Optimal. Leaf size=82 \[ -\frac{5}{21} (1-2 x)^{7/2}-\frac{2}{45} (1-2 x)^{5/2}-\frac{14}{81} (1-2 x)^{3/2}-\frac{98}{81} \sqrt{1-2 x}+\frac{98}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0954778, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5}{21} (1-2 x)^{7/2}-\frac{2}{45} (1-2 x)^{5/2}-\frac{14}{81} (1-2 x)^{3/2}-\frac{98}{81} \sqrt{1-2 x}+\frac{98}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x),x]
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Rubi in Sympy [A] time = 9.16496, size = 71, normalized size = 0.87 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{7}{2}}}{21} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{45} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{98 \sqrt{- 2 x + 1}}{81} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x),x)
[Out]
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Mathematica [A] time = 0.0685167, size = 56, normalized size = 0.68 \[ \frac{3 \sqrt{1-2 x} \left (5400 x^3-8604 x^2+5534 x-4721\right )+3430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{8505} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x),x]
[Out]
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Maple [A] time = 0.008, size = 56, normalized size = 0.7 \[ -{\frac{14}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2}{45} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{21} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{98\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{98}{81}\sqrt{1-2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)/(2+3*x),x)
[Out]
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Maxima [A] time = 1.50144, size = 99, normalized size = 1.21 \[ -\frac{5}{21} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2}{45} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{14}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{98}{81} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217915, size = 92, normalized size = 1.12 \[ \frac{1}{8505} \, \sqrt{3}{\left (\sqrt{3}{\left (5400 \, x^{3} - 8604 \, x^{2} + 5534 \, x - 4721\right )} \sqrt{-2 \, x + 1} + 1715 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.0818, size = 112, normalized size = 1.37 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{7}{2}}}{21} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{45} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{98 \sqrt{- 2 x + 1}}{81} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x),x)
[Out]
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GIAC/XCAS [A] time = 0.210562, size = 122, normalized size = 1.49 \[ \frac{5}{21} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2}{45} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{14}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{98}{81} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="giac")
[Out]