3.1184 \(\int \frac{(1-2 x) (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=37 \[ -\frac{125 x^4}{6}-\frac{475 x^3}{27}+\frac{545 x^2}{54}+\frac{1097 x}{81}-\frac{7}{243} \log (3 x+2) \]

[Out]

(1097*x)/81 + (545*x^2)/54 - (475*x^3)/27 - (125*x^4)/6 - (7*Log[2 + 3*x])/243

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Rubi [A]  time = 0.0137501, antiderivative size = 37, 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} \[ -\frac{125 x^4}{6}-\frac{475 x^3}{27}+\frac{545 x^2}{54}+\frac{1097 x}{81}-\frac{7}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1097*x)/81 + (545*x^2)/54 - (475*x^3)/27 - (125*x^4)/6 - (7*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+5 x)^3}{2+3 x} \, dx &=\int \left (\frac{1097}{81}+\frac{545 x}{27}-\frac{475 x^2}{9}-\frac{250 x^3}{3}-\frac{7}{81 (2+3 x)}\right ) \, dx\\ &=\frac{1097 x}{81}+\frac{545 x^2}{54}-\frac{475 x^3}{27}-\frac{125 x^4}{6}-\frac{7}{243} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.01178, size = 32, normalized size = 0.86 \[ \frac{-30375 x^4-25650 x^3+14715 x^2+19746 x-42 \log (3 x+2)+5024}{1458} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5024 + 19746*x + 14715*x^2 - 25650*x^3 - 30375*x^4 - 42*Log[2 + 3*x])/1458

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Maple [A]  time = 0.001, size = 28, normalized size = 0.8 \begin{align*}{\frac{1097\,x}{81}}+{\frac{545\,{x}^{2}}{54}}-{\frac{475\,{x}^{3}}{27}}-{\frac{125\,{x}^{4}}{6}}-{\frac{7\,\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+5*x)^3/(2+3*x),x)

[Out]

1097/81*x+545/54*x^2-475/27*x^3-125/6*x^4-7/243*ln(2+3*x)

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Maxima [A]  time = 1.07793, size = 36, normalized size = 0.97 \begin{align*} -\frac{125}{6} \, x^{4} - \frac{475}{27} \, x^{3} + \frac{545}{54} \, x^{2} + \frac{1097}{81} \, x - \frac{7}{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+5*x)^3/(2+3*x),x, algorithm="maxima")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(3*x + 2)

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Fricas [A]  time = 1.81489, size = 96, normalized size = 2.59 \begin{align*} -\frac{125}{6} \, x^{4} - \frac{475}{27} \, x^{3} + \frac{545}{54} \, x^{2} + \frac{1097}{81} \, x - \frac{7}{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+5*x)^3/(2+3*x),x, algorithm="fricas")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(3*x + 2)

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Sympy [A]  time = 0.086983, size = 34, normalized size = 0.92 \begin{align*} - \frac{125 x^{4}}{6} - \frac{475 x^{3}}{27} + \frac{545 x^{2}}{54} + \frac{1097 x}{81} - \frac{7 \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+5*x)**3/(2+3*x),x)

[Out]

-125*x**4/6 - 475*x**3/27 + 545*x**2/54 + 1097*x/81 - 7*log(3*x + 2)/243

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Giac [A]  time = 2.27813, size = 38, normalized size = 1.03 \begin{align*} -\frac{125}{6} \, x^{4} - \frac{475}{27} \, x^{3} + \frac{545}{54} \, x^{2} + \frac{1097}{81} \, x - \frac{7}{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+5*x)^3/(2+3*x),x, algorithm="giac")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(abs(3*x + 2))