3.1895 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{(1-2 x)^{5/2}}{55 (5 x+3)}+\frac{4}{55} (1-2 x)^{3/2}+\frac{12}{25} \sqrt{1-2 x}-\frac{12}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(12*Sqrt[1 - 2*x])/25 + (4*(1 - 2*x)^(3/2))/55 - (1 - 2*x)^(5/2)/(55*(3 + 5*x))
- (12*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.076643, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(1-2 x)^{5/2}}{55 (5 x+3)}+\frac{4}{55} (1-2 x)^{3/2}+\frac{12}{25} \sqrt{1-2 x}-\frac{12}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(12*Sqrt[1 - 2*x])/25 + (4*(1 - 2*x)^(3/2))/55 - (1 - 2*x)^(5/2)/(55*(3 + 5*x))
- (12*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi in Sympy [A]  time = 8.40946, size = 61, normalized size = 0.8 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{55 \left (5 x + 3\right )} + \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{55} + \frac{12 \sqrt{- 2 x + 1}}{25} - \frac{12 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)/(55*(5*x + 3)) + 4*(-2*x + 1)**(3/2)/55 + 12*sqrt(-2*x + 1)/2
5 - 12*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.0910931, size = 58, normalized size = 0.76 \[ \frac{1}{125} \left (\frac{5 \sqrt{1-2 x} \left (-20 x^2+60 x+41\right )}{5 x+3}-12 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(41 + 60*x - 20*x^2))/(3 + 5*x) - 12*Sqrt[55]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/125

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Maple [A]  time = 0.014, size = 54, normalized size = 0.7 \[{\frac{2}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{62}{125}\sqrt{1-2\,x}}+{\frac{22}{625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{12\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*(1-2*x)^(3/2)+62/125*(1-2*x)^(1/2)+22/625*(1-2*x)^(1/2)/(-6/5-2*x)-12/125*a
rctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48375, size = 96, normalized size = 1.26 \[ \frac{2}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{62}{125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*(-2*x + 1)^(3/2) + 6/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(
55) + 5*sqrt(-2*x + 1))) + 62/125*sqrt(-2*x + 1) - 11/125*sqrt(-2*x + 1)/(5*x +
3)

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Fricas [A]  time = 0.230226, size = 103, normalized size = 1.36 \[ \frac{\sqrt{5}{\left (6 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (20 \, x^{2} - 60 \, x - 41\right )} \sqrt{-2 \, x + 1}\right )}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/125*sqrt(5)*(6*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*
x + 1))/(5*x + 3)) - sqrt(5)*(20*x^2 - 60*x - 41)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 104.48, size = 187, normalized size = 2.46 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25} + \frac{62 \sqrt{- 2 x + 1}}{125} - \frac{484 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{125} + \frac{638 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)/25 + 62*sqrt(-2*x + 1)/125 - 484*Piecewise((sqrt(55)*(-log(s
qrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*
(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605,
(x <= 1/2) & (x > -3/5)))/125 + 638*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)/5
5, -2*x + 1 < 11/5))/125

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GIAC/XCAS [A]  time = 0.21077, size = 100, normalized size = 1.32 \[ \frac{2}{25} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{62}{125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*(-2*x + 1)^(3/2) + 6/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1
))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 62/125*sqrt(-2*x + 1) - 11/125*sqrt(-2*x + 1
)/(5*x + 3)