3.1249 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=37 \[ \frac{(1-2 x)^3}{84 (3 x+2)^4}-\frac{23 (1-2 x)^3}{294 (3 x+2)^3} \]

[Out]

(1 - 2*x)^3/(84*(2 + 3*x)^4) - (23*(1 - 2*x)^3)/(294*(2 + 3*x)^3)

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Rubi [A]  time = 0.0058116, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{(1-2 x)^3}{84 (3 x+2)^4}-\frac{23 (1-2 x)^3}{294 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^3/(84*(2 + 3*x)^4) - (23*(1 - 2*x)^3)/(294*(2 + 3*x)^3)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^5} \, dx &=\frac{(1-2 x)^3}{84 (2+3 x)^4}+\frac{23}{14} \int \frac{(1-2 x)^2}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^3}{84 (2+3 x)^4}-\frac{23 (1-2 x)^3}{294 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0116947, size = 26, normalized size = 0.7 \[ -\frac{2160 x^3+1728 x^2+516 x+167}{324 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(167 + 516*x + 1728*x^2 + 2160*x^3)/(324*(2 + 3*x)^4)

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Maple [A]  time = 0.004, size = 38, normalized size = 1. \begin{align*}{\frac{8}{9\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{49}{324\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{20}{162+243\,x}}-{\frac{91}{81\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/9/(2+3*x)^2+49/324/(2+3*x)^4-20/81/(2+3*x)-91/81/(2+3*x)^3

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Maxima [A]  time = 1.06652, size = 53, normalized size = 1.43 \begin{align*} -\frac{2160 \, x^{3} + 1728 \, x^{2} + 516 \, x + 167}{324 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/324*(2160*x^3 + 1728*x^2 + 516*x + 167)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Fricas [A]  time = 1.43102, size = 115, normalized size = 3.11 \begin{align*} -\frac{2160 \, x^{3} + 1728 \, x^{2} + 516 \, x + 167}{324 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/324*(2160*x^3 + 1728*x^2 + 516*x + 167)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.132608, size = 36, normalized size = 0.97 \begin{align*} - \frac{2160 x^{3} + 1728 x^{2} + 516 x + 167}{26244 x^{4} + 69984 x^{3} + 69984 x^{2} + 31104 x + 5184} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2160*x**3 + 1728*x**2 + 516*x + 167)/(26244*x**4 + 69984*x**3 + 69984*x**2 + 31104*x + 5184)

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Giac [A]  time = 2.91353, size = 50, normalized size = 1.35 \begin{align*} -\frac{20}{81 \,{\left (3 \, x + 2\right )}} + \frac{8}{9 \,{\left (3 \, x + 2\right )}^{2}} - \frac{91}{81 \,{\left (3 \, x + 2\right )}^{3}} + \frac{49}{324 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20/81/(3*x + 2) + 8/9/(3*x + 2)^2 - 91/81/(3*x + 2)^3 + 49/324/(3*x + 2)^4