3.1578 \(\int \frac{2+3 x}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=32 \[ \frac{7}{22 (1-2 x)}-\frac{1}{121} \log (1-2 x)+\frac{1}{121} \log (5 x+3) \]

[Out]

7/(22*(1 - 2*x)) - Log[1 - 2*x]/121 + Log[3 + 5*x]/121

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Rubi [A]  time = 0.0393525, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{7}{22 (1-2 x)}-\frac{1}{121} \log (1-2 x)+\frac{1}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

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

[Out]

7/(22*(1 - 2*x)) - Log[1 - 2*x]/121 + Log[3 + 5*x]/121

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Rubi in Sympy [A]  time = 6.27508, size = 22, normalized size = 0.69 \[ - \frac{\log{\left (- 2 x + 1 \right )}}{121} + \frac{\log{\left (5 x + 3 \right )}}{121} + \frac{7}{22 \left (- 2 x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-2*x + 1)/121 + log(5*x + 3)/121 + 7/(22*(-2*x + 1))

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Mathematica [A]  time = 0.0205567, size = 37, normalized size = 1.16 \[ \frac{(2-4 x) \log (1-2 x)+(4 x-2) \log (10 x+6)-77}{242 (2 x-1)} \]

Antiderivative was successfully verified.

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

[Out]

(-77 + (2 - 4*x)*Log[1 - 2*x] + (-2 + 4*x)*Log[6 + 10*x])/(242*(-1 + 2*x))

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Maple [A]  time = 0.01, size = 27, normalized size = 0.8 \[{\frac{\ln \left ( 3+5\,x \right ) }{121}}-{\frac{7}{-22+44\,x}}-{\frac{\ln \left ( -1+2\,x \right ) }{121}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/121*ln(3+5*x)-7/22/(-1+2*x)-1/121*ln(-1+2*x)

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Maxima [A]  time = 1.31746, size = 35, normalized size = 1.09 \[ -\frac{7}{22 \,{\left (2 \, x - 1\right )}} + \frac{1}{121} \, \log \left (5 \, x + 3\right ) - \frac{1}{121} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/22/(2*x - 1) + 1/121*log(5*x + 3) - 1/121*log(2*x - 1)

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Fricas [A]  time = 0.208951, size = 50, normalized size = 1.56 \[ \frac{2 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) - 2 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 77}{242 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/242*(2*(2*x - 1)*log(5*x + 3) - 2*(2*x - 1)*log(2*x - 1) - 77)/(2*x - 1)

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Sympy [A]  time = 0.240886, size = 22, normalized size = 0.69 \[ - \frac{\log{\left (x - \frac{1}{2} \right )}}{121} + \frac{\log{\left (x + \frac{3}{5} \right )}}{121} - \frac{7}{44 x - 22} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x - 1/2)/121 + log(x + 3/5)/121 - 7/(44*x - 22)

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GIAC/XCAS [A]  time = 0.212111, size = 34, normalized size = 1.06 \[ -\frac{7}{22 \,{\left (2 \, x - 1\right )}} + \frac{1}{121} \,{\rm ln}\left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/22/(2*x - 1) + 1/121*ln(abs(-11/(2*x - 1) - 5))