3.1782 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) - (23*Sqrt[1 - 2*x])/(42*(2 + 3*x)) + (23*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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Rubi [A]  time = 0.0624341, antiderivative size = 68, 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 \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) - (23*Sqrt[1 - 2*x])/(42*(2 + 3*x)) + (23*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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Rubi in Sympy [A]  time = 8.08846, size = 56, normalized size = 0.82 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{42 \left (3 x + 2\right )^{2}} - \frac{23 \sqrt{- 2 x + 1}}{42 \left (3 x + 2\right )} + \frac{23 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)/(42*(3*x + 2)**2) - 23*sqrt(-2*x + 1)/(42*(3*x + 2)) + 23*sqrt
(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/441

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Mathematica [A]  time = 0.0852665, size = 53, normalized size = 0.78 \[ \frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-\frac{\sqrt{1-2 x} (71 x+45)}{42 (3 x+2)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(45 + 71*x))/(42*(2 + 3*x)^2) + (23*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(21*Sqrt[21])

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Maple [A]  time = 0.015, size = 48, normalized size = 0.7 \[ -36\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{756}}+{\frac{23\,\sqrt{1-2\,x}}{108}} \right ) }+{\frac{23\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36*(-71/756*(1-2*x)^(3/2)+23/108*(1-2*x)^(1/2))/(-4-6*x)^2+23/441*arctanh(1/7*2
1^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48954, size = 100, normalized size = 1.47 \[ -\frac{23}{882} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{21 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23/882*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 1/21*(71*(-2*x + 1)^(3/2) - 161*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.215696, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (71 \, x + 45\right )} \sqrt{-2 \, x + 1} - 23 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/882*sqrt(21)*(sqrt(21)*(71*x + 45)*sqrt(-2*x + 1) - 23*(9*x^2 + 12*x + 4)*log
((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 95.0717, size = 313, normalized size = 4.6 \[ - \frac{148 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{20 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-148*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*s
qrt(-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 - 56*Piecewise((sqrt(21
)*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1
)/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7
+ 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x +
1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/9 - 20*Piecewise((-sqrt(21)*acot
h(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))/9

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GIAC/XCAS [A]  time = 0.219827, size = 92, normalized size = 1.35 \[ -\frac{23}{882} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{84 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23/882*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/84*(71*(-2*x + 1)^(3/2) - 161*sqrt(-2*x + 1))/(3*x + 2)^2