3.1825 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{(1-2 x)^{3/2}}{55 (5 x+3)}+\frac{64}{275} \sqrt{1-2 x}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

[Out]

(64*Sqrt[1 - 2*x])/275 - (1 - 2*x)^(3/2)/(55*(3 + 5*x)) - (64*ArcTanh[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi [A]  time = 0.0601267, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(1-2 x)^{3/2}}{55 (5 x+3)}+\frac{64}{275} \sqrt{1-2 x}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(64*Sqrt[1 - 2*x])/275 - (1 - 2*x)^(3/2)/(55*(3 + 5*x)) - (64*ArcTanh[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(25*Sqrt[55])

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Rubi in Sympy [A]  time = 7.02358, size = 49, normalized size = 0.8 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{55 \left (5 x + 3\right )} + \frac{64 \sqrt{- 2 x + 1}}{275} - \frac{64 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/(55*(5*x + 3)) + 64*sqrt(-2*x + 1)/275 - 64*sqrt(55)*atanh(sq
rt(55)*sqrt(-2*x + 1)/11)/1375

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Mathematica [A]  time = 0.072152, size = 53, normalized size = 0.87 \[ \frac{\sqrt{1-2 x} (30 x+17)}{25 (5 x+3)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(17 + 30*x))/(25*(3 + 5*x)) - (64*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(25*Sqrt[55])

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Maple [A]  time = 0.016, size = 45, normalized size = 0.7 \[{\frac{6}{25}\sqrt{1-2\,x}}+{\frac{2}{125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{64\,\sqrt{55}}{1375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/25*(1-2*x)^(1/2)+2/125*(1-2*x)^(1/2)/(-6/5-2*x)-64/1375*arctanh(1/11*55^(1/2)*
(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49081, size = 84, normalized size = 1.38 \[ \frac{32}{1375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 6/25*sqrt(-2*x + 1) - 1/25*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.21351, size = 86, normalized size = 1.41 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (30 \, x + 17\right )} \sqrt{-2 \, x + 1} + 32 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{1375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1375*sqrt(55)*(sqrt(55)*(30*x + 17)*sqrt(-2*x + 1) + 32*(5*x + 3)*log((sqrt(55
)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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Sympy [A]  time = 39.107, size = 175, normalized size = 2.87 \[ \frac{6 \sqrt{- 2 x + 1}}{25} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{25} + \frac{62 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(-2*x + 1)/25 - 44*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)))/2
5 + 62*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))/25

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GIAC/XCAS [A]  time = 0.21152, size = 88, normalized size = 1.44 \[ \frac{32}{1375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{6}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/1375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 6/25*sqrt(-2*x + 1) - 1/25*sqrt(-2*x + 1)/(5*x + 3)