3.1155 \(\int \frac{(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=45 \[ \frac{25}{81 (3 x+2)^2}-\frac{65}{81 (3 x+2)^3}+\frac{2}{9 (3 x+2)^4}-\frac{7}{405 (3 x+2)^5} \]

[Out]

-7/(405*(2 + 3*x)^5) + 2/(9*(2 + 3*x)^4) - 65/(81*(2 + 3*x)^3) + 25/(81*(2 + 3*x
)^2)

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Rubi [A]  time = 0.0510638, antiderivative size = 45, 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 \[ \frac{25}{81 (3 x+2)^2}-\frac{65}{81 (3 x+2)^3}+\frac{2}{9 (3 x+2)^4}-\frac{7}{405 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-7/(405*(2 + 3*x)^5) + 2/(9*(2 + 3*x)^4) - 65/(81*(2 + 3*x)^3) + 25/(81*(2 + 3*x
)^2)

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Rubi in Sympy [A]  time = 8.1492, size = 39, normalized size = 0.87 \[ \frac{25}{81 \left (3 x + 2\right )^{2}} - \frac{65}{81 \left (3 x + 2\right )^{3}} + \frac{2}{9 \left (3 x + 2\right )^{4}} - \frac{7}{405 \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)*(3+5*x)**2/(2+3*x)**6,x)

[Out]

25/(81*(3*x + 2)**2) - 65/(81*(3*x + 2)**3) + 2/(9*(3*x + 2)**4) - 7/(405*(3*x +
2)**5)

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Mathematica [A]  time = 0.0146415, size = 26, normalized size = 0.58 \[ \frac{3375 x^3+3825 x^2+870 x-127}{405 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

(-127 + 870*x + 3825*x^2 + 3375*x^3)/(405*(2 + 3*x)^5)

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Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[ -{\frac{7}{405\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{2}{9\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{65}{81\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{25}{81\, \left ( 2+3\,x \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)*(3+5*x)^2/(2+3*x)^6,x)

[Out]

-7/405/(2+3*x)^5+2/9/(2+3*x)^4-65/81/(2+3*x)^3+25/81/(2+3*x)^2

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Maxima [A]  time = 1.36609, size = 59, normalized size = 1.31 \[ \frac{3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)^2*(2*x - 1)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^
2 + 240*x + 32)

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Fricas [A]  time = 0.220526, size = 59, normalized size = 1.31 \[ \frac{3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)^2*(2*x - 1)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^
2 + 240*x + 32)

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Sympy [A]  time = 0.379965, size = 39, normalized size = 0.87 \[ \frac{3375 x^{3} + 3825 x^{2} + 870 x - 127}{98415 x^{5} + 328050 x^{4} + 437400 x^{3} + 291600 x^{2} + 97200 x + 12960} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)*(3+5*x)**2/(2+3*x)**6,x)

[Out]

(3375*x**3 + 3825*x**2 + 870*x - 127)/(98415*x**5 + 328050*x**4 + 437400*x**3 +
291600*x**2 + 97200*x + 12960)

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GIAC/XCAS [A]  time = 0.209218, size = 32, normalized size = 0.71 \[ \frac{3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)^2*(2*x - 1)/(3*x + 2)^6,x, algorithm="giac")

[Out]

1/405*(3375*x^3 + 3825*x^2 + 870*x - 127)/(3*x + 2)^5