3.1896 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=63 \[ -\frac{(1-2 x)^{3/2}}{5 (5 x+3)}-\frac{6}{25} \sqrt{1-2 x}+\frac{6}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-6*Sqrt[1 - 2*x])/25 - (1 - 2*x)^(3/2)/(5*(3 + 5*x)) + (6*Sqrt[11/5]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.0512542, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{(1-2 x)^{3/2}}{5 (5 x+3)}-\frac{6}{25} \sqrt{1-2 x}+\frac{6}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[1 - 2*x])/25 - (1 - 2*x)^(3/2)/(5*(3 + 5*x)) + (6*Sqrt[11/5]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi in Sympy [A]  time = 6.34716, size = 49, normalized size = 0.78 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{5 \left (5 x + 3\right )} - \frac{6 \sqrt{- 2 x + 1}}{25} + \frac{6 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/(5*(5*x + 3)) - 6*sqrt(-2*x + 1)/25 + 6*sqrt(55)*atanh(sqrt(5
5)*sqrt(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.0726717, size = 53, normalized size = 0.84 \[ \frac{1}{125} \left (6 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{5 \sqrt{1-2 x} (20 x+23)}{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-5*Sqrt[1 - 2*x]*(23 + 20*x))/(3 + 5*x) + 6*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/125

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Maple [A]  time = 0.013, size = 45, normalized size = 0.7 \[ -{\frac{4}{25}\sqrt{1-2\,x}}+{\frac{22}{125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{6\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(3+5*x)^2,x)

[Out]

-4/25*(1-2*x)^(1/2)+22/125*(1-2*x)^(1/2)/(-6/5-2*x)+6/125*arctanh(1/11*55^(1/2)*
(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49443, size = 84, normalized size = 1.33 \[ -\frac{3}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4}{25} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 4/25*sqrt(-2*x + 1) - 11/25*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.224538, size = 96, normalized size = 1.52 \[ \frac{\sqrt{5}{\left (3 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (20 \, x + 23\right )} \sqrt{-2 \, x + 1}\right )}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/125*sqrt(5)*(3*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) - 5*sqrt(11)*sqrt(-2*
x + 1))/(5*x + 3)) - sqrt(5)*(20*x + 23)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 3.49434, size = 240, normalized size = 3.81 \[ \begin{cases} \frac{6 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} + \frac{4 \sqrt{2} \sqrt{x + \frac{3}{5}}}{25 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} - \frac{11 \sqrt{2}}{125 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{121 \sqrt{2}}{1250 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11 \left |{\frac{1}{x + \frac{3}{5}}}\right |}{10} > 1 \\- \frac{6 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} - \frac{4 \sqrt{2} i \sqrt{x + \frac{3}{5}}}{25 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} + \frac{11 \sqrt{2} i}{125 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{121 \sqrt{2} i}{1250 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/125 + 4*sqrt(2)*sqrt(x
+ 3/5)/(25*sqrt(-1 + 11/(10*(x + 3/5)))) - 11*sqrt(2)/(125*sqrt(-1 + 11/(10*(x
+ 3/5)))*sqrt(x + 3/5)) - 121*sqrt(2)/(1250*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/
5)**(3/2)), 11*Abs(1/(x + 3/5))/10 > 1), (-6*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(
x + 3/5)))/125 - 4*sqrt(2)*I*sqrt(x + 3/5)/(25*sqrt(1 - 11/(10*(x + 3/5)))) + 11
*sqrt(2)*I/(125*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) + 121*sqrt(2)*I/(1250
*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True))

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GIAC/XCAS [A]  time = 0.211644, size = 88, normalized size = 1.4 \[ -\frac{3}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{4}{25} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 4/25*sqrt(-2*x + 1) - 11/25*sqrt(-2*x + 1)/(5*x + 3)