Optimal. Leaf size=95 \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.123307, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.586, size = 83, normalized size = 0.87 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{9}{2}}}{54} - \frac{155 \left (- 2 x + 1\right )^{\frac{7}{2}}}{126} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{135} + \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} + \frac{98 \sqrt{- 2 x + 1}}{243} - \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{729} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0850716, size = 61, normalized size = 0.64 \[ \frac{3 \sqrt{1-2 x} \left (63000 x^4-42300 x^3-30546 x^2+29791 x-2479\right )-3430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{25515} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 65, normalized size = 0.7 \[{\frac{14}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{135} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{155}{126} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{25}{54} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{98\,\sqrt{21}}{729}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{98}{243}\sqrt{1-2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.50745, size = 111, normalized size = 1.17 \[ \frac{25}{54} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{155}{126} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{135} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{98}{243} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22473, size = 99, normalized size = 1.04 \[ \frac{1}{25515} \, \sqrt{3}{\left (\sqrt{3}{\left (63000 \, x^{4} - 42300 \, x^{3} - 30546 \, x^{2} + 29791 \, x - 2479\right )} \sqrt{-2 \, x + 1} + 1715 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 13.7211, size = 122, normalized size = 1.28 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{9}{2}}}{54} - \frac{155 \left (- 2 x + 1\right )^{\frac{7}{2}}}{126} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{135} + \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} + \frac{98 \sqrt{- 2 x + 1}}{243} + \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.212845, size = 143, normalized size = 1.51 \[ \frac{25}{54} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{155}{126} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{135} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{98}{243} \, \sqrt{-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2),x, algorithm="giac")
[Out]