3.1781 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{(1-2 x)^{3/2}}{21 (3 x+2)}+\frac{8}{7} \sqrt{1-2 x}-\frac{8 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}} \]

[Out]

(8*Sqrt[1 - 2*x])/7 + (1 - 2*x)^(3/2)/(21*(2 + 3*x)) - (8*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/Sqrt[21]

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Rubi [A]  time = 0.0608249, antiderivative size = 59, 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 \[ \frac{(1-2 x)^{3/2}}{21 (3 x+2)}+\frac{8}{7} \sqrt{1-2 x}-\frac{8 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[1 - 2*x])/7 + (1 - 2*x)^(3/2)/(21*(2 + 3*x)) - (8*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/Sqrt[21]

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Rubi in Sympy [A]  time = 7.2672, size = 49, normalized size = 0.83 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{21 \left (3 x + 2\right )} + \frac{8 \sqrt{- 2 x + 1}}{7} - \frac{8 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)/(21*(3*x + 2)) + 8*sqrt(-2*x + 1)/7 - 8*sqrt(21)*atanh(sqrt(21
)*sqrt(-2*x + 1)/7)/21

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Mathematica [A]  time = 0.0618879, size = 55, normalized size = 0.93 \[ \frac{7 \sqrt{1-2 x} (10 x+7)-8 \sqrt{21} (3 x+2) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{63 x+42} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*(7 + 10*x) - 8*Sqrt[21]*(2 + 3*x)*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(42 + 63*x)

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Maple [A]  time = 0.014, size = 45, normalized size = 0.8 \[{\frac{10}{9}\sqrt{1-2\,x}}-{\frac{2}{27}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{8\,\sqrt{21}}{21}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/9*(1-2*x)^(1/2)-2/27*(1-2*x)^(1/2)/(-4/3-2*x)-8/21*arctanh(1/7*21^(1/2)*(1-2*
x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49533, size = 84, normalized size = 1.42 \[ \frac{4}{21} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{10}{9} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/21*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))
+ 10/9*sqrt(-2*x + 1) + 1/9*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.219032, size = 86, normalized size = 1.46 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (10 \, x + 7\right )} \sqrt{-2 \, x + 1} + 12 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{63 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*sqrt(21)*(sqrt(21)*(10*x + 7)*sqrt(-2*x + 1) + 12*(3*x + 2)*log((sqrt(21)*(
3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(3*x + 2)

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Sympy [A]  time = 42.6055, size = 175, normalized size = 2.97 \[ \frac{10 \sqrt{- 2 x + 1}}{9} + \frac{28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} + \frac{74 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*sqrt(-2*x + 1)/9 + 28*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1
)/4 + log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1
)) - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/9 + 7
4*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-s
qrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/9

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GIAC/XCAS [A]  time = 0.216274, size = 88, normalized size = 1.49 \[ \frac{4}{21} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{10}{9} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/21*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 10/9*sqrt(-2*x + 1) + 1/9*sqrt(-2*x + 1)/(3*x + 2)