3.1252 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=45 \[ -\frac{5}{81 (3 x+2)^4}+\frac{16}{45 (3 x+2)^5}-\frac{91}{162 (3 x+2)^6}+\frac{7}{81 (3 x+2)^7} \]

[Out]

7/(81*(2 + 3*x)^7) - 91/(162*(2 + 3*x)^6) + 16/(45*(2 + 3*x)^5) - 5/(81*(2 + 3*x)^4)

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Rubi [A]  time = 0.0170339, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{5}{81 (3 x+2)^4}+\frac{16}{45 (3 x+2)^5}-\frac{91}{162 (3 x+2)^6}+\frac{7}{81 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(81*(2 + 3*x)^7) - 91/(162*(2 + 3*x)^6) + 16/(45*(2 + 3*x)^5) - 5/(81*(2 + 3*x)^4)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^8} \, dx &=\int \left (-\frac{49}{27 (2+3 x)^8}+\frac{91}{9 (2+3 x)^7}-\frac{16}{3 (2+3 x)^6}+\frac{20}{27 (2+3 x)^5}\right ) \, dx\\ &=\frac{7}{81 (2+3 x)^7}-\frac{91}{162 (2+3 x)^6}+\frac{16}{45 (2+3 x)^5}-\frac{5}{81 (2+3 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0106047, size = 26, normalized size = 0.58 \[ -\frac{1350 x^3+108 x^2-291 x+88}{810 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(88 - 291*x + 108*x^2 + 1350*x^3)/(810*(2 + 3*x)^7)

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Maple [A]  time = 0.005, size = 38, normalized size = 0.8 \begin{align*}{\frac{7}{81\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{91}{162\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{16}{45\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{5}{81\, \left ( 2+3\,x \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/81/(2+3*x)^7-91/162/(2+3*x)^6+16/45/(2+3*x)^5-5/81/(2+3*x)^4

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Maxima [A]  time = 2.66469, size = 73, normalized size = 1.62 \begin{align*} -\frac{1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \,{\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)^2*(3+5*x)/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(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 = 1.39078, size = 171, normalized size = 3.8 \begin{align*} -\frac{1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \,{\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)^2*(3+5*x)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(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 = 0.161907, size = 51, normalized size = 1.13 \begin{align*} - \frac{1350 x^{3} + 108 x^{2} - 291 x + 88}{1771470 x^{7} + 8266860 x^{6} + 16533720 x^{5} + 18370800 x^{4} + 12247200 x^{3} + 4898880 x^{2} + 1088640 x + 103680} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1350*x**3 + 108*x**2 - 291*x + 88)/(1771470*x**7 + 8266860*x**6 + 16533720*x**5 + 18370800*x**4 + 12247200*x
**3 + 4898880*x**2 + 1088640*x + 103680)

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Giac [A]  time = 4.14265, size = 32, normalized size = 0.71 \begin{align*} -\frac{1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \,{\left (3 \, x + 2\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(3*x + 2)^7