Optimal. Leaf size=108 \[ \frac{81}{440} (1-2 x)^{11/2}-\frac{321}{200} (1-2 x)^{9/2}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{2 (1-2 x)^{3/2}}{9375}+\frac{22 \sqrt{1-2 x}}{15625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
[Out]
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Rubi [A] time = 0.109813, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{81}{440} (1-2 x)^{11/2}-\frac{321}{200} (1-2 x)^{9/2}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{2 (1-2 x)^{3/2}}{9375}+\frac{22 \sqrt{1-2 x}}{15625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]
[Out]
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Rubi in Sympy [A] time = 12.1008, size = 95, normalized size = 0.88 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{11}{2}}}{440} - \frac{321 \left (- 2 x + 1\right )^{\frac{9}{2}}}{200} + \frac{34371 \left (- 2 x + 1\right )^{\frac{7}{2}}}{7000} - \frac{136419 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9375} + \frac{22 \sqrt{- 2 x + 1}}{15625} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]
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),x)
[Out]
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Mathematica [A] time = 0.10164, size = 66, normalized size = 0.61 \[ \frac{-5 \sqrt{1-2 x} \left (21262500 x^5+39532500 x^4+9559125 x^3-21433590 x^2-12144995 x+7095688\right )-5082 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18046875} \]
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 = 0.01, size = 74, normalized size = 0.7 \[{\frac{2}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{136419}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{34371}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{321}{200} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{81}{440} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{22\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{15625}\sqrt{1-2\,x}} \]
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 = 1.56452, size = 123, normalized size = 1.14 \[ \frac{81}{440} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{321}{200} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{34371}{7000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{136419}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{15625} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21145, size = 105, normalized size = 0.97 \[ -\frac{1}{18046875} \, \sqrt{5}{\left (\sqrt{5}{\left (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\right )} \sqrt{-2 \, x + 1} - 2541 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.462, size = 134, normalized size = 1.24 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{11}{2}}}{440} - \frac{321 \left (- 2 x + 1\right )^{\frac{9}{2}}}{200} + \frac{34371 \left (- 2 x + 1\right )^{\frac{7}{2}}}{7000} - \frac{136419 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9375} + \frac{22 \sqrt{- 2 x + 1}}{15625} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{15625} \]
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 = 0.214573, size = 165, normalized size = 1.53 \[ -\frac{81}{440} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{321}{200} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{34371}{7000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{136419}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{78125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{15625} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3),x, algorithm="giac")
[Out]