3.1211 \(\int \frac{(1-2 x) (2+3 x)^3}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=41 \[ -\frac{18 x^3}{25}-\frac{81 x^2}{250}+\frac{522 x}{625}-\frac{11}{3125 (5 x+3)}+\frac{97 \log (5 x+3)}{3125} \]

[Out]

(522*x)/625 - (81*x^2)/250 - (18*x^3)/25 - 11/(3125*(3 + 5*x)) + (97*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0186173, 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} \[ -\frac{18 x^3}{25}-\frac{81 x^2}{250}+\frac{522 x}{625}-\frac{11}{3125 (5 x+3)}+\frac{97 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(522*x)/625 - (81*x^2)/250 - (18*x^3)/25 - 11/(3125*(3 + 5*x)) + (97*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)^2} \, dx &=\int \left (\frac{522}{625}-\frac{81 x}{125}-\frac{54 x^2}{25}+\frac{11}{625 (3+5 x)^2}+\frac{97}{625 (3+5 x)}\right ) \, dx\\ &=\frac{522 x}{625}-\frac{81 x^2}{250}-\frac{18 x^3}{25}-\frac{11}{3125 (3+5 x)}+\frac{97 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0131939, size = 46, normalized size = 1.12 \[ \frac{-67500 x^4-70875 x^3+60075 x^2+92680 x+582 (5 x+3) \log (-3 (5 x+3))+27354}{18750 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27354 + 92680*x + 60075*x^2 - 70875*x^3 - 67500*x^4 + 582*(3 + 5*x)*Log[-3*(3 + 5*x)])/(18750*(3 + 5*x))

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*}{\frac{522\,x}{625}}-{\frac{81\,{x}^{2}}{250}}-{\frac{18\,{x}^{3}}{25}}-{\frac{11}{9375+15625\,x}}+{\frac{97\,\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)^2,x)

[Out]

522/625*x-81/250*x^2-18/25*x^3-11/3125/(3+5*x)+97/3125*ln(3+5*x)

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Maxima [A]  time = 1.04108, size = 42, normalized size = 1.02 \begin{align*} -\frac{18}{25} \, x^{3} - \frac{81}{250} \, x^{2} + \frac{522}{625} \, x - \frac{11}{3125 \,{\left (5 \, x + 3\right )}} + \frac{97}{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)^2,x, algorithm="maxima")

[Out]

-18/25*x^3 - 81/250*x^2 + 522/625*x - 11/3125/(5*x + 3) + 97/3125*log(5*x + 3)

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Fricas [A]  time = 1.47549, size = 134, normalized size = 3.27 \begin{align*} -\frac{22500 \, x^{4} + 23625 \, x^{3} - 20025 \, x^{2} - 194 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 15660 \, x + 22}{6250 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/6250*(22500*x^4 + 23625*x^3 - 20025*x^2 - 194*(5*x + 3)*log(5*x + 3) - 15660*x + 22)/(5*x + 3)

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Sympy [A]  time = 0.099436, size = 34, normalized size = 0.83 \begin{align*} - \frac{18 x^{3}}{25} - \frac{81 x^{2}}{250} + \frac{522 x}{625} + \frac{97 \log{\left (5 x + 3 \right )}}{3125} - \frac{11}{15625 x + 9375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-18*x**3/25 - 81*x**2/250 + 522*x/625 + 97*log(5*x + 3)/3125 - 11/(15625*x + 9375)

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Giac [A]  time = 1.69013, size = 77, normalized size = 1.88 \begin{align*} \frac{9}{6250} \,{\left (5 \, x + 3\right )}^{3}{\left (\frac{27}{5 \, x + 3} + \frac{62}{{\left (5 \, x + 3\right )}^{2}} - 4\right )} - \frac{11}{3125 \,{\left (5 \, x + 3\right )}} - \frac{97}{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)*(2+3*x)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

9/6250*(5*x + 3)^3*(27/(5*x + 3) + 62/(5*x + 3)^2 - 4) - 11/3125/(5*x + 3) - 97/3125*log(1/5*abs(5*x + 3)/(5*x
+ 3)^2)