3.1816 \(\int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=55 \[ 2 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0702871, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ 2 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.11001, size = 49, normalized size = 0.89 \[ \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.045746, size = 55, normalized size = 1. \[ 2 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 38, normalized size = 0.7 \[ -{\frac{2\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/3*arctanh(1/7*21^(1/2)*(1-2
*x)^(1/2))*21^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.52145, size = 99, normalized size = 1.8 \[ \frac{1}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
1/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))

_______________________________________________________________________________________

Fricas [A]  time = 0.21917, size = 115, normalized size = 2.09 \[ \frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (\sqrt{11} \sqrt{3} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{7} \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*sqrt(5)*sqrt(3)*(sqrt(11)*sqrt(3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(
-2*x + 1))/(5*x + 3)) + sqrt(7)*sqrt(5)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(
-2*x + 1))/(3*x + 2)))

_______________________________________________________________________________________

Sympy [A]  time = 16.3111, size = 124, normalized size = 2.25 \[ - 14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) + 22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14*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)) + 22*Piecewise((
-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*ata
nh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220687, size = 107, normalized size = 1.95 \[ \frac{1}{5} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1}{3} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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