3.1352 \(\int \frac{(1-2 x)^3 (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=37 \[ \frac{(1-2 x)^4}{105 (3 x+2)^5}-\frac{173 (1-2 x)^4}{2940 (3 x+2)^4} \]

[Out]

(1 - 2*x)^4/(105*(2 + 3*x)^5) - (173*(1 - 2*x)^4)/(2940*(2 + 3*x)^4)

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Rubi [A]  time = 0.0061906, 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} \[ \frac{(1-2 x)^4}{105 (3 x+2)^5}-\frac{173 (1-2 x)^4}{2940 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1 - 2*x)^4/(105*(2 + 3*x)^5) - (173*(1 - 2*x)^4)/(2940*(2 + 3*x)^4)

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)^3 (3+5 x)}{(2+3 x)^6} \, dx &=\frac{(1-2 x)^4}{105 (2+3 x)^5}+\frac{173}{105} \int \frac{(1-2 x)^3}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^4}{105 (2+3 x)^5}-\frac{173 (1-2 x)^4}{2940 (2+3 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0107898, size = 31, normalized size = 0.84 \[ \frac{64800 x^4+57240 x^3+34920 x^2+16905 x+1282}{4860 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1282 + 16905*x + 34920*x^2 + 57240*x^3 + 64800*x^4)/(4860*(2 + 3*x)^5)

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Maple [A]  time = 0.005, size = 47, normalized size = 1.3 \begin{align*}{\frac{343}{1215\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{214}{243\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{2009}{972\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{40}{486+729\,x}}+{\frac{518}{243\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3+5*x)/(2+3*x)^6,x)

[Out]

343/1215/(2+3*x)^5-214/243/(2+3*x)^2-2009/972/(2+3*x)^4+40/243/(2+3*x)+518/243/(2+3*x)^3

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Maxima [A]  time = 1.36064, size = 66, normalized size = 1.78 \begin{align*} \frac{64800 \, x^{4} + 57240 \, x^{3} + 34920 \, x^{2} + 16905 \, x + 1282}{4860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^6,x, algorithm="maxima")

[Out]

1/4860*(64800*x^4 + 57240*x^3 + 34920*x^2 + 16905*x + 1282)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)

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Fricas [A]  time = 1.28856, size = 155, normalized size = 4.19 \begin{align*} \frac{64800 \, x^{4} + 57240 \, x^{3} + 34920 \, x^{2} + 16905 \, x + 1282}{4860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/4860*(64800*x^4 + 57240*x^3 + 34920*x^2 + 16905*x + 1282)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)

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Sympy [A]  time = 0.148133, size = 44, normalized size = 1.19 \begin{align*} \frac{64800 x^{4} + 57240 x^{3} + 34920 x^{2} + 16905 x + 1282}{1180980 x^{5} + 3936600 x^{4} + 5248800 x^{3} + 3499200 x^{2} + 1166400 x + 155520} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(3+5*x)/(2+3*x)**6,x)

[Out]

(64800*x**4 + 57240*x**3 + 34920*x**2 + 16905*x + 1282)/(1180980*x**5 + 3936600*x**4 + 5248800*x**3 + 3499200*
x**2 + 1166400*x + 155520)

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Giac [A]  time = 2.83585, size = 39, normalized size = 1.05 \begin{align*} \frac{64800 \, x^{4} + 57240 \, x^{3} + 34920 \, x^{2} + 16905 \, x + 1282}{4860 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^6,x, algorithm="giac")

[Out]

1/4860*(64800*x^4 + 57240*x^3 + 34920*x^2 + 16905*x + 1282)/(3*x + 2)^5