3.2022 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=54 \[ \frac{3}{10} (1-2 x)^{3/2}-\frac{111}{50} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

[Out]

(-111*Sqrt[1 - 2*x])/50 + (3*(1 - 2*x)^(3/2))/10 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(25*Sqrt[55])

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Rubi [A]  time = 0.0799516, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{3}{10} (1-2 x)^{3/2}-\frac{111}{50} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-111*Sqrt[1 - 2*x])/50 + (3*(1 - 2*x)^(3/2))/10 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(25*Sqrt[55])

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Rubi in Sympy [A]  time = 7.21052, size = 48, normalized size = 0.89 \[ \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}}}{10} - \frac{111 \sqrt{- 2 x + 1}}{50} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-2*x + 1)**(3/2)/10 - 111*sqrt(-2*x + 1)/50 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(
-2*x + 1)/11)/1375

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Mathematica [A]  time = 0.0544928, size = 46, normalized size = 0.85 \[ -\frac{3}{25} \sqrt{1-2 x} (5 x+16)-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*(16 + 5*x))/25 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25*Sqr
t[55])

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Maple [A]  time = 0.008, size = 38, normalized size = 0.7 \[{\frac{3}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{1375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{111}{50}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(1-2*x)^(3/2)-2/1375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-111/50*(
1-2*x)^(1/2)

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Maxima [A]  time = 1.54249, size = 74, normalized size = 1.37 \[ \frac{3}{10} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{111}{50} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(-2*x + 1)^(3/2) + 1/1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) - 111/50*sqrt(-2*x + 1)

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Fricas [A]  time = 0.248188, size = 72, normalized size = 1.33 \[ -\frac{1}{1375} \, \sqrt{55}{\left (3 \, \sqrt{55}{\left (5 \, x + 16\right )} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1375*sqrt(55)*(3*sqrt(55)*(5*x + 16)*sqrt(-2*x + 1) - log((sqrt(55)*(5*x - 8)
+ 55*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 4.21302, size = 90, normalized size = 1.67 \[ \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}}}{10} - \frac{111 \sqrt{- 2 x + 1}}{50} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-2*x + 1)**(3/2)/10 - 111*sqrt(-2*x + 1)/50 + 2*Piecewise((-sqrt(55)*acoth(sq
rt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) > 5/11), (-sqrt(55)*atanh(sqrt(55)/(
5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) < 5/11))/25

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GIAC/XCAS [A]  time = 0.211406, size = 78, normalized size = 1.44 \[ \frac{3}{10} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{111}{50} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(-2*x + 1)^(3/2) + 1/1375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 111/50*sqrt(-2*x + 1)