3.1991 \(\int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{\sqrt{1-2 x}}{21 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

Sqrt[1 - 2*x]/(21*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21
])

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Rubi [A]  time = 0.0540135, antiderivative size = 48, 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 \[ \frac{\sqrt{1-2 x}}{21 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(21*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21
])

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Rubi in Sympy [A]  time = 5.27715, size = 37, normalized size = 0.77 \[ \frac{\sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )} - \frac{68 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)/(21*(3*x + 2)) - 68*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/441

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Mathematica [A]  time = 0.0708318, size = 45, normalized size = 0.94 \[ \frac{\sqrt{1-2 x}}{63 x+42}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 36, normalized size = 0.8 \[ -{\frac{2}{63}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{68\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/63*(1-2*x)^(1/2)/(-4/3-2*x)-68/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/
2)

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Maxima [A]  time = 1.50246, size = 72, normalized size = 1.5 \[ \frac{34}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{\sqrt{-2 \, x + 1}}{21 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) + 1/21*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.231937, size = 80, normalized size = 1.67 \[ \frac{\sqrt{21}{\left (34 \,{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21} \sqrt{-2 \, x + 1}\right )}}{441 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/441*sqrt(21)*(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.23556, size = 76, normalized size = 1.58 \[ \frac{34}{441} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{\sqrt{-2 \, x + 1}}{21 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/441*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 1/21*sqrt(-2*x + 1)/(3*x + 2)