3.1884 \(\int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\)

Optimal. Leaf size=56 \[ \frac{2}{15} (1-2 x)^{3/2}+\frac{22}{25} \sqrt{1-2 x}-\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0490057, antiderivative size = 56, 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 \[ \frac{2}{15} (1-2 x)^{3/2}+\frac{22}{25} \sqrt{1-2 x}-\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.79157, size = 48, normalized size = 0.86 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15} + \frac{22 \sqrt{- 2 x + 1}}{25} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)/15 + 22*sqrt(-2*x + 1)/25 - 22*sqrt(55)*atanh(sqrt(55)*sqrt(
-2*x + 1)/11)/125

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Mathematica [A]  time = 0.046108, size = 46, normalized size = 0.82 \[ -\frac{2}{375} \left (10 \sqrt{1-2 x} (5 x-19)+33 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(10*Sqrt[1 - 2*x]*(-19 + 5*x) + 33*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
]))/375

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Maple [A]  time = 0.008, size = 38, normalized size = 0.7 \[{\frac{2}{15} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{22\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{25}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(3+5*x),x)

[Out]

2/15*(1-2*x)^(3/2)-22/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/25*(1
-2*x)^(1/2)

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Maxima [A]  time = 1.48653, size = 74, normalized size = 1.32 \[ \frac{2}{15} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{25} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(-2*x + 1)^(3/2) + 11/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) + 22/25*sqrt(-2*x + 1)

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Fricas [A]  time = 0.213309, size = 80, normalized size = 1.43 \[ -\frac{1}{375} \, \sqrt{5}{\left (4 \, \sqrt{5}{\left (5 \, x - 19\right )} \sqrt{-2 \, x + 1} - 33 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/375*sqrt(5)*(4*sqrt(5)*(5*x - 19)*sqrt(-2*x + 1) - 33*sqrt(11)*log((sqrt(5)*(
5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 3.01722, size = 155, normalized size = 2.77 \[ \begin{cases} - \frac{4 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{75} + \frac{88 \sqrt{5} i \sqrt{10 x - 5}}{375} + \frac{22 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{4 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{75} + \frac{88 \sqrt{5} \sqrt{- 10 x + 5}}{375} + \frac{11 \sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{125} - \frac{22 \sqrt{55} \log{\left (\sqrt{- \frac{10 x}{11} + \frac{5}{11}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/75 + 88*sqrt(5)*I*sqrt(10*x - 5
)/375 + 22*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/125, 10*Abs(x + 3/5)/11
> 1), (-4*sqrt(5)*sqrt(-10*x + 5)*(x + 3/5)/75 + 88*sqrt(5)*sqrt(-10*x + 5)/375
+ 11*sqrt(55)*log(x + 3/5)/125 - 22*sqrt(55)*log(sqrt(-10*x/11 + 5/11) + 1)/125
, True))

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GIAC/XCAS [A]  time = 0.209879, size = 78, normalized size = 1.39 \[ \frac{2}{15} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{25} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(-2*x + 1)^(3/2) + 11/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/25*sqrt(-2*x + 1)