3.1990 \(\int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]

[Out]

(-5*Sqrt[1 - 2*x])/3 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])

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Rubi [A]  time = 0.0497778, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x])/3 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])

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Rubi in Sympy [A]  time = 4.89695, size = 36, normalized size = 0.88 \[ - \frac{5 \sqrt{- 2 x + 1}}{3} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)/3 + 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/63

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Mathematica [A]  time = 0.0452158, size = 41, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-\frac{5}{3} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x])/3 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])

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Maple [A]  time = 0.008, size = 29, normalized size = 0.7 \[{\frac{2\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{5}{3}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-5/3*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49645, size = 62, normalized size = 1.51 \[ -\frac{1}{63} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5}{3} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/63*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))
- 5/3*sqrt(-2*x + 1)

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Fricas [A]  time = 0.228605, size = 65, normalized size = 1.59 \[ -\frac{1}{63} \, \sqrt{21}{\left (5 \, \sqrt{21} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/63*sqrt(21)*(5*sqrt(21)*sqrt(-2*x + 1) - log((sqrt(21)*(3*x - 5) - 21*sqrt(-2
*x + 1))/(3*x + 2)))

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Sympy [A]  time = 3.56448, size = 80, normalized size = 1.95 \[ - \frac{5 \sqrt{- 2 x + 1}}{3} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{3}{7} \end{cases}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)/3 - 2*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))/3

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GIAC/XCAS [A]  time = 0.230919, size = 66, normalized size = 1.61 \[ -\frac{1}{63} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5}{3} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/63*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 5/3*sqrt(-2*x + 1)