3.1250 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=45 \[ -\frac{10}{81 (3 x+2)^2}+\frac{16}{27 (3 x+2)^3}-\frac{91}{108 (3 x+2)^4}+\frac{49}{405 (3 x+2)^5} \]

[Out]

49/(405*(2 + 3*x)^5) - 91/(108*(2 + 3*x)^4) + 16/(27*(2 + 3*x)^3) - 10/(81*(2 + 3*x)^2)

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Rubi [A]  time = 0.0163791, 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} \[ -\frac{10}{81 (3 x+2)^2}+\frac{16}{27 (3 x+2)^3}-\frac{91}{108 (3 x+2)^4}+\frac{49}{405 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(405*(2 + 3*x)^5) - 91/(108*(2 + 3*x)^4) + 16/(27*(2 + 3*x)^3) - 10/(81*(2 + 3*x)^2)

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)^6} \, dx &=\int \left (-\frac{49}{27 (2+3 x)^6}+\frac{91}{9 (2+3 x)^5}-\frac{16}{3 (2+3 x)^4}+\frac{20}{27 (2+3 x)^3}\right ) \, dx\\ &=\frac{49}{405 (2+3 x)^5}-\frac{91}{108 (2+3 x)^4}+\frac{16}{27 (2+3 x)^3}-\frac{10}{81 (2+3 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0104485, size = 26, normalized size = 0.58 \[ -\frac{1800 x^3+720 x^2-75 x+98}{540 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(98 - 75*x + 720*x^2 + 1800*x^3)/(540*(2 + 3*x)^5)

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Maple [A]  time = 0.005, size = 38, normalized size = 0.8 \begin{align*}{\frac{49}{405\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{91}{108\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{16}{27\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{10}{81\, \left ( 2+3\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/405/(2+3*x)^5-91/108/(2+3*x)^4+16/27/(2+3*x)^3-10/81/(2+3*x)^2

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Maxima [A]  time = 1.13272, size = 59, normalized size = 1.31 \begin{align*} -\frac{1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \,{\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)^2*(3+5*x)/(2+3*x)^6,x, algorithm="maxima")

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Fricas [A]  time = 1.56981, size = 128, normalized size = 2.84 \begin{align*} -\frac{1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \,{\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)^2*(3+5*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.141443, size = 41, normalized size = 0.91 \begin{align*} - \frac{1800 x^{3} + 720 x^{2} - 75 x + 98}{131220 x^{5} + 437400 x^{4} + 583200 x^{3} + 388800 x^{2} + 129600 x + 17280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1800*x**3 + 720*x**2 - 75*x + 98)/(131220*x**5 + 437400*x**4 + 583200*x**3 + 388800*x**2 + 129600*x + 17280)

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Giac [A]  time = 3.17067, size = 32, normalized size = 0.71 \begin{align*} -\frac{1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(3*x + 2)^5