Optimal. Leaf size=108 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^3}{5 (5 x+3)}+\frac{27}{175} (1-2 x)^{3/2} (3 x+2)^2-\frac{6}{625} (1-2 x)^{3/2} (9 x+29)+\frac{192 \sqrt{1-2 x}}{3125}-\frac{192 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]
[Out]
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Rubi [A] time = 0.15843, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^3}{5 (5 x+3)}+\frac{27}{175} (1-2 x)^{3/2} (3 x+2)^2-\frac{6}{625} (1-2 x)^{3/2} (9 x+29)+\frac{192 \sqrt{1-2 x}}{3125}-\frac{192 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(3 + 5*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 21.0747, size = 90, normalized size = 0.83 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{5 \left (5 x + 3\right )} + \frac{27 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{175} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5670 x + 18270\right )}{65625} + \frac{192 \sqrt{- 2 x + 1}}{3125} - \frac{192 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**3/(3+5*x)**2,x)
[Out]
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Mathematica [A] time = 0.10631, size = 68, normalized size = 0.63 \[ \frac{-\frac{5 \sqrt{1-2 x} \left (67500 x^4+62100 x^3-57165 x^2-27640 x+8738\right )}{5 x+3}-1344 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{109375} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(3 + 5*x)^2,x]
[Out]
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Maple [A] time = 0.016, size = 72, normalized size = 0.7 \[{\frac{27}{350} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{351}{1250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{6}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{194}{3125}\sqrt{1-2\,x}}+{\frac{22}{15625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{192\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^3/(3+5*x)^2,x)
[Out]
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Maxima [A] time = 1.53305, size = 120, normalized size = 1.11 \[ \frac{27}{350} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{351}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{6}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{96}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{194}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213756, size = 116, normalized size = 1.07 \[ \frac{\sqrt{5}{\left (672 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (67500 \, x^{4} + 62100 \, x^{3} - 57165 \, x^{2} - 27640 \, x + 8738\right )} \sqrt{-2 \, x + 1}\right )}}{109375 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-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)**3/(3+5*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216055, size = 143, normalized size = 1.32 \[ -\frac{27}{350} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{351}{1250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{6}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{96}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{194}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(3/2)/(5*x + 3)^2,x, algorithm="giac")
[Out]