3.1882 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\)

Optimal. Leaf size=82 \[ \frac{9}{70} (1-2 x)^{7/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{2}{375} (1-2 x)^{3/2}+\frac{22}{625} \sqrt{1-2 x}-\frac{22}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(22*Sqrt[1 - 2*x])/625 + (2*(1 - 2*x)^(3/2))/375 - (111*(1 - 2*x)^(5/2))/250 + (
9*(1 - 2*x)^(7/2))/70 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi [A]  time = 0.0992437, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{9}{70} (1-2 x)^{7/2}-\frac{111}{250} (1-2 x)^{5/2}+\frac{2}{375} (1-2 x)^{3/2}+\frac{22}{625} \sqrt{1-2 x}-\frac{22}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[1 - 2*x])/625 + (2*(1 - 2*x)^(3/2))/375 - (111*(1 - 2*x)^(5/2))/250 + (
9*(1 - 2*x)^(7/2))/70 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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Rubi in Sympy [A]  time = 10.0775, size = 71, normalized size = 0.87 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{7}{2}}}{70} - \frac{111 \left (- 2 x + 1\right )^{\frac{5}{2}}}{250} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{22 \sqrt{- 2 x + 1}}{625} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(7/2)/70 - 111*(-2*x + 1)**(5/2)/250 + 2*(-2*x + 1)**(3/2)/375 + 2
2*sqrt(-2*x + 1)/625 - 22*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3125

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Mathematica [A]  time = 0.0787094, size = 56, normalized size = 0.68 \[ \frac{-5 \sqrt{1-2 x} \left (13500 x^3+3060 x^2-13045 x+3608\right )-462 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{65625} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(3608 - 13045*x + 3060*x^2 + 13500*x^3) - 462*Sqrt[55]*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/65625

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Maple [A]  time = 0.01, size = 56, normalized size = 0.7 \[{\frac{2}{375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{111}{250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{9}{70} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{22\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{625}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/375*(1-2*x)^(3/2)-111/250*(1-2*x)^(5/2)+9/70*(1-2*x)^(7/2)-22/3125*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/625*(1-2*x)^(1/2)

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Maxima [A]  time = 1.4964, size = 99, normalized size = 1.21 \[ \frac{9}{70} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{111}{250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/70*(-2*x + 1)^(7/2) - 111/250*(-2*x + 1)^(5/2) + 2/375*(-2*x + 1)^(3/2) + 11/3
125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) +
22/625*sqrt(-2*x + 1)

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Fricas [A]  time = 0.211381, size = 92, normalized size = 1.12 \[ -\frac{1}{65625} \, \sqrt{5}{\left (\sqrt{5}{\left (13500 \, x^{3} + 3060 \, x^{2} - 13045 \, x + 3608\right )} \sqrt{-2 \, x + 1} - 231 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/65625*sqrt(5)*(sqrt(5)*(13500*x^3 + 3060*x^2 - 13045*x + 3608)*sqrt(-2*x + 1)
- 231*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 = 9.75902, size = 110, normalized size = 1.34 \[ \frac{9 \left (- 2 x + 1\right )^{\frac{7}{2}}}{70} - \frac{111 \left (- 2 x + 1\right )^{\frac{5}{2}}}{250} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{22 \sqrt{- 2 x + 1}}{625} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(-2*x + 1)**(7/2)/70 - 111*(-2*x + 1)**(5/2)/250 + 2*(-2*x + 1)**(3/2)/375 + 2
2*sqrt(-2*x + 1)/625 + 242*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))/625

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GIAC/XCAS [A]  time = 0.213922, size = 122, normalized size = 1.49 \[ -\frac{9}{70} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{111}{250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{3125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/70*(2*x - 1)^3*sqrt(-2*x + 1) - 111/250*(2*x - 1)^2*sqrt(-2*x + 1) + 2/375*(-
2*x + 1)^(3/2) + 11/3125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(s
qrt(55) + 5*sqrt(-2*x + 1))) + 22/625*sqrt(-2*x + 1)