3.1171 \(\int \frac{(1-2 x) (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=37 \[ \frac{3 (5 x+3)^3}{4 (3 x+2)^3}+\frac{7 (5 x+3)^3}{12 (3 x+2)^4} \]

[Out]

(7*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (3*(3 + 5*x)^3)/(4*(2 + 3*x)^3)

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Rubi [A]  time = 0.006027, 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} \[ \frac{3 (5 x+3)^3}{4 (3 x+2)^3}+\frac{7 (5 x+3)^3}{12 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(3 + 5*x)^3)/(12*(2 + 3*x)^4) + (3*(3 + 5*x)^3)/(4*(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) (3+5 x)^2}{(2+3 x)^5} \, dx &=\frac{7 (3+5 x)^3}{12 (2+3 x)^4}+\frac{9}{4} \int \frac{(3+5 x)^2}{(2+3 x)^4} \, dx\\ &=\frac{7 (3+5 x)^3}{12 (2+3 x)^4}+\frac{3 (3+5 x)^3}{4 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0086455, size = 26, normalized size = 0.7 \[ \frac{600 x^3+810 x^2+312 x+25}{36 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25 + 312*x + 810*x^2 + 600*x^3)/(36*(2 + 3*x)^4)

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Maple [A]  time = 0.005, size = 38, normalized size = 1. \begin{align*} -{\frac{65}{54\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{7}{324\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{50}{162+243\,x}}+{\frac{8}{27\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-65/54/(2+3*x)^2-7/324/(2+3*x)^4+50/81/(2+3*x)+8/27/(2+3*x)^3

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Maxima [A]  time = 1.14649, size = 53, normalized size = 1.43 \begin{align*} \frac{600 \, x^{3} + 810 \, x^{2} + 312 \, x + 25}{36 \,{\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)*(3+5*x)^2/(2+3*x)^5,x, algorithm="maxima")

[Out]

1/36*(600*x^3 + 810*x^2 + 312*x + 25)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Fricas [A]  time = 1.73086, size = 108, normalized size = 2.92 \begin{align*} \frac{600 \, x^{3} + 810 \, x^{2} + 312 \, x + 25}{36 \,{\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)*(3+5*x)^2/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/36*(600*x^3 + 810*x^2 + 312*x + 25)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.13291, size = 34, normalized size = 0.92 \begin{align*} \frac{600 x^{3} + 810 x^{2} + 312 x + 25}{2916 x^{4} + 7776 x^{3} + 7776 x^{2} + 3456 x + 576} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(600*x**3 + 810*x**2 + 312*x + 25)/(2916*x**4 + 7776*x**3 + 7776*x**2 + 3456*x + 576)

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Giac [A]  time = 2.32661, size = 50, normalized size = 1.35 \begin{align*} \frac{50}{81 \,{\left (3 \, x + 2\right )}} - \frac{65}{54 \,{\left (3 \, x + 2\right )}^{2}} + \frac{8}{27 \,{\left (3 \, x + 2\right )}^{3}} - \frac{7}{324 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/81/(3*x + 2) - 65/54/(3*x + 2)^2 + 8/27/(3*x + 2)^3 - 7/324/(3*x + 2)^4