3.2434 \(\int \frac{(1-2 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{2 (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(5/2))/(15*(3 + 5*x)^(3/2)) + (4*(1 - 2*x)^(3/2))/(15*Sqrt[3 + 5*x
]) + (4*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 + (22*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/25

_______________________________________________________________________________________

Rubi [A]  time = 0.084375, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2 (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{4 (1-2 x)^{3/2}}{15 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2))/(15*(3 + 5*x)^(3/2)) + (4*(1 - 2*x)^(3/2))/(15*Sqrt[3 + 5*x
]) + (4*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 + (22*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.91471, size = 85, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \sqrt{5 x + 3}} + \frac{4 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{25} + \frac{22 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/(15*(5*x + 3)**(3/2)) + 4*(-2*x + 1)**(3/2)/(15*sqrt(5*x +
3)) + 4*sqrt(-2*x + 1)*sqrt(5*x + 3)/25 + 22*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3
)/11)/125

_______________________________________________________________________________________

Mathematica [A]  time = 0.16204, size = 60, normalized size = 0.62 \[ \frac{2}{375} \left (\frac{5 \sqrt{1-2 x} \left (30 x^2+190 x+79\right )}{(5 x+3)^{3/2}}-33 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((5*Sqrt[1 - 2*x]*(79 + 190*x + 30*x^2))/(3 + 5*x)^(3/2) - 33*Sqrt[10]*ArcSin
[Sqrt[5/11]*Sqrt[1 - 2*x]]))/375

_______________________________________________________________________________________

Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{1 \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.49762, size = 174, normalized size = 1.81 \[ \frac{11}{125} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{30 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{150 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{77 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/125*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/5*(-10*x^2 - x + 3)^(5/2)/(625
*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) - 11/30*(-10*x^2 - x + 3)^(3/2)/(125*x^
3 + 225*x^2 + 135*x + 27) - 121/150*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) +
77/75*sqrt(-10*x^2 - x + 3)/(5*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.219522, size = 122, normalized size = 1.27 \[ \frac{\sqrt{5}{\left (2 \, \sqrt{5}{\left (30 \, x^{2} + 190 \, x + 79\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 33 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/375*sqrt(5)*(2*sqrt(5)*(30*x^2 + 190*x + 79)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 33
*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))/(25*x^2 + 30*x + 9)

_______________________________________________________________________________________

Sympy [A]  time = 29.1668, size = 258, normalized size = 2.69 \[ \begin{cases} \frac{4 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{125} + \frac{308 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875} - \frac{242 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{9375 \left (x + \frac{3}{5}\right )} + \frac{11 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} + \frac{11 \sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{22 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\\frac{4 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{125} + \frac{308 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875} - \frac{242 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{9375 \left (x + \frac{3}{5}\right )} + \frac{11 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} - \frac{22 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((4*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/125 + 308*sqrt(10)*
sqrt(-1 + 11/(10*(x + 3/5)))/1875 - 242*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(9
375*(x + 3/5)) + 11*sqrt(10)*I*log(1/(x + 3/5))/125 + 11*sqrt(10)*I*log(x + 3/5)
/125 + 22*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/125, 11*Abs(1/(x + 3/5))/10
> 1), (4*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/125 + 308*sqrt(10)*I*s
qrt(1 - 11/(10*(x + 3/5)))/1875 - 242*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(93
75*(x + 3/5)) + 11*sqrt(10)*I*log(1/(x + 3/5))/125 - 22*sqrt(10)*I*log(sqrt(1 -
11/(10*(x + 3/5))) + 1)/125, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288613, size = 220, normalized size = 2.29 \[ -\frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{30000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{4}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{22}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{99 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2500 \, \sqrt{5 \, x + 3}} - \frac{11 \,{\left (\frac{27 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/30000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 4/62
5*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 22/125*sqrt(10)*arcsin(1/11*sqrt(22)*s
qrt(5*x + 3)) + 99/2500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x +
3) - 11/1875*(27*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4*
sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3