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

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

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {(1-2 x)^3}{84 (3 x+2)^4}-\frac {23 (1-2 x)^3}{294 (3 x+2)^3} \end {gather*}

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 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]

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]))))

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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.70 \begin {gather*} -\frac {2160 x^3+1728 x^2+516 x+167}{324 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2 (3+5 x)}{(2+3 x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.24, size = 39, normalized size = 1.05 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.17, size = 37, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 38, normalized size = 1.03 \begin {gather*} -\frac {91}{81 \left (3 x +2\right )^{3}}+\frac {49}{324 \left (3 x +2\right )^{4}}+\frac {8}{9 \left (3 x +2\right )^{2}}-\frac {20}{81 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 39, normalized size = 1.05 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 37, normalized size = 1.00 \begin {gather*} \frac {8}{9\,{\left (3\,x+2\right )}^2}-\frac {20}{81\,\left (3\,x+2\right )}-\frac {91}{81\,{\left (3\,x+2\right )}^3}+\frac {49}{324\,{\left (3\,x+2\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 36, normalized size = 0.97 \begin {gather*} \frac {- 2160 x^{3} - 1728 x^{2} - 516 x - 167}{26244 x^{4} + 69984 x^{3} + 69984 x^{2} + 31104 x + 5184} \end {gather*}

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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