3.1354 \(\int \frac{(1-2 x)^3 (3+5 x)}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=56 \[ \frac{40}{729 (3 x+2)^3}-\frac{107}{243 (3 x+2)^4}+\frac{518}{405 (3 x+2)^5}-\frac{2009}{1458 (3 x+2)^6}+\frac{49}{243 (3 x+2)^7} \]

[Out]

49/(243*(2 + 3*x)^7) - 2009/(1458*(2 + 3*x)^6) + 518/(405*(2 + 3*x)^5) - 107/(243*(2 + 3*x)^4) + 40/(729*(2 +
3*x)^3)

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Rubi [A]  time = 0.0192513, antiderivative size = 56, 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} \[ \frac{40}{729 (3 x+2)^3}-\frac{107}{243 (3 x+2)^4}+\frac{518}{405 (3 x+2)^5}-\frac{2009}{1458 (3 x+2)^6}+\frac{49}{243 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(243*(2 + 3*x)^7) - 2009/(1458*(2 + 3*x)^6) + 518/(405*(2 + 3*x)^5) - 107/(243*(2 + 3*x)^4) + 40/(729*(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 (3+5 x)}{(2+3 x)^8} \, dx &=\int \left (-\frac{343}{81 (2+3 x)^8}+\frac{2009}{81 (2+3 x)^7}-\frac{518}{27 (2+3 x)^6}+\frac{428}{81 (2+3 x)^5}-\frac{40}{81 (2+3 x)^4}\right ) \, dx\\ &=\frac{49}{243 (2+3 x)^7}-\frac{2009}{1458 (2+3 x)^6}+\frac{518}{405 (2+3 x)^5}-\frac{107}{243 (2+3 x)^4}+\frac{40}{729 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0106357, size = 31, normalized size = 0.55 \[ \frac{32400 x^4-270 x^3-3024 x^2+4593 x-604}{7290 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-604 + 4593*x - 3024*x^2 - 270*x^3 + 32400*x^4)/(7290*(2 + 3*x)^7)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.8 \begin{align*}{\frac{49}{243\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{2009}{1458\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{518}{405\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{107}{243\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{40}{729\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/243/(2+3*x)^7-2009/1458/(2+3*x)^6+518/405/(2+3*x)^5-107/243/(2+3*x)^4+40/729/(2+3*x)^3

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Maxima [A]  time = 1.08285, size = 80, normalized size = 1.43 \begin{align*} \frac{32400 \, x^{4} - 270 \, x^{3} - 3024 \, x^{2} + 4593 \, x - 604}{7290 \,{\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+3*x)^8,x, algorithm="maxima")

[Out]

1/7290*(32400*x^4 - 270*x^3 - 3024*x^2 + 4593*x - 604)/(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 = 1.26189, size = 190, normalized size = 3.39 \begin{align*} \frac{32400 \, x^{4} - 270 \, x^{3} - 3024 \, x^{2} + 4593 \, x - 604}{7290 \,{\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+3*x)^8,x, algorithm="fricas")

[Out]

1/7290*(32400*x^4 - 270*x^3 - 3024*x^2 + 4593*x - 604)/(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 = 0.170338, size = 54, normalized size = 0.96 \begin{align*} \frac{32400 x^{4} - 270 x^{3} - 3024 x^{2} + 4593 x - 604}{15943230 x^{7} + 74401740 x^{6} + 148803480 x^{5} + 165337200 x^{4} + 110224800 x^{3} + 44089920 x^{2} + 9797760 x + 933120} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(32400*x**4 - 270*x**3 - 3024*x**2 + 4593*x - 604)/(15943230*x**7 + 74401740*x**6 + 148803480*x**5 + 165337200
*x**4 + 110224800*x**3 + 44089920*x**2 + 9797760*x + 933120)

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Giac [A]  time = 1.92169, size = 39, normalized size = 0.7 \begin{align*} \frac{32400 \, x^{4} - 270 \, x^{3} - 3024 \, x^{2} + 4593 \, x - 604}{7290 \,{\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+3*x)^8,x, algorithm="giac")

[Out]

1/7290*(32400*x^4 - 270*x^3 - 3024*x^2 + 4593*x - 604)/(3*x + 2)^7