3.1491 \(\int \frac{2+3 x}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{55 (5 x+3)}-\frac{7}{121} \log (1-2 x)+\frac{7}{121} \log (5 x+3) \]

[Out]

-1/(55*(3 + 5*x)) - (7*Log[1 - 2*x])/121 + (7*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0394648, 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 \[ -\frac{1}{55 (5 x+3)}-\frac{7}{121} \log (1-2 x)+\frac{7}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

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

[Out]

-1/(55*(3 + 5*x)) - (7*Log[1 - 2*x])/121 + (7*Log[3 + 5*x])/121

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Rubi in Sympy [A]  time = 6.26449, size = 26, normalized size = 0.81 \[ - \frac{7 \log{\left (- 2 x + 1 \right )}}{121} + \frac{7 \log{\left (5 x + 3 \right )}}{121} - \frac{1}{55 \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*log(-2*x + 1)/121 + 7*log(5*x + 3)/121 - 1/(55*(5*x + 3))

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Mathematica [A]  time = 0.0229537, size = 30, normalized size = 0.94 \[ \frac{1}{605} \left (-\frac{11}{5 x+3}-35 \log (5-10 x)+35 \log (5 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-11/(3 + 5*x) - 35*Log[5 - 10*x] + 35*Log[3 + 5*x])/605

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Maple [A]  time = 0.01, size = 27, normalized size = 0.8 \[ -{\frac{1}{165+275\,x}}+{\frac{7\,\ln \left ( 3+5\,x \right ) }{121}}-{\frac{7\,\ln \left ( -1+2\,x \right ) }{121}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/55/(3+5*x)+7/121*ln(3+5*x)-7/121*ln(-1+2*x)

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Maxima [A]  time = 1.34882, size = 35, normalized size = 1.09 \[ -\frac{1}{55 \,{\left (5 \, x + 3\right )}} + \frac{7}{121} \, \log \left (5 \, x + 3\right ) - \frac{7}{121} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/55/(5*x + 3) + 7/121*log(5*x + 3) - 7/121*log(2*x - 1)

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Fricas [A]  time = 0.211308, size = 50, normalized size = 1.56 \[ \frac{35 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 35 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 11}{605 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/605*(35*(5*x + 3)*log(5*x + 3) - 35*(5*x + 3)*log(2*x - 1) - 11)/(5*x + 3)

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Sympy [A]  time = 0.269817, size = 26, normalized size = 0.81 \[ - \frac{7 \log{\left (x - \frac{1}{2} \right )}}{121} + \frac{7 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{1}{275 x + 165} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*log(x - 1/2)/121 + 7*log(x + 3/5)/121 - 1/(275*x + 165)

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GIAC/XCAS [A]  time = 0.207384, size = 34, normalized size = 1.06 \[ -\frac{1}{55 \,{\left (5 \, x + 3\right )}} - \frac{7}{121} \,{\rm ln}\left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/55/(5*x + 3) - 7/121*ln(abs(-11/(5*x + 3) + 2))