3.1393 \(\int \frac{(1-2 x)^3 (2+3 x)}{3+5 x} \, dx\)

Optimal. Leaf size=37 \[ -\frac{6 x^4}{5}+\frac{172 x^3}{75}-\frac{183 x^2}{125}-\frac{27 x}{625}+\frac{1331 \log (5 x+3)}{3125} \]

[Out]

(-27*x)/625 - (183*x^2)/125 + (172*x^3)/75 - (6*x^4)/5 + (1331*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0131486, 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} \[ -\frac{6 x^4}{5}+\frac{172 x^3}{75}-\frac{183 x^2}{125}-\frac{27 x}{625}+\frac{1331 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-27*x)/625 - (183*x^2)/125 + (172*x^3)/75 - (6*x^4)/5 + (1331*Log[3 + 5*x])/3125

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 (2+3 x)}{3+5 x} \, dx &=\int \left (-\frac{27}{625}-\frac{366 x}{125}+\frac{172 x^2}{25}-\frac{24 x^3}{5}+\frac{1331}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{27 x}{625}-\frac{183 x^2}{125}+\frac{172 x^3}{75}-\frac{6 x^4}{5}+\frac{1331 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0128692, size = 35, normalized size = 0.95 \[ \frac{3993 \log (5 x+3)-5 \left (2250 x^4-4300 x^3+2745 x^2+81 x-2160\right )}{9375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(-2160 + 81*x + 2745*x^2 - 4300*x^3 + 2250*x^4) + 3993*Log[3 + 5*x])/9375

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Maple [A]  time = 0.003, size = 28, normalized size = 0.8 \begin{align*} -{\frac{27\,x}{625}}-{\frac{183\,{x}^{2}}{125}}+{\frac{172\,{x}^{3}}{75}}-{\frac{6\,{x}^{4}}{5}}+{\frac{1331\,\ln \left ( 3+5\,x \right ) }{3125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/625*x-183/125*x^2+172/75*x^3-6/5*x^4+1331/3125*ln(3+5*x)

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Maxima [A]  time = 1.05946, size = 36, normalized size = 0.97 \begin{align*} -\frac{6}{5} \, x^{4} + \frac{172}{75} \, x^{3} - \frac{183}{125} \, x^{2} - \frac{27}{625} \, x + \frac{1331}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*x^4 + 172/75*x^3 - 183/125*x^2 - 27/625*x + 1331/3125*log(5*x + 3)

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Fricas [A]  time = 1.52552, size = 99, normalized size = 2.68 \begin{align*} -\frac{6}{5} \, x^{4} + \frac{172}{75} \, x^{3} - \frac{183}{125} \, x^{2} - \frac{27}{625} \, x + \frac{1331}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*x^4 + 172/75*x^3 - 183/125*x^2 - 27/625*x + 1331/3125*log(5*x + 3)

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Sympy [A]  time = 0.085847, size = 34, normalized size = 0.92 \begin{align*} - \frac{6 x^{4}}{5} + \frac{172 x^{3}}{75} - \frac{183 x^{2}}{125} - \frac{27 x}{625} + \frac{1331 \log{\left (5 x + 3 \right )}}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*x**4/5 + 172*x**3/75 - 183*x**2/125 - 27*x/625 + 1331*log(5*x + 3)/3125

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Giac [A]  time = 2.23316, size = 38, normalized size = 1.03 \begin{align*} -\frac{6}{5} \, x^{4} + \frac{172}{75} \, x^{3} - \frac{183}{125} \, x^{2} - \frac{27}{625} \, x + \frac{1331}{3125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*x^4 + 172/75*x^3 - 183/125*x^2 - 27/625*x + 1331/3125*log(abs(5*x + 3))