3.1661 \(\int \frac{2+3 x}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{121 (1-2 x)}+\frac{7}{44 (1-2 x)^2}-\frac{5 \log (1-2 x)}{1331}+\frac{5 \log (5 x+3)}{1331} \]

[Out]

7/(44*(1 - 2*x)^2) + 1/(121*(1 - 2*x)) - (5*Log[1 - 2*x])/1331 + (5*Log[3 + 5*x]
)/1331

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Rubi [A]  time = 0.0459588, antiderivative size = 43, 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 \[ \frac{1}{121 (1-2 x)}+\frac{7}{44 (1-2 x)^2}-\frac{5 \log (1-2 x)}{1331}+\frac{5 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

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

[Out]

7/(44*(1 - 2*x)^2) + 1/(121*(1 - 2*x)) - (5*Log[1 - 2*x])/1331 + (5*Log[3 + 5*x]
)/1331

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Rubi in Sympy [A]  time = 7.34944, size = 36, normalized size = 0.84 \[ - \frac{5 \log{\left (- 2 x + 1 \right )}}{1331} + \frac{5 \log{\left (5 x + 3 \right )}}{1331} + \frac{1}{121 \left (- 2 x + 1\right )} + \frac{7}{44 \left (- 2 x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*log(-2*x + 1)/1331 + 5*log(5*x + 3)/1331 + 1/(121*(-2*x + 1)) + 7/(44*(-2*x +
1)**2)

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Mathematica [A]  time = 0.0257871, size = 46, normalized size = 1.07 \[ \frac{-88 x-20 (1-2 x)^2 \log (1-2 x)+20 (1-2 x)^2 \log (10 x+6)+891}{5324 (1-2 x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(891 - 88*x - 20*(1 - 2*x)^2*Log[1 - 2*x] + 20*(1 - 2*x)^2*Log[6 + 10*x])/(5324*
(1 - 2*x)^2)

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Maple [A]  time = 0.012, size = 36, normalized size = 0.8 \[{\frac{5\,\ln \left ( 3+5\,x \right ) }{1331}}+{\frac{7}{44\, \left ( -1+2\,x \right ) ^{2}}}-{\frac{1}{-121+242\,x}}-{\frac{5\,\ln \left ( -1+2\,x \right ) }{1331}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/1331*ln(3+5*x)+7/44/(-1+2*x)^2-1/121/(-1+2*x)-5/1331*ln(-1+2*x)

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Maxima [A]  time = 1.35605, size = 49, normalized size = 1.14 \[ -\frac{8 \, x - 81}{484 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5}{1331} \, \log \left (5 \, x + 3\right ) - \frac{5}{1331} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/484*(8*x - 81)/(4*x^2 - 4*x + 1) + 5/1331*log(5*x + 3) - 5/1331*log(2*x - 1)

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Fricas [A]  time = 0.213874, size = 74, normalized size = 1.72 \[ \frac{20 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 20 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 88 \, x + 891}{5324 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5324*(20*(4*x^2 - 4*x + 1)*log(5*x + 3) - 20*(4*x^2 - 4*x + 1)*log(2*x - 1) -
88*x + 891)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.316571, size = 34, normalized size = 0.79 \[ - \frac{8 x - 81}{1936 x^{2} - 1936 x + 484} - \frac{5 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{5 \log{\left (x + \frac{3}{5} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(8*x - 81)/(1936*x**2 - 1936*x + 484) - 5*log(x - 1/2)/1331 + 5*log(x + 3/5)/13
31

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GIAC/XCAS [A]  time = 0.229772, size = 45, normalized size = 1.05 \[ -\frac{8 \, x - 81}{484 \,{\left (2 \, x - 1\right )}^{2}} + \frac{5}{1331} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - \frac{5}{1331} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/484*(8*x - 81)/(2*x - 1)^2 + 5/1331*ln(abs(5*x + 3)) - 5/1331*ln(abs(2*x - 1)
)