3.1817 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.122492, antiderivative size = 68, 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 \[ \frac{\sqrt{1-2 x}}{3 x+2}+\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-2 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.9269, size = 61, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1}}{3 x + 2} + \frac{68 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} - 2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)/(3*x + 2) + 68*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21 - 2*s
qrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.121971, size = 81, normalized size = 1.19 \[ \frac{68 \sqrt{21} (3 x+2) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+21 \left (\sqrt{1-2 x}-2 \sqrt{55} (3 x+2) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{63 x+42} \]

Antiderivative was successfully verified.

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

[Out]

(68*Sqrt[21]*(2 + 3*x)*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 21*(Sqrt[1 - 2*x] - 2*
Sqrt[55]*(2 + 3*x)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/(42 + 63*x)

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Maple [A]  time = 0.015, size = 54, normalized size = 0.8 \[ -{\frac{2}{3}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{68\,\sqrt{21}}{21}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-2\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(1-2*x)^(1/2)/(-4/3-2*x)+68/21*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
-2*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.679, size = 117, normalized size = 1.72 \[ \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{34}{21} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{\sqrt{-2 \, x + 1}}{3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219221, size = 130, normalized size = 1.91 \[ \frac{\sqrt{21}{\left (\sqrt{55} \sqrt{21}{\left (3 \, x + 2\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 34 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21} \sqrt{-2 \, x + 1}\right )}}{21 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 27.4837, size = 223, normalized size = 3.28 \[ 28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right ) - 66 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) + 110 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqr
t(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sq
rt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 66*Piecewise((-sqrt(21)*a
coth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*s
qrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3)) + 110*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 = 0.219452, size = 126, normalized size = 1.85 \[ \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{34}{21} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{\sqrt{-2 \, x + 1}}{3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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