3.1967 \(\int \frac{(1-2 x)^{5/2}}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{(1-2 x)^{5/2}}{5 (5 x+3)}-\frac{2}{15} (1-2 x)^{3/2}-\frac{22}{25} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0675084, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{(1-2 x)^{5/2}}{5 (5 x+3)}-\frac{2}{15} (1-2 x)^{3/2}-\frac{22}{25} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.36659, size = 61, normalized size = 0.8 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{5 \left (5 x + 3\right )} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15} - \frac{22 \sqrt{- 2 x + 1}}{25} + \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/(5*(5*x + 3)) - 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 = 0.0778733, size = 58, normalized size = 0.76 \[ \frac{1}{375} \left (\frac{5 \sqrt{1-2 x} \left (40 x^2-260 x-243\right )}{5 x+3}+66 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(-243 - 260*x + 40*x^2))/(3 + 5*x) + 66*Sqrt[55]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/375

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Maple [A]  time = 0.013, size = 54, normalized size = 0.7 \[ -{\frac{4}{75} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{88}{125}\sqrt{1-2\,x}}+{\frac{242}{625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{22\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(3+5*x)^2,x)

[Out]

-4/75*(1-2*x)^(3/2)-88/125*(1-2*x)^(1/2)+242/625*(1-2*x)^(1/2)/(-6/5-2*x)+22/125
*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52051, size = 96, normalized size = 1.26 \[ -\frac{4}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{88}{125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/75*(-2*x + 1)^(3/2) - 11/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqr
t(55) + 5*sqrt(-2*x + 1))) - 88/125*sqrt(-2*x + 1) - 121/125*sqrt(-2*x + 1)/(5*x
+ 3)

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Fricas [A]  time = 0.213493, size = 101, normalized size = 1.33 \[ \frac{\sqrt{5}{\left (33 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (40 \, x^{2} - 260 \, x - 243\right )} \sqrt{-2 \, x + 1}\right )}}{375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/375*sqrt(5)*(33*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) - 5*sqrt(11)*sqrt(-2
*x + 1))/(5*x + 3)) + sqrt(5)*(40*x^2 - 260*x - 243)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 4.83603, size = 197, normalized size = 2.59 \[ \begin{cases} \frac{8 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{375} - \frac{308 \sqrt{5} i \sqrt{10 x - 5}}{1875} - \frac{22 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} - \frac{121 \sqrt{5} i \sqrt{10 x - 5}}{3125 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{8 \sqrt{5} \sqrt{- 10 x + 5} \left (x + \frac{3}{5}\right )}{375} - \frac{308 \sqrt{5} \sqrt{- 10 x + 5}}{1875} - \frac{121 \sqrt{5} \sqrt{- 10 x + 5}}{3125 \left (x + \frac{3}{5}\right )} - \frac{11 \sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{22 \sqrt{55} \log{\left (\sqrt{- \frac{10 x}{11} + \frac{5}{11}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/375 - 308*sqrt(5)*I*sqrt(10*x -
5)/1875 - 22*sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/125 - 121*sqrt(5)*I*s
qrt(10*x - 5)/(3125*(x + 3/5)), 10*Abs(x + 3/5)/11 > 1), (8*sqrt(5)*sqrt(-10*x +
5)*(x + 3/5)/375 - 308*sqrt(5)*sqrt(-10*x + 5)/1875 - 121*sqrt(5)*sqrt(-10*x +
5)/(3125*(x + 3/5)) - 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 = 0.212034, size = 100, normalized size = 1.32 \[ -\frac{4}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{88}{125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/75*(-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))) - 88/125*sqrt(-2*x + 1) - 121/125*sqrt(-2*x
+ 1)/(5*x + 3)