3.1350 \(\int \frac{(1-2 x)^3 (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=49 \[ -\frac{40 x}{81}+\frac{518}{81 (3 x+2)}-\frac{2009}{486 (3 x+2)^2}+\frac{343}{729 (3 x+2)^3}+\frac{428}{243} \log (3 x+2) \]

[Out]

(-40*x)/81 + 343/(729*(2 + 3*x)^3) - 2009/(486*(2 + 3*x)^2) + 518/(81*(2 + 3*x)) + (428*Log[2 + 3*x])/243

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Rubi [A]  time = 0.0202781, antiderivative size = 49, 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} \[ -\frac{40 x}{81}+\frac{518}{81 (3 x+2)}-\frac{2009}{486 (3 x+2)^2}+\frac{343}{729 (3 x+2)^3}+\frac{428}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*x)/81 + 343/(729*(2 + 3*x)^3) - 2009/(486*(2 + 3*x)^2) + 518/(81*(2 + 3*x)) + (428*Log[2 + 3*x])/243

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 (3+5 x)}{(2+3 x)^4} \, dx &=\int \left (-\frac{40}{81}-\frac{343}{81 (2+3 x)^4}+\frac{2009}{81 (2+3 x)^3}-\frac{518}{27 (2+3 x)^2}+\frac{428}{81 (2+3 x)}\right ) \, dx\\ &=-\frac{40 x}{81}+\frac{343}{729 (2+3 x)^3}-\frac{2009}{486 (2+3 x)^2}+\frac{518}{81 (2+3 x)}+\frac{428}{243} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0147109, size = 46, normalized size = 0.94 \[ \frac{-19440 x^4-51840 x^3+32076 x^2+70767 x+2568 (3 x+2)^3 \log (3 x+2)+22088}{1458 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(22088 + 70767*x + 32076*x^2 - 51840*x^3 - 19440*x^4 + 2568*(2 + 3*x)^3*Log[2 + 3*x])/(1458*(2 + 3*x)^3)

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Maple [A]  time = 0.007, size = 40, normalized size = 0.8 \begin{align*} -{\frac{40\,x}{81}}+{\frac{343}{729\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{2009}{486\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{518}{162+243\,x}}+{\frac{428\,\ln \left ( 2+3\,x \right ) }{243}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40/81*x+343/729/(2+3*x)^3-2009/486/(2+3*x)^2+518/81/(2+3*x)+428/243*ln(2+3*x)

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Maxima [A]  time = 1.04761, size = 55, normalized size = 1.12 \begin{align*} -\frac{40}{81} \, x + \frac{7 \,{\left (11988 \, x^{2} + 13401 \, x + 3704\right )}}{1458 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{428}{243} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40/81*x + 7/1458*(11988*x^2 + 13401*x + 3704)/(27*x^3 + 54*x^2 + 36*x + 8) + 428/243*log(3*x + 2)

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Fricas [A]  time = 1.32111, size = 190, normalized size = 3.88 \begin{align*} -\frac{19440 \, x^{4} + 38880 \, x^{3} - 57996 \, x^{2} - 2568 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 88047 \, x - 25928}{1458 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1458*(19440*x^4 + 38880*x^3 - 57996*x^2 - 2568*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 88047*x - 25928)
/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.129777, size = 39, normalized size = 0.8 \begin{align*} - \frac{40 x}{81} + \frac{83916 x^{2} + 93807 x + 25928}{39366 x^{3} + 78732 x^{2} + 52488 x + 11664} + \frac{428 \log{\left (3 x + 2 \right )}}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40*x/81 + (83916*x**2 + 93807*x + 25928)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 428*log(3*x + 2)/243

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Giac [A]  time = 3.22229, size = 43, normalized size = 0.88 \begin{align*} -\frac{40}{81} \, x + \frac{7 \,{\left (11988 \, x^{2} + 13401 \, x + 3704\right )}}{1458 \,{\left (3 \, x + 2\right )}^{3}} + \frac{428}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-40/81*x + 7/1458*(11988*x^2 + 13401*x + 3704)/(3*x + 2)^3 + 428/243*log(abs(3*x + 2))