3.1445 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{68}{441 (3 x+2)}-\frac{1}{126 (3 x+2)^2}-\frac{121}{343} \log (1-2 x)+\frac{121}{343} \log (3 x+2) \]

[Out]

-1/(126*(2 + 3*x)^2) + 68/(441*(2 + 3*x)) - (121*Log[1 - 2*x])/343 + (121*Log[2
+ 3*x])/343

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Rubi [A]  time = 0.0502732, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{68}{441 (3 x+2)}-\frac{1}{126 (3 x+2)^2}-\frac{121}{343} \log (1-2 x)+\frac{121}{343} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-1/(126*(2 + 3*x)^2) + 68/(441*(2 + 3*x)) - (121*Log[1 - 2*x])/343 + (121*Log[2
+ 3*x])/343

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Rubi in Sympy [A]  time = 7.89337, size = 36, normalized size = 0.84 \[ - \frac{121 \log{\left (- 2 x + 1 \right )}}{343} + \frac{121 \log{\left (3 x + 2 \right )}}{343} + \frac{68}{441 \left (3 x + 2\right )} - \frac{1}{126 \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-121*log(-2*x + 1)/343 + 121*log(3*x + 2)/343 + 68/(441*(3*x + 2)) - 1/(126*(3*x
+ 2)**2)

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Mathematica [A]  time = 0.0330904, size = 35, normalized size = 0.81 \[ \frac{\frac{7 (408 x+265)}{(3 x+2)^2}-2178 \log (1-2 x)+2178 \log (6 x+4)}{6174} \]

Antiderivative was successfully verified.

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

[Out]

((7*(265 + 408*x))/(2 + 3*x)^2 - 2178*Log[1 - 2*x] + 2178*Log[4 + 6*x])/6174

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Maple [A]  time = 0.012, size = 36, normalized size = 0.8 \[ -{\frac{1}{126\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{68}{882+1323\,x}}+{\frac{121\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{121\,\ln \left ( -1+2\,x \right ) }{343}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/126/(2+3*x)^2+68/441/(2+3*x)+121/343*ln(2+3*x)-121/343*ln(-1+2*x)

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Maxima [A]  time = 1.34055, size = 49, normalized size = 1.14 \[ \frac{408 \, x + 265}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{121}{343} \, \log \left (3 \, x + 2\right ) - \frac{121}{343} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/882*(408*x + 265)/(9*x^2 + 12*x + 4) + 121/343*log(3*x + 2) - 121/343*log(2*x
- 1)

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Fricas [A]  time = 0.20316, size = 74, normalized size = 1.72 \[ \frac{2178 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 2178 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x - 1\right ) + 2856 \, x + 1855}{6174 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6174*(2178*(9*x^2 + 12*x + 4)*log(3*x + 2) - 2178*(9*x^2 + 12*x + 4)*log(2*x -
1) + 2856*x + 1855)/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 0.380362, size = 34, normalized size = 0.79 \[ \frac{408 x + 265}{7938 x^{2} + 10584 x + 3528} - \frac{121 \log{\left (x - \frac{1}{2} \right )}}{343} + \frac{121 \log{\left (x + \frac{2}{3} \right )}}{343} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(408*x + 265)/(7938*x**2 + 10584*x + 3528) - 121*log(x - 1/2)/343 + 121*log(x +
2/3)/343

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GIAC/XCAS [A]  time = 0.211206, size = 45, normalized size = 1.05 \[ \frac{408 \, x + 265}{882 \,{\left (3 \, x + 2\right )}^{2}} + \frac{121}{343} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{121}{343} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/882*(408*x + 265)/(3*x + 2)^2 + 121/343*ln(abs(3*x + 2)) - 121/343*ln(abs(2*x
- 1))