3.1955 \(\int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx\)

Optimal. Leaf size=69 \[ \frac{2}{25} (1-2 x)^{5/2}+\frac{22}{75} (1-2 x)^{3/2}+\frac{242}{125} \sqrt{1-2 x}-\frac{242}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(242*Sqrt[1 - 2*x])/125 + (22*(1 - 2*x)^(3/2))/75 + (2*(1 - 2*x)^(5/2))/25 - (24
2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Rubi [A]  time = 0.0708609, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2}{25} (1-2 x)^{5/2}+\frac{22}{75} (1-2 x)^{3/2}+\frac{242}{125} \sqrt{1-2 x}-\frac{242}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(242*Sqrt[1 - 2*x])/125 + (22*(1 - 2*x)^(3/2))/75 + (2*(1 - 2*x)^(5/2))/25 - (24
2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Rubi in Sympy [A]  time = 7.06808, size = 60, normalized size = 0.87 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{75} + \frac{242 \sqrt{- 2 x + 1}}{125} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)/25 + 22*(-2*x + 1)**(3/2)/75 + 242*sqrt(-2*x + 1)/125 - 242*
sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/625

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Mathematica [A]  time = 0.0555151, size = 51, normalized size = 0.74 \[ \frac{2 \left (5 \sqrt{1-2 x} \left (60 x^2-170 x+433\right )-363 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{1875} \]

Antiderivative was successfully verified.

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

[Out]

(2*(5*Sqrt[1 - 2*x]*(433 - 170*x + 60*x^2) - 363*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]]))/1875

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Maple [A]  time = 0.007, size = 47, normalized size = 0.7 \[{\frac{22}{75} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{25} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{242\,\sqrt{55}}{625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(3+5*x),x)

[Out]

22/75*(1-2*x)^(3/2)+2/25*(1-2*x)^(5/2)-242/625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/
2))*55^(1/2)+242/125*(1-2*x)^(1/2)

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Maxima [A]  time = 1.48976, size = 86, normalized size = 1.25 \[ \frac{2}{25} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*(-2*x + 1)^(5/2) + 22/75*(-2*x + 1)^(3/2) + 121/625*sqrt(55)*log(-(sqrt(55)
- 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.209991, size = 86, normalized size = 1.25 \[ \frac{1}{1875} \, \sqrt{5}{\left (2 \, \sqrt{5}{\left (60 \, x^{2} - 170 \, x + 433\right )} \sqrt{-2 \, x + 1} + 363 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1875*sqrt(5)*(2*sqrt(5)*(60*x^2 - 170*x + 433)*sqrt(-2*x + 1) + 363*sqrt(11)*l
og((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 4.6883, size = 204, normalized size = 2.96 \[ \begin{cases} \frac{8 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{125} - \frac{484 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{1875} + \frac{5566 \sqrt{5} i \sqrt{10 x - 5}}{9375} + \frac{242 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{625} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{8 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )^{2}}{125} - \frac{484 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{1875} + \frac{5566 \sqrt{5} \sqrt{- 10 x + 5}}{9375} + \frac{121 \sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{625} - \frac{242 \sqrt{55} \log{\left (\sqrt{- \frac{10 x}{11} + \frac{5}{11}} + 1 \right )}}{625} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/125 - 484*sqrt(5)*I*(x + 3/5)
*sqrt(10*x - 5)/1875 + 5566*sqrt(5)*I*sqrt(10*x - 5)/9375 + 242*sqrt(55)*I*asin(
sqrt(110)/(10*sqrt(x + 3/5)))/625, 10*Abs(x + 3/5)/11 > 1), (8*sqrt(5)*sqrt(-10*
x + 5)*(x + 3/5)**2/125 - 484*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/1875 + 5566*sqrt
(5)*sqrt(-10*x + 5)/9375 + 121*sqrt(55)*log(x + 3/5)/625 - 242*sqrt(55)*log(sqrt
(-10*x/11 + 5/11) + 1)/625, True))

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GIAC/XCAS [A]  time = 0.210593, size = 100, normalized size = 1.45 \[ \frac{2}{25} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*(2*x - 1)^2*sqrt(-2*x + 1) + 22/75*(-2*x + 1)^(3/2) + 121/625*sqrt(55)*ln(1
/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/125
*sqrt(-2*x + 1)