3.1172 \(\int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx\)

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 = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {25}{81 (3 x+2)^2}-\frac {65}{81 (3 x+2)^3}+\frac {2}{9 (3 x+2)^4}-\frac {7}{405 (3 x+2)^5} \]

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)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^6} \, dx &=\int \left (\frac {7}{27 (2+3 x)^6}-\frac {8}{3 (2+3 x)^5}+\frac {65}{9 (2+3 x)^4}-\frac {50}{27 (2+3 x)^3}\right ) \, dx\\ &=-\frac {7}{405 (2+3 x)^5}+\frac {2}{9 (2+3 x)^4}-\frac {65}{81 (2+3 x)^3}+\frac {25}{81 (2+3 x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.74, size = 44, normalized size = 0.98 \[ \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^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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giac [A]  time = 1.22, size = 24, normalized size = 0.53 \[ \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 38, normalized size = 0.84 \[ -\frac {7}{405 \left (3 x +2\right )^{5}}+\frac {2}{9 \left (3 x +2\right )^{4}}-\frac {65}{81 \left (3 x +2\right )^{3}}+\frac {25}{81 \left (3 x +2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 44, normalized size = 0.98 \[ \frac {3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^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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mupad [B]  time = 1.10, size = 37, normalized size = 0.82 \[ \frac {25}{81\,{\left (3\,x+2\right )}^2}-\frac {65}{81\,{\left (3\,x+2\right )}^3}+\frac {2}{9\,{\left (3\,x+2\right )}^4}-\frac {7}{405\,{\left (3\,x+2\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)*(5*x + 3)^2)/(3*x + 2)^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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sympy [A]  time = 0.15, size = 41, normalized size = 0.91 \[ - \frac {- 3375 x^{3} - 3825 x^{2} - 870 x + 127}{98415 x^{5} + 328050 x^{4} + 437400 x^{3} + 291600 x^{2} + 97200 x + 12960} \]

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 + 1296
0)

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