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 = 0.0163552, 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, 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*}

Mathematica [A]  time = 0.0090066, 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 = 0.005, size = 38, normalized size = 0.8 \begin{align*} -{\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}}} \end{align*}

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 = 1.04076, size = 59, normalized size = 1.31 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.89596, size = 131, normalized size = 2.91 \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 0.143193, size = 39, normalized size = 0.87 \begin{align*} \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} \end{align*}

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 [A]  time = 1.63445, size = 32, normalized size = 0.71 \begin{align*} \frac{3375 \, x^{3} + 3825 \, x^{2} + 870 \, x - 127}{405 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

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