Optimal. Leaf size=80 \[ \frac{81}{280} (1-2 x)^{7/2}-\frac{2889 (1-2 x)^{5/2}}{1000}+\frac{11457 (1-2 x)^{3/2}}{1000}-\frac{136419 \sqrt{1-2 x}}{5000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]
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Rubi [A] time = 0.0940398, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{81}{280} (1-2 x)^{7/2}-\frac{2889 (1-2 x)^{5/2}}{1000}+\frac{11457 (1-2 x)^{3/2}}{1000}-\frac{136419 \sqrt{1-2 x}}{5000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)),x]
[Out]
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Rubi in Sympy [A] time = 9.00939, size = 71, normalized size = 0.89 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{7}{2}}}{280} - \frac{2889 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1000} + \frac{11457 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1000} - \frac{136419 \sqrt{- 2 x + 1}}{5000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{34375} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**4/(3+5*x)/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0884446, size = 56, normalized size = 0.7 \[ -\frac{3 \sqrt{1-2 x} \left (3375 x^3+11790 x^2+19095 x+26872\right )}{4375}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)),x]
[Out]
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Maple [A] time = 0.01, size = 56, normalized size = 0.7 \[{\frac{11457}{1000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2889}{1000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{81}{280} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{34375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{136419}{5000}\sqrt{1-2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^4/(3+5*x)/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.55474, size = 99, normalized size = 1.24 \[ \frac{81}{280} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2889}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{11457}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{34375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{136419}{5000} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255179, size = 85, normalized size = 1.06 \[ -\frac{1}{240625} \, \sqrt{55}{\left (3 \, \sqrt{55}{\left (3375 \, x^{3} + 11790 \, x^{2} + 19095 \, x + 26872\right )} \sqrt{-2 \, x + 1} - 7 \, \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)*sqrt(-2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.19791, size = 114, normalized size = 1.42 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{7}{2}}}{280} - \frac{2889 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1000} + \frac{11457 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1000} - \frac{136419 \sqrt{- 2 x + 1}}{5000} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**4/(3+5*x)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236888, size = 122, normalized size = 1.52 \[ -\frac{81}{280} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2889}{1000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{11457}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{34375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{136419}{5000} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]