3.1152 \(\int \frac{(1-2 x) (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=23 \[ -\frac{5 x^2}{3}+\frac{17 x}{9}-\frac{7}{27} \log (3 x+2) \]

[Out]

(17*x)/9 - (5*x^2)/3 - (7*Log[2 + 3*x])/27

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Rubi [A]  time = 0.0101749, antiderivative size = 23, 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} \[ -\frac{5 x^2}{3}+\frac{17 x}{9}-\frac{7}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(17*x)/9 - (5*x^2)/3 - (7*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} \, dx &=\int \left (\frac{17}{9}-\frac{10 x}{3}-\frac{7}{9 (2+3 x)}\right ) \, dx\\ &=\frac{17 x}{9}-\frac{5 x^2}{3}-\frac{7}{27} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0053774, size = 22, normalized size = 0.96 \[ \frac{1}{27} \left (-45 x^2+51 x-7 \log (3 x+2)+54\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54 + 51*x - 45*x^2 - 7*Log[2 + 3*x])/27

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Maple [A]  time = 0.003, size = 18, normalized size = 0.8 \begin{align*}{\frac{17\,x}{9}}-{\frac{5\,{x}^{2}}{3}}-{\frac{7\,\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),x)

[Out]

17/9*x-5/3*x^2-7/27*ln(2+3*x)

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Maxima [A]  time = 1.10713, size = 23, normalized size = 1. \begin{align*} -\frac{5}{3} \, x^{2} + \frac{17}{9} \, x - \frac{7}{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),x, algorithm="maxima")

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(3*x + 2)

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Fricas [A]  time = 1.46491, size = 53, normalized size = 2.3 \begin{align*} -\frac{5}{3} \, x^{2} + \frac{17}{9} \, x - \frac{7}{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),x, algorithm="fricas")

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(3*x + 2)

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Sympy [A]  time = 0.076814, size = 20, normalized size = 0.87 \begin{align*} - \frac{5 x^{2}}{3} + \frac{17 x}{9} - \frac{7 \log{\left (3 x + 2 \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*x**2/3 + 17*x/9 - 7*log(3*x + 2)/27

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Giac [A]  time = 2.33405, size = 24, normalized size = 1.04 \begin{align*} -\frac{5}{3} \, x^{2} + \frac{17}{9} \, x - \frac{7}{27} \, \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+5*x)/(2+3*x),x, algorithm="giac")

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(abs(3*x + 2))