3.1886 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=77 \[ \frac{7 \sqrt{1-2 x}}{3 (3 x+2)}+\frac{64}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(3*(2 + 3*x)) + (64*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/3 - 22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.125472, antiderivative size = 77, 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 \[ \frac{7 \sqrt{1-2 x}}{3 (3 x+2)}+\frac{64}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x])/(3*(2 + 3*x)) + (64*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/3 - 22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 14.7761, size = 65, normalized size = 0.84 \[ \frac{7 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )} + \frac{64 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

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]

7*sqrt(-2*x + 1)/(3*(3*x + 2)) + 64*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/9
- 22*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/5

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Mathematica [A]  time = 0.135607, size = 75, normalized size = 0.97 \[ \frac{7 \sqrt{1-2 x}}{9 x+6}+\frac{64}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x])/(6 + 9*x) + (64*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3
- 22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.016, size = 54, normalized size = 0.7 \[ -{\frac{14}{9}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{64\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{22\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14/9*(1-2*x)^(1/2)/(-4/3-2*x)+64/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
-22/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49016, size = 120, normalized size = 1.56 \[ \frac{11}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{32}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7 \, \sqrt{-2 \, x + 1}}{3 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 32/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) + 7/3*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.236376, size = 161, normalized size = 2.09 \[ \frac{\sqrt{5} \sqrt{3}{\left (33 \, \sqrt{11} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 32 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 7 \, \sqrt{5} \sqrt{3} \sqrt{-2 \, x + 1}\right )}}{45 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*sqrt(5)*sqrt(3)*(33*sqrt(11)*sqrt(3)*(3*x + 2)*log((sqrt(5)*(5*x - 8) + 5*s
qrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 32*sqrt(7)*sqrt(5)*(3*x + 2)*log((sqrt(3)*(
3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) + 7*sqrt(5)*sqrt(3)*sqrt(-2*x +
1))/(3*x + 2)

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Sympy [A]  time = 56.7161, size = 226, normalized size = 2.94 \[ \frac{196 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{3} - \frac{434 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{3} + 242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

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]

196*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sq
rt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/3 - 434*Piecewise((-sqrt(2
1)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(2
1)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/3 + 242*Piecewise((-sqrt(55)*acoth(sqr
t(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))

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GIAC/XCAS [A]  time = 0.212343, size = 128, normalized size = 1.66 \[ \frac{11}{5} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{32}{9} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7 \, \sqrt{-2 \, x + 1}}{3 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/5*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 32/9*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 7/3*sqrt(-2*x + 1)/(3*x + 2)