3.1823 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{5 (5 x+3)}+\frac{21}{125} \sqrt{1-2 x} (3 x+2)^2-\frac{294}{625} \sqrt{1-2 x}-\frac{196 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

[Out]

(-294*Sqrt[1 - 2*x])/625 + (21*Sqrt[1 - 2*x]*(2 + 3*x)^2)/125 - (Sqrt[1 - 2*x]*(
2 + 3*x)^3)/(5*(3 + 5*x)) - (196*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(625*Sqrt[55
])

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Rubi [A]  time = 0.138456, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{5 (5 x+3)}+\frac{21}{125} \sqrt{1-2 x} (3 x+2)^2-\frac{294}{625} \sqrt{1-2 x}-\frac{196 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-294*Sqrt[1 - 2*x])/625 + (21*Sqrt[1 - 2*x]*(2 + 3*x)^2)/125 - (Sqrt[1 - 2*x]*(
2 + 3*x)^3)/(5*(3 + 5*x)) - (196*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(625*Sqrt[55
])

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Rubi in Sympy [A]  time = 15.9742, size = 75, normalized size = 0.85 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{5 \left (5 x + 3\right )} + \frac{21 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{125} - \frac{294 \sqrt{- 2 x + 1}}{625} - \frac{196 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{34375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**3/(5*(5*x + 3)) + 21*sqrt(-2*x + 1)*(3*x + 2)**2/125
- 294*sqrt(-2*x + 1)/625 - 196*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/34375

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Mathematica [A]  time = 0.103634, size = 63, normalized size = 0.72 \[ \frac{\sqrt{1-2 x} \left (1350 x^3+2385 x^2-90 x-622\right )}{625 (5 x+3)}-\frac{196 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-622 - 90*x + 2385*x^2 + 1350*x^3))/(625*(3 + 5*x)) - (196*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(625*Sqrt[55])

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Maple [A]  time = 0.016, size = 63, normalized size = 0.7 \[{\frac{27}{250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{117}{250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{18}{625}\sqrt{1-2\,x}}+{\frac{2}{3125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{196\,\sqrt{55}}{34375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(1-2*x)^(5/2)-117/250*(1-2*x)^(3/2)+18/625*(1-2*x)^(1/2)+2/3125*(1-2*x)^(
1/2)/(-6/5-2*x)-196/34375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52692, size = 108, normalized size = 1.23 \[ \frac{27}{250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{117}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{98}{34375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18}{625} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(-2*x + 1)^(5/2) - 117/250*(-2*x + 1)^(3/2) + 98/34375*sqrt(55)*log(-(sqr
t(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 18/625*sqrt(-2*x + 1)
- 1/625*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.21411, size = 100, normalized size = 1.14 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (1350 \, x^{3} + 2385 \, x^{2} - 90 \, x - 622\right )} \sqrt{-2 \, x + 1} + 98 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{34375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/34375*sqrt(55)*(sqrt(55)*(1350*x^3 + 2385*x^2 - 90*x - 622)*sqrt(-2*x + 1) + 9
8*(5*x + 3)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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Sympy [A]  time = 60.7099, size = 199, normalized size = 2.26 \[ \frac{27 \left (- 2 x + 1\right )^{\frac{5}{2}}}{250} - \frac{117 \left (- 2 x + 1\right )^{\frac{3}{2}}}{250} + \frac{18 \sqrt{- 2 x + 1}}{625} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{625} + \frac{194 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*(-2*x + 1)**(5/2)/250 - 117*(-2*x + 1)**(3/2)/250 + 18*sqrt(-2*x + 1)/625 - 4
4*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqr
t(-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 + 194*Piecewise((-sq
rt(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 = 0.218737, size = 122, normalized size = 1.39 \[ \frac{27}{250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{117}{250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{98}{34375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{18}{625} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/250*(2*x - 1)^2*sqrt(-2*x + 1) - 117/250*(-2*x + 1)^(3/2) + 98/34375*sqrt(55)
*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 18
/625*sqrt(-2*x + 1) - 1/625*sqrt(-2*x + 1)/(5*x + 3)