3.1966 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=89 \[ -\frac{(1-2 x)^{7/2}}{55 (5 x+3)}+\frac{56 (1-2 x)^{5/2}}{1375}+\frac{56}{375} (1-2 x)^{3/2}+\frac{616}{625} \sqrt{1-2 x}-\frac{616}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(616*Sqrt[1 - 2*x])/625 + (56*(1 - 2*x)^(3/2))/375 + (56*(1 - 2*x)^(5/2))/1375 -
(1 - 2*x)^(7/2)/(55*(3 + 5*x)) - (616*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/625

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Rubi [A]  time = 0.0979778, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(1-2 x)^{7/2}}{55 (5 x+3)}+\frac{56 (1-2 x)^{5/2}}{1375}+\frac{56}{375} (1-2 x)^{3/2}+\frac{616}{625} \sqrt{1-2 x}-\frac{616}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(616*Sqrt[1 - 2*x])/625 + (56*(1 - 2*x)^(3/2))/375 + (56*(1 - 2*x)^(5/2))/1375 -
(1 - 2*x)^(7/2)/(55*(3 + 5*x)) - (616*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/625

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Rubi in Sympy [A]  time = 9.74244, size = 73, normalized size = 0.82 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{55 \left (5 x + 3\right )} + \frac{56 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1375} + \frac{56 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{616 \sqrt{- 2 x + 1}}{625} - \frac{616 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)/(55*(5*x + 3)) + 56*(-2*x + 1)**(5/2)/1375 + 56*(-2*x + 1)**(
3/2)/375 + 616*sqrt(-2*x + 1)/625 - 616*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/1
1)/3125

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Mathematica [A]  time = 0.107007, size = 63, normalized size = 0.71 \[ \frac{\frac{5 \sqrt{1-2 x} \left (1800 x^3-3820 x^2+8630 x+6579\right )}{5 x+3}-1848 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{9375} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(6579 + 8630*x - 3820*x^2 + 1800*x^3))/(3 + 5*x) - 1848*Sqrt[5
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/9375

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Maple [A]  time = 0.014, size = 63, normalized size = 0.7 \[{\frac{6}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{62}{375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{638}{625}\sqrt{1-2\,x}}+{\frac{242}{3125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{616\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(1-2*x)^(5/2)+62/375*(1-2*x)^(3/2)+638/625*(1-2*x)^(1/2)+242/3125*(1-2*x)^
(1/2)/(-6/5-2*x)-616/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5087, size = 108, normalized size = 1.21 \[ \frac{6}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{62}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{308}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{638}{625} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(-2*x + 1)^(5/2) + 62/375*(-2*x + 1)^(3/2) + 308/3125*sqrt(55)*log(-(sqrt(
55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 638/625*sqrt(-2*x + 1)
- 121/625*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.211835, size = 108, normalized size = 1.21 \[ \frac{\sqrt{5}{\left (924 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (1800 \, x^{3} - 3820 \, x^{2} + 8630 \, x + 6579\right )} \sqrt{-2 \, x + 1}\right )}}{9375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9375*sqrt(5)*(924*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(
-2*x + 1))/(5*x + 3)) + sqrt(5)*(1800*x^3 - 3820*x^2 + 8630*x + 6579)*sqrt(-2*x
+ 1))/(5*x + 3)

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Sympy [A]  time = 153.886, size = 199, normalized size = 2.24 \[ \frac{6 \left (- 2 x + 1\right )^{\frac{5}{2}}}{125} + \frac{62 \left (- 2 x + 1\right )^{\frac{3}{2}}}{375} + \frac{638 \sqrt{- 2 x + 1}}{625} - \frac{5324 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{625} + \frac{6534 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(-2*x + 1)**(5/2)/125 + 62*(-2*x + 1)**(3/2)/375 + 638*sqrt(-2*x + 1)/625 - 53
24*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sq
rt(-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)))/625 + 6534*Piecewise((-
sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atan
h(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/625

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GIAC/XCAS [A]  time = 0.212356, size = 122, normalized size = 1.37 \[ \frac{6}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{62}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{308}{3125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{638}{625} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/125*(2*x - 1)^2*sqrt(-2*x + 1) + 62/375*(-2*x + 1)^(3/2) + 308/3125*sqrt(55)*l
n(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 638/
625*sqrt(-2*x + 1) - 121/625*sqrt(-2*x + 1)/(5*x + 3)