3.1153 \(\int \frac{(1-2 x) (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{10 x}{9}+\frac{7}{27 (3 x+2)}+\frac{37}{27} \log (3 x+2) \]

[Out]

(-10*x)/9 + 7/(27*(2 + 3*x)) + (37*Log[2 + 3*x])/27

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Rubi [A]  time = 0.012331, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{10 x}{9}+\frac{7}{27 (3 x+2)}+\frac{37}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*x)/9 + 7/(27*(2 + 3*x)) + (37*Log[2 + 3*x])/27

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+3 x)^2} \, dx &=\int \left (-\frac{10}{9}-\frac{7}{9 (2+3 x)^2}+\frac{37}{9 (2+3 x)}\right ) \, dx\\ &=-\frac{10 x}{9}+\frac{7}{27 (2+3 x)}+\frac{37}{27} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0099498, size = 26, normalized size = 0.96 \[ \frac{1}{27} \left (-30 x+\frac{7}{3 x+2}+37 \log (3 x+2)-20\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-20 - 30*x + 7/(2 + 3*x) + 37*Log[2 + 3*x])/27

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -{\frac{10\,x}{9}}+{\frac{7}{54+81\,x}}+{\frac{37\,\ln \left ( 2+3\,x \right ) }{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*x+7/27/(2+3*x)+37/27*ln(2+3*x)

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Maxima [A]  time = 1.13441, size = 28, normalized size = 1.04 \begin{align*} -\frac{10}{9} \, x + \frac{7}{27 \,{\left (3 \, x + 2\right )}} + \frac{37}{27} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*x + 7/27/(3*x + 2) + 37/27*log(3*x + 2)

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Fricas [A]  time = 1.4763, size = 88, normalized size = 3.26 \begin{align*} -\frac{90 \, x^{2} - 37 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 60 \, x - 7}{27 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(90*x^2 - 37*(3*x + 2)*log(3*x + 2) + 60*x - 7)/(3*x + 2)

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Sympy [A]  time = 0.091926, size = 20, normalized size = 0.74 \begin{align*} - \frac{10 x}{9} + \frac{37 \log{\left (3 x + 2 \right )}}{27} + \frac{7}{81 x + 54} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*x/9 + 37*log(3*x + 2)/27 + 7/(81*x + 54)

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Giac [A]  time = 2.95895, size = 43, normalized size = 1.59 \begin{align*} -\frac{10}{9} \, x + \frac{7}{27 \,{\left (3 \, x + 2\right )}} - \frac{37}{27} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) - \frac{20}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*x + 7/27/(3*x + 2) - 37/27*log(1/3*abs(3*x + 2)/(3*x + 2)^2) - 20/27