Optimal. Leaf size=76 \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
[Out]
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Rubi [A] time = 0.0871042, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{60}{343 \sqrt{1-2 x}}+\frac{1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac{20}{147 (1-2 x)^{3/2}}-\frac{60}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 8.37323, size = 65, normalized size = 0.86 \[ - \frac{60 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{60}{343 \sqrt{- 2 x + 1}} + \frac{20}{147 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{1}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)/(1-2*x)**(5/2)/(2+3*x)**2,x)
[Out]
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Mathematica [A] time = 0.116484, size = 58, normalized size = 0.76 \[ \frac{\frac{7 \left (-1080 x^2+240 x+689\right )}{(1-2 x)^{3/2} (3 x+2)}-180 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)/((1 - 2*x)^(5/2)*(2 + 3*x)^2),x]
[Out]
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Maple [A] time = 0.018, size = 54, normalized size = 0.7 \[{\frac{22}{147} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{62}{343}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{60\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)/(1-2*x)^(5/2)/(2+3*x)^2,x)
[Out]
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Maxima [A] time = 1.4917, size = 100, normalized size = 1.32 \[ \frac{30}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (270 \,{\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2222, size = 119, normalized size = 1.57 \[ \frac{\sqrt{7}{\left (90 \, \sqrt{3}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1080 \, x^{2} - 240 \, x - 689\right )}\right )}}{7203 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)/(1-2*x)**(5/2)/(2+3*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.244749, size = 104, normalized size = 1.37 \[ \frac{30}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (93 \, x - 85\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{3 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)/((3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]