3.1157 \(\int \frac{(1-2 x) (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=34 \[ \frac{10}{81 (3 x+2)^3}-\frac{37}{108 (3 x+2)^4}+\frac{7}{135 (3 x+2)^5} \]

[Out]

7/(135*(2 + 3*x)^5) - 37/(108*(2 + 3*x)^4) + 10/(81*(2 + 3*x)^3)

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Rubi [A]  time = 0.0143678, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{10}{81 (3 x+2)^3}-\frac{37}{108 (3 x+2)^4}+\frac{7}{135 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(135*(2 + 3*x)^5) - 37/(108*(2 + 3*x)^4) + 10/(81*(2 + 3*x)^3)

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) (3+5 x)}{(2+3 x)^6} \, dx &=\int \left (-\frac{7}{9 (2+3 x)^6}+\frac{37}{9 (2+3 x)^5}-\frac{10}{9 (2+3 x)^4}\right ) \, dx\\ &=\frac{7}{135 (2+3 x)^5}-\frac{37}{108 (2+3 x)^4}+\frac{10}{81 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0065319, size = 21, normalized size = 0.62 \[ \frac{1800 x^2+735 x-226}{1620 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-226 + 735*x + 1800*x^2)/(1620*(2 + 3*x)^5)

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Maple [A]  time = 0.004, size = 29, normalized size = 0.9 \begin{align*}{\frac{7}{135\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{37}{108\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{10}{81\, \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+3*x)^6,x)

[Out]

7/135/(2+3*x)^5-37/108/(2+3*x)^4+10/81/(2+3*x)^3

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Maxima [A]  time = 1.17689, size = 53, normalized size = 1.56 \begin{align*} \frac{1800 \, x^{2} + 735 \, x - 226}{1620 \,{\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)*(3+5*x)/(2+3*x)^6,x, algorithm="maxima")

[Out]

1/1620*(1800*x^2 + 735*x - 226)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Fricas [A]  time = 1.44389, size = 117, normalized size = 3.44 \begin{align*} \frac{1800 \, x^{2} + 735 \, x - 226}{1620 \,{\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)*(3+5*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/1620*(1800*x^2 + 735*x - 226)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.13582, size = 34, normalized size = 1. \begin{align*} \frac{1800 x^{2} + 735 x - 226}{393660 x^{5} + 1312200 x^{4} + 1749600 x^{3} + 1166400 x^{2} + 388800 x + 51840} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1800*x**2 + 735*x - 226)/(393660*x**5 + 1312200*x**4 + 1749600*x**3 + 1166400*x**2 + 388800*x + 51840)

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Giac [A]  time = 2.04798, size = 26, normalized size = 0.76 \begin{align*} \frac{1800 \, x^{2} + 735 \, x - 226}{1620 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1620*(1800*x^2 + 735*x - 226)/(3*x + 2)^5