3.1792 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{(1-2 x)^{3/2}}{63 (3 x+2)}-\frac{25}{27} (1-2 x)^{3/2}-\frac{142}{189} \sqrt{1-2 x}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-142*Sqrt[1 - 2*x])/189 - (25*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(3/2)/(63*(2 + 3*
x)) + (142*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi [A]  time = 0.0896951, antiderivative size = 74, 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 \[ -\frac{(1-2 x)^{3/2}}{63 (3 x+2)}-\frac{25}{27} (1-2 x)^{3/2}-\frac{142}{189} \sqrt{1-2 x}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-142*Sqrt[1 - 2*x])/189 - (25*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(3/2)/(63*(2 + 3*
x)) + (142*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi in Sympy [A]  time = 8.92098, size = 61, normalized size = 0.82 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{63 \left (3 x + 2\right )} - \frac{142 \sqrt{- 2 x + 1}}{189} + \frac{142 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{567} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(3/2)/27 - (-2*x + 1)**(3/2)/(63*(3*x + 2)) - 142*sqrt(-2*x + 1)
/189 + 142*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/567

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Mathematica [A]  time = 0.0977814, size = 55, normalized size = 0.74 \[ \frac{\sqrt{1-2 x} \left (150 x^2-35 x-91\right )}{81 x+54}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-91 - 35*x + 150*x^2))/(54 + 81*x) + (142*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/(27*Sqrt[21])

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Maple [A]  time = 0.017, size = 54, normalized size = 0.7 \[ -{\frac{25}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{20}{27}\sqrt{1-2\,x}}+{\frac{2}{81}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{142\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(1-2*x)^(3/2)-20/27*(1-2*x)^(1/2)+2/81*(1-2*x)^(1/2)/(-4/3-2*x)+142/567*a
rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51001, size = 96, normalized size = 1.3 \[ -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{71}{567} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) - 20/27*sqrt(-2*x + 1) - 1/27*sqrt(-2*x + 1)/(3*x +
2)

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Fricas [A]  time = 0.215588, size = 93, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (150 \, x^{2} - 35 \, x - 91\right )} \sqrt{-2 \, x + 1} + 71 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{567 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/567*sqrt(21)*(sqrt(21)*(150*x^2 - 35*x - 91)*sqrt(-2*x + 1) + 71*(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 = 51.18, size = 189, normalized size = 2.55 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{20 \sqrt{- 2 x + 1}}{27} - \frac{28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{27} - \frac{16 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(3/2)/27 - 20*sqrt(-2*x + 1)/27 - 28*Piecewise((sqrt(21)*(-log(s
qrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*(s
qrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)))/147, (x
<= 1/2) & (x > -2/3)))/27 - 16*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1
)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x +
1 < 7/3))/3

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GIAC/XCAS [A]  time = 0.212287, size = 100, normalized size = 1.35 \[ -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{71}{567} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x +
1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 20/27*sqrt(-2*x + 1) - 1/27*sqrt(-2*x + 1)
/(3*x + 2)