3.1349 \(\int \frac{(1-2 x)^3 (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{20 x^2}{27}+\frac{116 x}{27}-\frac{2009}{243 (3 x+2)}+\frac{343}{486 (3 x+2)^2}-\frac{518}{81} \log (3 x+2) \]

[Out]

(116*x)/27 - (20*x^2)/27 + 343/(486*(2 + 3*x)^2) - 2009/(243*(2 + 3*x)) - (518*Log[2 + 3*x])/81

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Rubi [A]  time = 0.0183089, 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} \[ -\frac{20 x^2}{27}+\frac{116 x}{27}-\frac{2009}{243 (3 x+2)}+\frac{343}{486 (3 x+2)^2}-\frac{518}{81} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(116*x)/27 - (20*x^2)/27 + 343/(486*(2 + 3*x)^2) - 2009/(243*(2 + 3*x)) - (518*Log[2 + 3*x])/81

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)^3} \, dx &=\int \left (\frac{116}{27}-\frac{40 x}{27}-\frac{343}{81 (2+3 x)^3}+\frac{2009}{81 (2+3 x)^2}-\frac{518}{27 (2+3 x)}\right ) \, dx\\ &=\frac{116 x}{27}-\frac{20 x^2}{27}+\frac{343}{486 (2+3 x)^2}-\frac{2009}{243 (2+3 x)}-\frac{518}{81} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0180574, size = 46, normalized size = 1.02 \[ -\frac{3240 x^4-14472 x^3-15030 x^2+15150 x+3108 (3 x+2)^2 \log (6 x+4)+11509}{486 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(11509 + 15150*x - 15030*x^2 - 14472*x^3 + 3240*x^4 + 3108*(2 + 3*x)^2*Log[4 + 6*x])/(486*(2 + 3*x)^2)

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{116\,x}{27}}-{\frac{20\,{x}^{2}}{27}}+{\frac{343}{486\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{2009}{486+729\,x}}-{\frac{518\,\ln \left ( 2+3\,x \right ) }{81}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

116/27*x-20/27*x^2+343/486/(2+3*x)^2-2009/243/(2+3*x)-518/81*ln(2+3*x)

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Maxima [A]  time = 0.998895, size = 49, normalized size = 1.09 \begin{align*} -\frac{20}{27} \, x^{2} + \frac{116}{27} \, x - \frac{49 \,{\left (246 \, x + 157\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{518}{81} \, \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)^3,x, algorithm="maxima")

[Out]

-20/27*x^2 + 116/27*x - 49/486*(246*x + 157)/(9*x^2 + 12*x + 4) - 518/81*log(3*x + 2)

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Fricas [A]  time = 1.21595, size = 158, normalized size = 3.51 \begin{align*} -\frac{3240 \, x^{4} - 14472 \, x^{3} - 23616 \, x^{2} + 3108 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 3702 \, x + 7693}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

-1/486*(3240*x^4 - 14472*x^3 - 23616*x^2 + 3108*(9*x^2 + 12*x + 4)*log(3*x + 2) + 3702*x + 7693)/(9*x^2 + 12*x
+ 4)

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Sympy [A]  time = 0.116981, size = 36, normalized size = 0.8 \begin{align*} - \frac{20 x^{2}}{27} + \frac{116 x}{27} - \frac{12054 x + 7693}{4374 x^{2} + 5832 x + 1944} - \frac{518 \log{\left (3 x + 2 \right )}}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*x**2/27 + 116*x/27 - (12054*x + 7693)/(4374*x**2 + 5832*x + 1944) - 518*log(3*x + 2)/81

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Giac [A]  time = 1.92835, size = 43, normalized size = 0.96 \begin{align*} -\frac{20}{27} \, x^{2} + \frac{116}{27} \, x - \frac{49 \,{\left (246 \, x + 157\right )}}{486 \,{\left (3 \, x + 2\right )}^{2}} - \frac{518}{81} \, \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)^3,x, algorithm="giac")

[Out]

-20/27*x^2 + 116/27*x - 49/486*(246*x + 157)/(3*x + 2)^2 - 518/81*log(abs(3*x + 2))