3.2025 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0800213, 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 \[ 2 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.35821, size = 49, normalized size = 0.89 \[ \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0448507, size = 55, normalized size = 1. \[ 2 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-2 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50539, size = 99, normalized size = 1.8 \[ \frac{1}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224545, size = 115, normalized size = 2.09 \[ \frac{1}{77} \, \sqrt{11} \sqrt{7}{\left (\sqrt{7} \sqrt{5} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11} \sqrt{3} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/77*sqrt(11)*sqrt(7)*(sqrt(7)*sqrt(5)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt
(-2*x + 1))/(5*x + 3)) + sqrt(11)*sqrt(3)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqr
t(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 8.30647, size = 131, normalized size = 2.38 \[ - 6 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{3}{7} \end{cases}\right ) + 10 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*Piecewise((-sqrt(21)*acoth(sqrt(21)/(3*sqrt(-2*x + 1)))/21, 1/(-2*x + 1) > 3/
7), (-sqrt(21)*atanh(sqrt(21)/(3*sqrt(-2*x + 1)))/21, 1/(-2*x + 1) < 3/7)) + 10*
Piecewise((-sqrt(55)*acoth(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) > 5/11)
, (-sqrt(55)*atanh(sqrt(55)/(5*sqrt(-2*x + 1)))/55, 1/(-2*x + 1) < 5/11))

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GIAC/XCAS [A]  time = 0.216553, size = 107, normalized size = 1.95 \[ \frac{1}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1}{7} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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