3.1475 \(\int \frac{2+3 x}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

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Rubi [A]  time = 0.0301171, antiderivative size = 21, 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 \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

Antiderivative was successfully verified.

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

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

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Rubi in Sympy [A]  time = 5.29214, size = 17, normalized size = 0.81 \[ - \frac{7 \log{\left (- 2 x + 1 \right )}}{22} + \frac{\log{\left (5 x + 3 \right )}}{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*log(-2*x + 1)/22 + log(5*x + 3)/55

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Mathematica [A]  time = 0.0066902, size = 21, normalized size = 1. \[ \frac{1}{55} \log (5 x+3)-\frac{7}{22} \log (1-2 x) \]

Antiderivative was successfully verified.

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

[Out]

(-7*Log[1 - 2*x])/22 + Log[3 + 5*x]/55

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Maple [A]  time = 0.008, size = 18, normalized size = 0.9 \[{\frac{\ln \left ( 3+5\,x \right ) }{55}}-{\frac{7\,\ln \left ( -1+2\,x \right ) }{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)/(1-2*x)/(3+5*x),x)

[Out]

1/55*ln(3+5*x)-7/22*ln(-1+2*x)

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Maxima [A]  time = 1.34146, size = 23, normalized size = 1.1 \[ \frac{1}{55} \, \log \left (5 \, x + 3\right ) - \frac{7}{22} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/55*log(5*x + 3) - 7/22*log(2*x - 1)

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Fricas [A]  time = 0.212682, size = 23, normalized size = 1.1 \[ \frac{1}{55} \, \log \left (5 \, x + 3\right ) - \frac{7}{22} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/55*log(5*x + 3) - 7/22*log(2*x - 1)

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Sympy [A]  time = 0.252636, size = 17, normalized size = 0.81 \[ - \frac{7 \log{\left (x - \frac{1}{2} \right )}}{22} + \frac{\log{\left (x + \frac{3}{5} \right )}}{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*log(x - 1/2)/22 + log(x + 3/5)/55

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GIAC/XCAS [A]  time = 0.209539, size = 26, normalized size = 1.24 \[ \frac{1}{55} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - \frac{7}{22} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/55*ln(abs(5*x + 3)) - 7/22*ln(abs(2*x - 1))