3.1920 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=96 \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-365*Sqrt[1 - 2*x])/81 - (365*(1 - 2*x)^(3/2))/567 + (1 - 2*x)^(7/2)/(42*(2 + 3
*x)^2) - (73*(1 - 2*x)^(5/2))/(126*(2 + 3*x)) + (365*Sqrt[7/3]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/81

_______________________________________________________________________________________

Rubi [A]  time = 0.10505, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^3,x]

[Out]

(-365*Sqrt[1 - 2*x])/81 - (365*(1 - 2*x)^(3/2))/567 + (1 - 2*x)^(7/2)/(42*(2 + 3
*x)^2) - (73*(1 - 2*x)^(5/2))/(126*(2 + 3*x)) + (365*Sqrt[7/3]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/81

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4319, size = 80, normalized size = 0.83 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{42 \left (3 x + 2\right )^{2}} - \frac{73 \left (- 2 x + 1\right )^{\frac{5}{2}}}{126 \left (3 x + 2\right )} - \frac{365 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567} - \frac{365 \sqrt{- 2 x + 1}}{81} + \frac{365 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**3,x)

[Out]

(-2*x + 1)**(7/2)/(42*(3*x + 2)**2) - 73*(-2*x + 1)**(5/2)/(126*(3*x + 2)) - 365
*(-2*x + 1)**(3/2)/567 - 365*sqrt(-2*x + 1)/81 + 365*sqrt(21)*atanh(sqrt(21)*sqr
t(-2*x + 1)/7)/243

_______________________________________________________________________________________

Mathematica [A]  time = 0.104901, size = 63, normalized size = 0.66 \[ \frac{1}{486} \left (\frac{3 \sqrt{1-2 x} \left (720 x^3-4584 x^2-8731 x-3521\right )}{(3 x+2)^2}+730 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^3,x]

[Out]

((3*Sqrt[1 - 2*x]*(-3521 - 8731*x - 4584*x^2 + 720*x^3))/(2 + 3*x)^2 + 730*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/486

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 66, normalized size = 0.7 \[ -{\frac{20}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{32}{9}\sqrt{1-2\,x}}-{\frac{28}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{79}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{539}{108}\sqrt{1-2\,x}} \right ) }+{\frac{365\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^3,x)

[Out]

-20/81*(1-2*x)^(3/2)-32/9*(1-2*x)^(1/2)-28/3*(-79/36*(1-2*x)^(3/2)+539/108*(1-2*
x)^(1/2))/(-4-6*x)^2+365/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50192, size = 124, normalized size = 1.29 \[ -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="maxima")

[Out]

-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 32/9*sqrt(-2*x + 1) + 7/81*(237*(-2*x + 1)^(3/2)
- 539*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

_______________________________________________________________________________________

Fricas [A]  time = 0.213401, size = 122, normalized size = 1.27 \[ \frac{\sqrt{3}{\left (365 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\right )} \sqrt{-2 \, x + 1}\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="fricas")

[Out]

1/486*sqrt(3)*(365*sqrt(7)*(9*x^2 + 12*x + 4)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)
*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(3)*(720*x^3 - 4584*x^2 - 8731*x - 3521)*sqrt(
-2*x + 1))/(9*x^2 + 12*x + 4)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212099, size = 116, normalized size = 1.21 \[ -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="giac")

[Out]

-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 32/9*sqrt(-2*x + 1) + 7/324*(237*(-2*x +
1)^(3/2) - 539*sqrt(-2*x + 1))/(3*x + 2)^2