3.1827 \(\int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

[Out]

-(Sqrt[1 - 2*x]/(3 + 5*x)) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (68*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Rubi [A]  time = 0.118421, antiderivative size = 69, 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 \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]/(3 + 5*x)) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (68*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Rubi in Sympy [A]  time = 14.7759, size = 61, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1}}{5 x + 3} - 2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )} + \frac{68 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)/(5*x + 3) - 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7) + 68*sqr
t(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55

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Mathematica [A]  time = 0.119936, size = 69, normalized size = 1. \[ -\frac{\sqrt{1-2 x}}{5 x+3}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]/(3 + 5*x)) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (68*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Maple [A]  time = 0.017, size = 54, normalized size = 0.8 \[ -2\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+{\frac{2}{5}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{68\,\sqrt{55}}{55}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/5*(1-2*x)^(1/2)/(-6/5-2*x)+68/
55*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50384, size = 119, normalized size = 1.72 \[ -\frac{34}{55} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{\sqrt{-2 \, x + 1}}{5 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-34/55*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) -
sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.2205, size = 131, normalized size = 1.9 \[ \frac{\sqrt{55}{\left (\sqrt{55} \sqrt{21}{\left (5 \, x + 3\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 34 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55} \sqrt{-2 \, x + 1}\right )}}{55 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/55*sqrt(55)*(sqrt(55)*sqrt(21)*(5*x + 3)*log((3*x + sqrt(21)*sqrt(-2*x + 1) -
5)/(3*x + 2)) + 34*(5*x + 3)*log((sqrt(55)*(5*x - 8) - 55*sqrt(-2*x + 1))/(5*x +
3)) - sqrt(55)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 25.0942, size = 223, normalized size = 3.23 \[ - 44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right ) + 42 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) - 70 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-44*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*s
qrt(-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))) + 42*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)) - 70*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))

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GIAC/XCAS [A]  time = 0.214093, size = 127, normalized size = 1.84 \[ -\frac{34}{55} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{\sqrt{-2 \, x + 1}}{5 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-34/55*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - sqrt(-2*x + 1)/(5*x + 3)