3.1814 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{3+5 x} \, dx\)

Optimal. Leaf size=56 \[ -\frac{1}{5} (1-2 x)^{3/2}+\frac{2}{25} \sqrt{1-2 x}-\frac{2}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0591537, 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 \[ -\frac{1}{5} (1-2 x)^{3/2}+\frac{2}{25} \sqrt{1-2 x}-\frac{2}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.16112, size = 46, normalized size = 0.82 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{5} + \frac{2 \sqrt{- 2 x + 1}}{25} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/5 + 2*sqrt(-2*x + 1)/25 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x
+ 1)/11)/125

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Mathematica [A]  time = 0.0425222, size = 46, normalized size = 0.82 \[ \frac{1}{125} \left (5 \sqrt{1-2 x} (10 x-3)-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-3 + 10*x) - 2*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/125

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Maple [A]  time = 0.007, size = 38, normalized size = 0.7 \[ -{\frac{1}{5} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{25}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(1-2*x)^(3/2)-2/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/25*(1-2
*x)^(1/2)

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Maxima [A]  time = 1.6326, size = 74, normalized size = 1.32 \[ -\frac{1}{5} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{25} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-2*x + 1)^(3/2) + 1/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(
55) + 5*sqrt(-2*x + 1))) + 2/25*sqrt(-2*x + 1)

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Fricas [A]  time = 0.210886, size = 77, normalized size = 1.38 \[ \frac{1}{125} \, \sqrt{5}{\left (\sqrt{5}{\left (10 \, x - 3\right )} \sqrt{-2 \, x + 1} + \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/125*sqrt(5)*(sqrt(5)*(10*x - 3)*sqrt(-2*x + 1) + sqrt(11)*log((sqrt(5)*(5*x -
8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 7.89633, size = 85, normalized size = 1.52 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{5} + \frac{2 \sqrt{- 2 x + 1}}{25} + \frac{22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/5 + 2*sqrt(-2*x + 1)/25 + 22*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 = 0.209601, size = 78, normalized size = 1.39 \[ -\frac{1}{5} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2}{25} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(-2*x + 1)^(3/2) + 1/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1
))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/25*sqrt(-2*x + 1)