3.1906 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{50 (5 x+3)}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

[Out]

-(1 - 2*x)^(3/2)/(10*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(50*(3 + 5*x)) - (3*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.0531767, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{(1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{50 (5 x+3)}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(3/2)/(10*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(50*(3 + 5*x)) - (3*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.22817, size = 56, normalized size = 0.82 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{10 \left (5 x + 3\right )^{2}} + \frac{3 \sqrt{- 2 x + 1}}{50 \left (5 x + 3\right )} - \frac{3 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/(10*(5*x + 3)**2) + 3*sqrt(-2*x + 1)/(50*(5*x + 3)) - 3*sqrt(
55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1375

_______________________________________________________________________________________

Mathematica [A]  time = 0.0756638, size = 53, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} (25 x+4)}{50 (5 x+3)^2}-\frac{3 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(4 + 25*x))/(50*(3 + 5*x)^2) - (3*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(25*Sqrt[55])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 48, normalized size = 0.7 \[ 200\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{200}}+{\frac{33\,\sqrt{1-2\,x}}{5000}} \right ) }-{\frac{3\,\sqrt{55}}{1375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(3+5*x)^3,x)

[Out]

200*(-1/200*(1-2*x)^(3/2)+33/5000*(1-2*x)^(1/2))/(-6-10*x)^2-3/1375*arctanh(1/11
*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50583, size = 100, normalized size = 1.47 \[ \frac{3}{2750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 33 \, \sqrt{-2 \, x + 1}}{25 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 1/25*(25*(-2*x + 1)^(3/2) - 33*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

_______________________________________________________________________________________

Fricas [A]  time = 0.216255, size = 100, normalized size = 1.47 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (25 \, x + 4\right )} \sqrt{-2 \, x + 1} + 3 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{2750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2750*sqrt(55)*(sqrt(55)*(25*x + 4)*sqrt(-2*x + 1) + 3*(25*x^2 + 30*x + 9)*log(
(sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 30*x + 9)

_______________________________________________________________________________________

Sympy [A]  time = 4.26169, size = 236, normalized size = 3.47 \[ \begin{cases} - \frac{3 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1375} - \frac{\sqrt{2}}{50 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{77 \sqrt{2}}{2500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} - \frac{121 \sqrt{2}}{12500 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{3 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1375} + \frac{\sqrt{2} i}{50 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{77 \sqrt{2} i}{2500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} + \frac{121 \sqrt{2} i}{12500 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/1375 - sqrt(2)/(50*sq
rt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) + 77*sqrt(2)/(2500*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(3/2)) - 121*sqrt(2)/(12500*sqrt(-1 + 11/(10*(x + 3/5)))*(x
+ 3/5)**(5/2)), 11*Abs(1/(x + 3/5))/10 > 1), (3*sqrt(55)*I*asin(sqrt(110)/(10*s
qrt(x + 3/5)))/1375 + sqrt(2)*I/(50*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) -
77*sqrt(2)*I/(2500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + 121*sqrt(2)*
I/(12500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211105, size = 92, normalized size = 1.35 \[ \frac{3}{2750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 33 \, \sqrt{-2 \, x + 1}}{100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 1/100*(25*(-2*x + 1)^(3/2) - 33*sqrt(-2*x + 1))/(5*x + 3)^2