3.1885 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=72 \[ -\frac{4}{15} \sqrt{1-2 x}+\frac{14}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{22}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-4*Sqrt[1 - 2*x])/15 + (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 - (22*
Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi [A]  time = 0.122085, antiderivative size = 72, 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 \[ -\frac{4}{15} \sqrt{1-2 x}+\frac{14}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{22}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[1 - 2*x])/15 + (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 - (22*
Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi in Sympy [A]  time = 13.6936, size = 61, normalized size = 0.85 \[ - \frac{4 \sqrt{- 2 x + 1}}{15} + \frac{14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(-2*x + 1)/15 + 14*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/9 - 22*sqrt(
55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/25

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Mathematica [A]  time = 0.0649297, size = 66, normalized size = 0.92 \[ -\frac{2}{225} \left (30 \sqrt{1-2 x}-175 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+99 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(30*Sqrt[1 - 2*x] - 175*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 99*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/225

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Maple [A]  time = 0.013, size = 47, normalized size = 0.7 \[ -{\frac{22\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{14\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{4}{15}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+14/9*arctanh(1/7*21^(1/2)*(
1-2*x)^(1/2))*21^(1/2)-4/15*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49164, size = 111, normalized size = 1.54 \[ \frac{11}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4}{15} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 7/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) - 4/15*sqrt(-2*x + 1)

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Fricas [A]  time = 0.2364, size = 138, normalized size = 1.92 \[ \frac{1}{225} \, \sqrt{5} \sqrt{3}{\left (33 \, \sqrt{11} \sqrt{3} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 35 \, \sqrt{7} \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - 4 \, \sqrt{5} \sqrt{3} \sqrt{-2 \, x + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/225*sqrt(5)*sqrt(3)*(33*sqrt(11)*sqrt(3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*s
qrt(-2*x + 1))/(5*x + 3)) + 35*sqrt(7)*sqrt(5)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7
)*sqrt(-2*x + 1))/(3*x + 2)) - 4*sqrt(5)*sqrt(3)*sqrt(-2*x + 1))

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Sympy [A]  time = 6.71963, size = 139, normalized size = 1.93 \[ - \frac{4 \sqrt{- 2 x + 1}}{15} - \frac{98 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{3} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(-2*x + 1)/15 - 98*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/
21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 <
7/3))/3 + 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))/5

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GIAC/XCAS [A]  time = 0.212118, size = 119, normalized size = 1.65 \[ \frac{11}{25} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{7}{9} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{4}{15} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 7/9*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) - 4/15*sqrt(-2*x + 1)