3.1173 \(\int \frac{(1-2 x) (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=45 \[ \frac{50}{243 (3 x+2)^3}-\frac{65}{108 (3 x+2)^4}+\frac{8}{45 (3 x+2)^5}-\frac{7}{486 (3 x+2)^6} \]

[Out]

-7/(486*(2 + 3*x)^6) + 8/(45*(2 + 3*x)^5) - 65/(108*(2 + 3*x)^4) + 50/(243*(2 + 3*x)^3)

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Rubi [A]  time = 0.0167643, 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} \[ \frac{50}{243 (3 x+2)^3}-\frac{65}{108 (3 x+2)^4}+\frac{8}{45 (3 x+2)^5}-\frac{7}{486 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-7/(486*(2 + 3*x)^6) + 8/(45*(2 + 3*x)^5) - 65/(108*(2 + 3*x)^4) + 50/(243*(2 + 3*x)^3)

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)^7} \, dx &=\int \left (\frac{7}{27 (2+3 x)^7}-\frac{8}{3 (2+3 x)^6}+\frac{65}{9 (2+3 x)^5}-\frac{50}{27 (2+3 x)^4}\right ) \, dx\\ &=-\frac{7}{486 (2+3 x)^6}+\frac{8}{45 (2+3 x)^5}-\frac{65}{108 (2+3 x)^4}+\frac{50}{243 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0087142, size = 26, normalized size = 0.58 \[ \frac{27000 x^3+27675 x^2+3492 x-2042}{4860 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

(-2042 + 3492*x + 27675*x^2 + 27000*x^3)/(4860*(2 + 3*x)^6)

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*} -{\frac{7}{486\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{8}{45\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{65}{108\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{50}{243\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3+5*x)^2/(2+3*x)^7,x)

[Out]

-7/486/(2+3*x)^6+8/45/(2+3*x)^5-65/108/(2+3*x)^4+50/243/(2+3*x)^3

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Maxima [A]  time = 2.08019, size = 66, normalized size = 1.47 \begin{align*} \frac{27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="maxima")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x +
64)

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Fricas [A]  time = 1.74405, size = 155, normalized size = 3.44 \begin{align*} \frac{27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x +
64)

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Sympy [A]  time = 0.154109, size = 44, normalized size = 0.98 \begin{align*} \frac{27000 x^{3} + 27675 x^{2} + 3492 x - 2042}{3542940 x^{6} + 14171760 x^{5} + 23619600 x^{4} + 20995200 x^{3} + 10497600 x^{2} + 2799360 x + 311040} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**2/(2+3*x)**7,x)

[Out]

(27000*x**3 + 27675*x**2 + 3492*x - 2042)/(3542940*x**6 + 14171760*x**5 + 23619600*x**4 + 20995200*x**3 + 1049
7600*x**2 + 2799360*x + 311040)

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Giac [A]  time = 3.61151, size = 32, normalized size = 0.71 \begin{align*} \frac{27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^7,x, algorithm="giac")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(3*x + 2)^6