3.1833 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=81 \[ -\frac{133 (1-2 x)^{3/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{3/2}}{550 (5 x+3)^2}+\frac{409 \sqrt{1-2 x}}{3025}-\frac{409 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

[Out]

(409*Sqrt[1 - 2*x])/3025 - (1 - 2*x)^(3/2)/(550*(3 + 5*x)^2) - (133*(1 - 2*x)^(3
/2))/(6050*(3 + 5*x)) - (409*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi [A]  time = 0.0932296, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{133 (1-2 x)^{3/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{3/2}}{550 (5 x+3)^2}+\frac{409 \sqrt{1-2 x}}{3025}-\frac{409 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(409*Sqrt[1 - 2*x])/3025 - (1 - 2*x)^(3/2)/(550*(3 + 5*x)^2) - (133*(1 - 2*x)^(3
/2))/(6050*(3 + 5*x)) - (409*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi in Sympy [A]  time = 9.82156, size = 68, normalized size = 0.84 \[ - \frac{133 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6050 \left (5 x + 3\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{550 \left (5 x + 3\right )^{2}} + \frac{409 \sqrt{- 2 x + 1}}{3025} - \frac{409 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-133*(-2*x + 1)**(3/2)/(6050*(5*x + 3)) - (-2*x + 1)**(3/2)/(550*(5*x + 3)**2) +
409*sqrt(-2*x + 1)/3025 - 409*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/15125

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Mathematica [A]  time = 0.0895959, size = 58, normalized size = 0.72 \[ \frac{\sqrt{1-2 x} \left (1980 x^2+2245 x+632\right )}{550 (5 x+3)^2}-\frac{409 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(632 + 2245*x + 1980*x^2))/(550*(3 + 5*x)^2) - (409*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Maple [A]  time = 0.015, size = 57, normalized size = 0.7 \[{\frac{18}{125}\sqrt{1-2\,x}}+{\frac{2}{5\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{131}{110} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{133}{50}\sqrt{1-2\,x}} \right ) }-{\frac{409\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/125*(1-2*x)^(1/2)+2/5*(131/110*(1-2*x)^(3/2)-133/50*(1-2*x)^(1/2))/(-6-10*x)^
2-409/15125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.47934, size = 112, normalized size = 1.38 \[ \frac{409}{30250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18}{125} \, \sqrt{-2 \, x + 1} + \frac{655 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{1375 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

409/30250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 18/125*sqrt(-2*x + 1) + 1/1375*(655*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1
))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.219777, size = 107, normalized size = 1.32 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (1980 \, x^{2} + 2245 \, x + 632\right )} \sqrt{-2 \, x + 1} + 409 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{30250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30250*sqrt(55)*(sqrt(55)*(1980*x^2 + 2245*x + 632)*sqrt(-2*x + 1) + 409*(25*x^
2 + 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 +
30*x + 9)

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Sympy [A]  time = 123.005, size = 323, normalized size = 3.99 \[ \frac{18 \sqrt{- 2 x + 1}}{125} - \frac{256 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{125} + \frac{88 \left (\begin{cases} \frac{\sqrt{55} \left (\frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )^{2}}\right )}{6655} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{125} + \frac{174 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18*sqrt(-2*x + 1)/125 - 256*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11
- 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/
11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5))
)/125 + 88*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/16 - 3*log
(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) +
1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(-2*x + 1)/11 -
1)) - 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)**2))/6655, (x <= 1/2) & (x > -3/5)
))/125 + 174*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1
> 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/125

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GIAC/XCAS [A]  time = 0.224905, size = 104, normalized size = 1.28 \[ \frac{409}{30250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{18}{125} \, \sqrt{-2 \, x + 1} + \frac{655 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{5500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

409/30250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) + 18/125*sqrt(-2*x + 1) + 1/5500*(655*(-2*x + 1)^(3/2) - 1463*sqrt
(-2*x + 1))/(5*x + 3)^2