}\right )}}{546875 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216547, size = 165, normalized size = 1.29 \[ -\frac{9}{100} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{999}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{12393}{12500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="giac")
[Out]