3.1196 \(\int \frac{(1-2 x) (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=37 \[ -\frac{27 x^4}{10}-\frac{81 x^3}{25}+\frac{279 x^2}{250}+\frac{1663 x}{625}+\frac{11 \log (5 x+3)}{3125} \]

[Out]

(1663*x)/625 + (279*x^2)/250 - (81*x^3)/25 - (27*x^4)/10 + (11*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0127903, 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} \[ -\frac{27 x^4}{10}-\frac{81 x^3}{25}+\frac{279 x^2}{250}+\frac{1663 x}{625}+\frac{11 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1663*x)/625 + (279*x^2)/250 - (81*x^3)/25 - (27*x^4)/10 + (11*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) (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac{1663}{625}+\frac{279 x}{125}-\frac{243 x^2}{25}-\frac{54 x^3}{5}+\frac{11}{625 (3+5 x)}\right ) \, dx\\ &=\frac{1663 x}{625}+\frac{279 x^2}{250}-\frac{81 x^3}{25}-\frac{27 x^4}{10}+\frac{11 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0118916, size = 35, normalized size = 0.95 \[ \frac{5 \left (-3375 x^4-4050 x^3+1395 x^2+3326 x+1056\right )+22 \log (5 x+3)}{6250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(1056 + 3326*x + 1395*x^2 - 4050*x^3 - 3375*x^4) + 22*Log[3 + 5*x])/6250

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Maple [A]  time = 0.003, size = 28, normalized size = 0.8 \begin{align*}{\frac{1663\,x}{625}}+{\frac{279\,{x}^{2}}{250}}-{\frac{81\,{x}^{3}}{25}}-{\frac{27\,{x}^{4}}{10}}+{\frac{11\,\ln \left ( 3+5\,x \right ) }{3125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1663/625*x+279/250*x^2-81/25*x^3-27/10*x^4+11/3125*ln(3+5*x)

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Maxima [A]  time = 1.14828, size = 36, normalized size = 0.97 \begin{align*} -\frac{27}{10} \, x^{4} - \frac{81}{25} \, x^{3} + \frac{279}{250} \, x^{2} + \frac{1663}{625} \, x + \frac{11}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/10*x^4 - 81/25*x^3 + 279/250*x^2 + 1663/625*x + 11/3125*log(5*x + 3)

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Fricas [A]  time = 1.48107, size = 100, normalized size = 2.7 \begin{align*} -\frac{27}{10} \, x^{4} - \frac{81}{25} \, x^{3} + \frac{279}{250} \, x^{2} + \frac{1663}{625} \, x + \frac{11}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/10*x^4 - 81/25*x^3 + 279/250*x^2 + 1663/625*x + 11/3125*log(5*x + 3)

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Sympy [A]  time = 0.085246, size = 34, normalized size = 0.92 \begin{align*} - \frac{27 x^{4}}{10} - \frac{81 x^{3}}{25} + \frac{279 x^{2}}{250} + \frac{1663 x}{625} + \frac{11 \log{\left (5 x + 3 \right )}}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*x**4/10 - 81*x**3/25 + 279*x**2/250 + 1663*x/625 + 11*log(5*x + 3)/3125

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Giac [A]  time = 3.10521, size = 38, normalized size = 1.03 \begin{align*} -\frac{27}{10} \, x^{4} - \frac{81}{25} \, x^{3} + \frac{279}{250} \, x^{2} + \frac{1663}{625} \, x + \frac{11}{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)*(2+3*x)^3/(3+5*x),x, algorithm="giac")

[Out]

-27/10*x^4 - 81/25*x^3 + 279/250*x^2 + 1663/625*x + 11/3125*log(abs(5*x + 3))