Optimal. Leaf size=85 \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.187728, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)),x]
[Out]
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Rubi in Sympy [A] time = 20.724, size = 73, normalized size = 0.86 \[ - \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{45} - \frac{272 \sqrt{- 2 x + 1}}{225} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{27} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)/(2+3*x)/(3+5*x),x)
[Out]
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Mathematica [A] time = 0.113769, size = 71, normalized size = 0.84 \[ \frac{2 \left (30 \sqrt{1-2 x} (10 x-73)+6125 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-3267 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{3375} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)),x]
[Out]
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Maple [A] time = 0.014, size = 56, normalized size = 0.7 \[ -{\frac{4}{45} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{242\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{98\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{272}{225}\sqrt{1-2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)/(2+3*x)/(3+5*x),x)
[Out]
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Maxima [A] time = 1.47872, size = 123, normalized size = 1.45 \[ -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{272}{225} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)*(3*x + 2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217301, size = 144, normalized size = 1.69 \[ \frac{1}{3375} \, \sqrt{5} \sqrt{3}{\left (4 \, \sqrt{5} \sqrt{3}{\left (10 \, x - 73\right )} \sqrt{-2 \, x + 1} + 1089 \, \sqrt{11} \sqrt{3} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1225 \, \sqrt{7} \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)*(3*x + 2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.0773, size = 151, normalized size = 1.78 \[ - \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{45} - \frac{272 \sqrt{- 2 x + 1}}{225} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)/(2+3*x)/(3+5*x),x)
[Out]
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GIAC/XCAS [A] time = 0.215343, size = 131, normalized size = 1.54 \[ -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{49}{27} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{272}{225} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)*(3*x + 2)),x, algorithm="giac")
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