Optimal. Leaf size=105 \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.281731, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{1370}{41503 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190}{1617 (1-2 x)^{3/2}}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)),x]
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Rubi in Sympy [A] time = 28.6037, size = 90, normalized size = 0.86 \[ \frac{720 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} - \frac{1370}{41503 \sqrt{- 2 x + 1}} - \frac{190}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{3}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(2+3*x)**2/(3+5*x),x)
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Mathematica [A] time = 0.282215, size = 105, normalized size = 1. \[ \frac{11 \left (24660 x^2-39780 x+15881\right )+257250 \sqrt{55} \sqrt{1-2 x} \left (6 x^2+x-2\right ) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1369599 (1-2 x)^{3/2} (3 x+2)}+\frac{720}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)),x]
[Out]
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Maple [A] time = 0.022, size = 72, normalized size = 0.7 \[{\frac{8}{1617} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{808}{41503}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{18}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{720\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{250\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x),x)
[Out]
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Maxima [A] time = 1.507, size = 149, normalized size = 1.42 \[ \frac{125}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{360}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (6165 \,{\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="maxima")
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Fricas [A] time = 0.229615, size = 205, normalized size = 1.95 \[ \frac{\sqrt{11} \sqrt{7}{\left (128625 \, \sqrt{7} \sqrt{5}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 130680 \, \sqrt{11} \sqrt{3}{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (24660 \, x^{2} - 39780 \, x + 15881\right )}\right )}}{9587193 \,{\left (6 \, x^{2} + x - 2\right )} \sqrt{-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="fricas")
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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(1/(1-2*x)**(5/2)/(2+3*x)**2/(3+5*x),x)
[Out]
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GIAC/XCAS [A] time = 0.217633, size = 157, normalized size = 1.5 \[ \frac{125}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{360}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (303 \, x - 190\right )}}{124509 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{27 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="giac")
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