3.1371 \(\int \frac{(1-2 x)^3 (3+5 x)^2}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=67 \[ \frac{100}{729 (3 x+2)^2}-\frac{2180}{2187 (3 x+2)^3}+\frac{4099}{1458 (3 x+2)^4}-\frac{11599}{3645 (3 x+2)^5}+\frac{1862}{2187 (3 x+2)^6}-\frac{49}{729 (3 x+2)^7} \]

[Out]

-49/(729*(2 + 3*x)^7) + 1862/(2187*(2 + 3*x)^6) - 11599/(3645*(2 + 3*x)^5) + 4099/(1458*(2 + 3*x)^4) - 2180/(2
187*(2 + 3*x)^3) + 100/(729*(2 + 3*x)^2)

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Rubi [A]  time = 0.023736, antiderivative size = 67, 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} \[ \frac{100}{729 (3 x+2)^2}-\frac{2180}{2187 (3 x+2)^3}+\frac{4099}{1458 (3 x+2)^4}-\frac{11599}{3645 (3 x+2)^5}+\frac{1862}{2187 (3 x+2)^6}-\frac{49}{729 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

-49/(729*(2 + 3*x)^7) + 1862/(2187*(2 + 3*x)^6) - 11599/(3645*(2 + 3*x)^5) + 4099/(1458*(2 + 3*x)^4) - 2180/(2
187*(2 + 3*x)^3) + 100/(729*(2 + 3*x)^2)

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)^3 (3+5 x)^2}{(2+3 x)^8} \, dx &=\int \left (\frac{343}{243 (2+3 x)^8}-\frac{3724}{243 (2+3 x)^7}+\frac{11599}{243 (2+3 x)^6}-\frac{8198}{243 (2+3 x)^5}+\frac{2180}{243 (2+3 x)^4}-\frac{200}{243 (2+3 x)^3}\right ) \, dx\\ &=-\frac{49}{729 (2+3 x)^7}+\frac{1862}{2187 (2+3 x)^6}-\frac{11599}{3645 (2+3 x)^5}+\frac{4099}{1458 (2+3 x)^4}-\frac{2180}{2187 (2+3 x)^3}+\frac{100}{729 (2+3 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0269496, size = 36, normalized size = 0.54 \[ \frac{729000 x^5+664200 x^4+191295 x^3+145044 x^2+61392 x-3526}{21870 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

(-3526 + 61392*x + 145044*x^2 + 191295*x^3 + 664200*x^4 + 729000*x^5)/(21870*(2 + 3*x)^7)

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Maple [A]  time = 0.005, size = 56, normalized size = 0.8 \begin{align*} -{\frac{49}{729\, \left ( 2+3\,x \right ) ^{7}}}+{\frac{1862}{2187\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{11599}{3645\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{4099}{1458\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{2180}{2187\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{100}{729\, \left ( 2+3\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3+5*x)^2/(2+3*x)^8,x)

[Out]

-49/729/(2+3*x)^7+1862/2187/(2+3*x)^6-11599/3645/(2+3*x)^5+4099/1458/(2+3*x)^4-2180/2187/(2+3*x)^3+100/729/(2+
3*x)^2

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Maxima [A]  time = 1.01787, size = 86, normalized size = 1.28 \begin{align*} \frac{729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^8,x, algorithm="maxima")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(2187*x^7 + 10206*x^6 + 20412*x^5
+ 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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Fricas [A]  time = 1.16695, size = 220, normalized size = 3.28 \begin{align*} \frac{729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^8,x, algorithm="fricas")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(2187*x^7 + 10206*x^6 + 20412*x^5
+ 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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Sympy [A]  time = 0.183176, size = 60, normalized size = 0.9 \begin{align*} \frac{729000 x^{5} + 664200 x^{4} + 191295 x^{3} + 145044 x^{2} + 61392 x - 3526}{47829690 x^{7} + 223205220 x^{6} + 446410440 x^{5} + 496011600 x^{4} + 330674400 x^{3} + 132269760 x^{2} + 29393280 x + 2799360} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(3+5*x)**2/(2+3*x)**8,x)

[Out]

(729000*x**5 + 664200*x**4 + 191295*x**3 + 145044*x**2 + 61392*x - 3526)/(47829690*x**7 + 223205220*x**6 + 446
410440*x**5 + 496011600*x**4 + 330674400*x**3 + 132269760*x**2 + 29393280*x + 2799360)

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Giac [A]  time = 2.45509, size = 46, normalized size = 0.69 \begin{align*} \frac{729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \,{\left (3 \, x + 2\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^8,x, algorithm="giac")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(3*x + 2)^7