3.1285 \(\int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=60 \[ \frac{500 x}{243}-\frac{8285}{729 (3 x+2)}+\frac{4099}{1458 (3 x+2)^2}-\frac{763}{2187 (3 x+2)^3}+\frac{49}{2916 (3 x+2)^4}-\frac{3800}{729} \log (3 x+2) \]

[Out]

(500*x)/243 + 49/(2916*(2 + 3*x)^4) - 763/(2187*(2 + 3*x)^3) + 4099/(1458*(2 + 3*x)^2) - 8285/(729*(2 + 3*x))
- (3800*Log[2 + 3*x])/729

________________________________________________________________________________________

Rubi [A]  time = 0.0251481, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{500 x}{243}-\frac{8285}{729 (3 x+2)}+\frac{4099}{1458 (3 x+2)^2}-\frac{763}{2187 (3 x+2)^3}+\frac{49}{2916 (3 x+2)^4}-\frac{3800}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(500*x)/243 + 49/(2916*(2 + 3*x)^4) - 763/(2187*(2 + 3*x)^3) + 4099/(1458*(2 + 3*x)^2) - 8285/(729*(2 + 3*x))
- (3800*Log[2 + 3*x])/729

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^5} \, dx &=\int \left (\frac{500}{243}-\frac{49}{243 (2+3 x)^5}+\frac{763}{243 (2+3 x)^4}-\frac{4099}{243 (2+3 x)^3}+\frac{8285}{243 (2+3 x)^2}-\frac{3800}{243 (2+3 x)}\right ) \, dx\\ &=\frac{500 x}{243}+\frac{49}{2916 (2+3 x)^4}-\frac{763}{2187 (2+3 x)^3}+\frac{4099}{1458 (2+3 x)^2}-\frac{8285}{729 (2+3 x)}-\frac{3800}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0326124, size = 51, normalized size = 0.85 \[ \frac{1458000 x^5+4860000 x^4+3795660 x^3-827334 x^2-1853148 x-45600 (3 x+2)^4 \log (30 x+20)-510941}{8748 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-510941 - 1853148*x - 827334*x^2 + 3795660*x^3 + 4860000*x^4 + 1458000*x^5 - 45600*(2 + 3*x)^4*Log[20 + 30*x]
)/(8748*(2 + 3*x)^4)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 49, normalized size = 0.8 \begin{align*}{\frac{500\,x}{243}}+{\frac{49}{2916\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{763}{2187\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{4099}{1458\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{8285}{1458+2187\,x}}-{\frac{3800\,\ln \left ( 2+3\,x \right ) }{729}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x+49/2916/(2+3*x)^4-763/2187/(2+3*x)^3+4099/1458/(2+3*x)^2-8285/729/(2+3*x)-3800/729*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.06971, size = 69, normalized size = 1.15 \begin{align*} \frac{500}{243} \, x - \frac{2684340 \, x^{3} + 5147334 \, x^{2} + 3293148 \, x + 702941}{8748 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{3800}{729} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x - 1/8748*(2684340*x^3 + 5147334*x^2 + 3293148*x + 702941)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) -
3800/729*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.54781, size = 254, normalized size = 4.23 \begin{align*} \frac{1458000 \, x^{5} + 3888000 \, x^{4} + 1203660 \, x^{3} - 3419334 \, x^{2} - 45600 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 3005148 \, x - 702941}{8748 \,{\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)^3/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/8748*(1458000*x^5 + 3888000*x^4 + 1203660*x^3 - 3419334*x^2 - 45600*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
*log(3*x + 2) - 3005148*x - 702941)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

________________________________________________________________________________________

Sympy [A]  time = 0.146273, size = 49, normalized size = 0.82 \begin{align*} \frac{500 x}{243} - \frac{2684340 x^{3} + 5147334 x^{2} + 3293148 x + 702941}{708588 x^{4} + 1889568 x^{3} + 1889568 x^{2} + 839808 x + 139968} - \frac{3800 \log{\left (3 x + 2 \right )}}{729} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500*x/243 - (2684340*x**3 + 5147334*x**2 + 3293148*x + 702941)/(708588*x**4 + 1889568*x**3 + 1889568*x**2 + 83
9808*x + 139968) - 3800*log(3*x + 2)/729

________________________________________________________________________________________

Giac [A]  time = 1.65923, size = 80, normalized size = 1.33 \begin{align*} \frac{500}{243} \, x - \frac{8285}{729 \,{\left (3 \, x + 2\right )}} + \frac{4099}{1458 \,{\left (3 \, x + 2\right )}^{2}} - \frac{763}{2187 \,{\left (3 \, x + 2\right )}^{3}} + \frac{49}{2916 \,{\left (3 \, x + 2\right )}^{4}} + \frac{3800}{729} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) + \frac{1000}{729} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

500/243*x - 8285/729/(3*x + 2) + 4099/1458/(3*x + 2)^2 - 763/2187/(3*x + 2)^3 + 49/2916/(3*x + 2)^4 + 3800/729
*log(1/3*abs(3*x + 2)/(3*x + 2)^2) + 1000/729