3.1231 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=44 \[ \frac{16}{9 (3 x+2)}-\frac{91}{54 (3 x+2)^2}+\frac{49}{243 (3 x+2)^3}+\frac{20}{81} \log (3 x+2) \]

[Out]

49/(243*(2 + 3*x)^3) - 91/(54*(2 + 3*x)^2) + 16/(9*(2 + 3*x)) + (20*Log[2 + 3*x]
)/81

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Rubi [A]  time = 0.0453486, antiderivative size = 44, 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 \[ \frac{16}{9 (3 x+2)}-\frac{91}{54 (3 x+2)^2}+\frac{49}{243 (3 x+2)^3}+\frac{20}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

49/(243*(2 + 3*x)^3) - 91/(54*(2 + 3*x)^2) + 16/(9*(2 + 3*x)) + (20*Log[2 + 3*x]
)/81

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Rubi in Sympy [A]  time = 7.64278, size = 36, normalized size = 0.82 \[ \frac{20 \log{\left (3 x + 2 \right )}}{81} + \frac{16}{9 \left (3 x + 2\right )} - \frac{91}{54 \left (3 x + 2\right )^{2}} + \frac{49}{243 \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**2*(3+5*x)/(2+3*x)**4,x)

[Out]

20*log(3*x + 2)/81 + 16/(9*(3*x + 2)) - 91/(54*(3*x + 2)**2) + 49/(243*(3*x + 2)
**3)

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Mathematica [A]  time = 0.0230535, size = 36, normalized size = 0.82 \[ \frac{7776 x^2+7911 x+120 (3 x+2)^3 \log (3 x+2)+1916}{486 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^4,x]

[Out]

(1916 + 7911*x + 7776*x^2 + 120*(2 + 3*x)^3*Log[2 + 3*x])/(486*(2 + 3*x)^3)

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Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \[{\frac{49}{243\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{91}{54\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{16}{18+27\,x}}+{\frac{20\,\ln \left ( 2+3\,x \right ) }{81}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^2*(3+5*x)/(2+3*x)^4,x)

[Out]

49/243/(2+3*x)^3-91/54/(2+3*x)^2+16/9/(2+3*x)+20/81*ln(2+3*x)

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Maxima [A]  time = 1.34288, size = 51, normalized size = 1.16 \[ \frac{7776 \, x^{2} + 7911 \, x + 1916}{486 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{20}{81} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(2*x - 1)^2/(3*x + 2)^4,x, algorithm="maxima")

[Out]

1/486*(7776*x^2 + 7911*x + 1916)/(27*x^3 + 54*x^2 + 36*x + 8) + 20/81*log(3*x +
2)

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Fricas [A]  time = 0.214561, size = 70, normalized size = 1.59 \[ \frac{7776 \, x^{2} + 120 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 7911 \, x + 1916}{486 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(2*x - 1)^2/(3*x + 2)^4,x, algorithm="fricas")

[Out]

1/486*(7776*x^2 + 120*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 7911*x + 1916)
/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.307259, size = 34, normalized size = 0.77 \[ \frac{7776 x^{2} + 7911 x + 1916}{13122 x^{3} + 26244 x^{2} + 17496 x + 3888} + \frac{20 \log{\left (3 x + 2 \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**2*(3+5*x)/(2+3*x)**4,x)

[Out]

(7776*x**2 + 7911*x + 1916)/(13122*x**3 + 26244*x**2 + 17496*x + 3888) + 20*log(
3*x + 2)/81

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GIAC/XCAS [A]  time = 0.207762, size = 39, normalized size = 0.89 \[ \frac{7776 \, x^{2} + 7911 \, x + 1916}{486 \,{\left (3 \, x + 2\right )}^{3}} + \frac{20}{81} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(2*x - 1)^2/(3*x + 2)^4,x, algorithm="giac")

[Out]

1/486*(7776*x^2 + 7911*x + 1916)/(3*x + 2)^3 + 20/81*ln(abs(3*x + 2))