3.1408 \(\int \frac{(1-2 x)^3 (2+3 x)}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=41 \[ -\frac{8 x^3}{25}+\frac{122 x^2}{125}-\frac{1098 x}{625}-\frac{1331}{3125 (5 x+3)}+\frac{3267 \log (5 x+3)}{3125} \]

[Out]

(-1098*x)/625 + (122*x^2)/125 - (8*x^3)/25 - 1331/(3125*(3 + 5*x)) + (3267*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0185232, antiderivative size = 41, 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} \[ -\frac{8 x^3}{25}+\frac{122 x^2}{125}-\frac{1098 x}{625}-\frac{1331}{3125 (5 x+3)}+\frac{3267 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1098*x)/625 + (122*x^2)/125 - (8*x^3)/25 - 1331/(3125*(3 + 5*x)) + (3267*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)^2} \, dx &=\int \left (-\frac{1098}{625}+\frac{244 x}{125}-\frac{24 x^2}{25}+\frac{1331}{625 (3+5 x)^2}+\frac{3267}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{1098 x}{625}+\frac{122 x^2}{125}-\frac{8 x^3}{25}-\frac{1331}{3125 (3+5 x)}+\frac{3267 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0149918, size = 44, normalized size = 1.07 \[ \frac{-10000 x^4+24500 x^3-36600 x^2-11865 x+6534 (5 x+3) \log (10 x+6)+9983}{6250 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9983 - 11865*x - 36600*x^2 + 24500*x^3 - 10000*x^4 + 6534*(3 + 5*x)*Log[6 + 10*x])/(6250*(3 + 5*x))

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -{\frac{1098\,x}{625}}+{\frac{122\,{x}^{2}}{125}}-{\frac{8\,{x}^{3}}{25}}-{\frac{1331}{9375+15625\,x}}+{\frac{3267\,\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)^2,x)

[Out]

-1098/625*x+122/125*x^2-8/25*x^3-1331/3125/(3+5*x)+3267/3125*ln(3+5*x)

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Maxima [A]  time = 1.00383, size = 42, normalized size = 1.02 \begin{align*} -\frac{8}{25} \, x^{3} + \frac{122}{125} \, x^{2} - \frac{1098}{625} \, x - \frac{1331}{3125 \,{\left (5 \, x + 3\right )}} + \frac{3267}{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)^2,x, algorithm="maxima")

[Out]

-8/25*x^3 + 122/125*x^2 - 1098/625*x - 1331/3125/(5*x + 3) + 3267/3125*log(5*x + 3)

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Fricas [A]  time = 1.44526, size = 136, normalized size = 3.32 \begin{align*} -\frac{5000 \, x^{4} - 12250 \, x^{3} + 18300 \, x^{2} - 3267 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 16470 \, x + 1331}{3125 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/3125*(5000*x^4 - 12250*x^3 + 18300*x^2 - 3267*(5*x + 3)*log(5*x + 3) + 16470*x + 1331)/(5*x + 3)

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Sympy [A]  time = 0.100907, size = 34, normalized size = 0.83 \begin{align*} - \frac{8 x^{3}}{25} + \frac{122 x^{2}}{125} - \frac{1098 x}{625} + \frac{3267 \log{\left (5 x + 3 \right )}}{3125} - \frac{1331}{15625 x + 9375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*x**3/25 + 122*x**2/125 - 1098*x/625 + 3267*log(5*x + 3)/3125 - 1331/(15625*x + 9375)

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Giac [A]  time = 3.11655, size = 77, normalized size = 1.88 \begin{align*} \frac{2}{3125} \,{\left (5 \, x + 3\right )}^{3}{\left (\frac{97}{5 \, x + 3} - \frac{1023}{{\left (5 \, x + 3\right )}^{2}} - 4\right )} - \frac{1331}{3125 \,{\left (5 \, x + 3\right )}} - \frac{3267}{3125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\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)^2,x, algorithm="giac")

[Out]

2/3125*(5*x + 3)^3*(97/(5*x + 3) - 1023/(5*x + 3)^2 - 4) - 1331/3125/(5*x + 3) - 3267/3125*log(1/5*abs(5*x + 3
)/(5*x + 3)^2)