3.1883 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{3+5 x} \, dx\)

Optimal. Leaf size=69 \[ -\frac{3}{25} (1-2 x)^{5/2}+\frac{2}{75} (1-2 x)^{3/2}+\frac{22}{125} \sqrt{1-2 x}-\frac{22}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(22*Sqrt[1 - 2*x])/125 + (2*(1 - 2*x)^(3/2))/75 - (3*(1 - 2*x)^(5/2))/25 - (22*S
qrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

_______________________________________________________________________________________

Rubi [A]  time = 0.0730236, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3}{25} (1-2 x)^{5/2}+\frac{2}{75} (1-2 x)^{3/2}+\frac{22}{125} \sqrt{1-2 x}-\frac{22}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[1 - 2*x])/125 + (2*(1 - 2*x)^(3/2))/75 - (3*(1 - 2*x)^(5/2))/25 - (22*S
qrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.90258, size = 60, normalized size = 0.87 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{75} + \frac{22 \sqrt{- 2 x + 1}}{125} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)/25 + 2*(-2*x + 1)**(3/2)/75 + 22*sqrt(-2*x + 1)/125 - 22*sq
rt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/625

_______________________________________________________________________________________

Mathematica [A]  time = 0.0506408, size = 51, normalized size = 0.74 \[ \frac{5 \sqrt{1-2 x} \left (-180 x^2+160 x+31\right )-66 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1875} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(31 + 160*x - 180*x^2) - 66*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/1875

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 47, normalized size = 0.7 \[{\frac{2}{75} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3}{25} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{22\,\sqrt{55}}{625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*(1-2*x)^(3/2)-3/25*(1-2*x)^(5/2)-22/625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2)
)*55^(1/2)+22/125*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.48614, size = 86, normalized size = 1.25 \[ -\frac{3}{25} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/25*(-2*x + 1)^(5/2) + 2/75*(-2*x + 1)^(3/2) + 11/625*sqrt(55)*log(-(sqrt(55)
- 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/125*sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.213753, size = 85, normalized size = 1.23 \[ -\frac{1}{1875} \, \sqrt{5}{\left (\sqrt{5}{\left (180 \, x^{2} - 160 \, x - 31\right )} \sqrt{-2 \, x + 1} - 33 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1875*sqrt(5)*(sqrt(5)*(180*x^2 - 160*x - 31)*sqrt(-2*x + 1) - 33*sqrt(11)*log
((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

_______________________________________________________________________________________

Sympy [A]  time = 6.73885, size = 99, normalized size = 1.43 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{75} + \frac{22 \sqrt{- 2 x + 1}}{125} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)/25 + 2*(-2*x + 1)**(3/2)/75 + 22*sqrt(-2*x + 1)/125 + 242*P
iecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sq
rt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/125

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211935, size = 100, normalized size = 1.45 \[ -\frac{3}{25} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/25*(2*x - 1)^2*sqrt(-2*x + 1) + 2/75*(-2*x + 1)^(3/2) + 11/625*sqrt(55)*ln(1/
2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/125*s
qrt(-2*x + 1)