3.1780 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=56 \[ -\frac{5}{9} (1-2 x)^{3/2}-\frac{2}{9} \sqrt{1-2 x}+\frac{2}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/9 - (5*(1 - 2*x)^(3/2))/9 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/9

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Rubi [A]  time = 0.0637262, antiderivative size = 56, 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 \[ -\frac{5}{9} (1-2 x)^{3/2}-\frac{2}{9} \sqrt{1-2 x}+\frac{2}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/9 - (5*(1 - 2*x)^(3/2))/9 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/9

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Rubi in Sympy [A]  time = 7.00967, size = 48, normalized size = 0.86 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9} - \frac{2 \sqrt{- 2 x + 1}}{9} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(3/2)/9 - 2*sqrt(-2*x + 1)/9 + 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*
x + 1)/7)/27

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Mathematica [A]  time = 0.0499141, size = 46, normalized size = 0.82 \[ \frac{1}{27} \left (3 \sqrt{1-2 x} (10 x-7)+2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-7 + 10*x) + 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Maple [A]  time = 0.009, size = 38, normalized size = 0.7 \[ -{\frac{5}{9} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2}{9}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(1-2*x)^(3/2)+2/27*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2/9*(1-2*x)
^(1/2)

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Maxima [A]  time = 1.51873, size = 74, normalized size = 1.32 \[ -\frac{5}{9} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2}{9} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(-2*x + 1)^(3/2) - 1/27*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) - 2/9*sqrt(-2*x + 1)

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Fricas [A]  time = 0.21255, size = 77, normalized size = 1.38 \[ \frac{1}{27} \, \sqrt{3}{\left (\sqrt{3}{\left (10 \, x - 7\right )} \sqrt{-2 \, x + 1} + \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(sqrt(3)*(10*x - 7)*sqrt(-2*x + 1) + sqrt(7)*log((sqrt(3)*(3*x - 5)
- 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 8.39125, size = 88, normalized size = 1.57 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9} - \frac{2 \sqrt{- 2 x + 1}}{9} - \frac{14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(3/2)/9 - 2*sqrt(-2*x + 1)/9 - 14*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))/9

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GIAC/XCAS [A]  time = 0.223767, size = 78, normalized size = 1.39 \[ -\frac{5}{9} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{27} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2}{9} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*(-2*x + 1)^(3/2) - 1/27*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))
/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/9*sqrt(-2*x + 1)