Optimal. Leaf size=70 \[ -\frac{15 \sqrt{1-2 x}}{121 (5 x+3)}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0609609, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{15 \sqrt{1-2 x}}{121 (5 x+3)}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(3 + 5*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 5.93499, size = 56, normalized size = 0.8 \[ - \frac{15 \sqrt{- 2 x + 1}}{121 \left (5 x + 3\right )} - \frac{6 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{2}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(3+5*x)**2,x)
[Out]
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Mathematica [A] time = 0.0863881, size = 56, normalized size = 0.8 \[ \frac{-\frac{11 \sqrt{1-2 x} (30 x+7)}{10 x^2+x-3}-6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(3 + 5*x)^2),x]
[Out]
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Maple [A] time = 0.015, size = 45, normalized size = 0.6 \[{\frac{4}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{121}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{6\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(3+5*x)^2,x)
[Out]
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Maxima [A] time = 1.52113, size = 88, normalized size = 1.26 \[ \frac{3}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.250521, size = 104, normalized size = 1.49 \[ \frac{\sqrt{11}{\left (3 \, \sqrt{5}{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (30 \, x + 7\right )}\right )}}{1331 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.32505, size = 177, normalized size = 2.53 \[ \begin{cases} - \frac{6 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} + \frac{3 \sqrt{2}}{121 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{110 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{6 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} - \frac{3 \sqrt{2} i}{121 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{110 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-2*x)**(3/2)/(3+5*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217487, size = 92, normalized size = 1.31 \[ \frac{3}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(3/2)),x, algorithm="giac")
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