3.1269 \(\int \frac{(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=56 \[ -\frac{100}{729 (3 x+2)^3}+\frac{185}{243 (3 x+2)^4}-\frac{503}{405 (3 x+2)^5}+\frac{259}{729 (3 x+2)^6}-\frac{7}{243 (3 x+2)^7} \]

[Out]

-7/(243*(2 + 3*x)^7) + 259/(729*(2 + 3*x)^6) - 503/(405*(2 + 3*x)^5) + 185/(243*(2 + 3*x)^4) - 100/(729*(2 + 3
*x)^3)

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Rubi [A]  time = 0.0197464, antiderivative size = 56, 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} \[ -\frac{100}{729 (3 x+2)^3}+\frac{185}{243 (3 x+2)^4}-\frac{503}{405 (3 x+2)^5}+\frac{259}{729 (3 x+2)^6}-\frac{7}{243 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(243*(2 + 3*x)^7) + 259/(729*(2 + 3*x)^6) - 503/(405*(2 + 3*x)^5) + 185/(243*(2 + 3*x)^4) - 100/(729*(2 + 3
*x)^3)

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)^2 (3+5 x)^2}{(2+3 x)^8} \, dx &=\int \left (\frac{49}{81 (2+3 x)^8}-\frac{518}{81 (2+3 x)^7}+\frac{503}{27 (2+3 x)^6}-\frac{740}{81 (2+3 x)^5}+\frac{100}{81 (2+3 x)^4}\right ) \, dx\\ &=-\frac{7}{243 (2+3 x)^7}+\frac{259}{729 (2+3 x)^6}-\frac{503}{405 (2+3 x)^5}+\frac{185}{243 (2+3 x)^4}-\frac{100}{729 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0113724, size = 31, normalized size = 0.55 \[ \frac{-40500 x^4-33075 x^3+1107 x^2+1461 x-1423}{3645 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1423 + 1461*x + 1107*x^2 - 33075*x^3 - 40500*x^4)/(3645*(2 + 3*x)^7)

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Maple [A]  time = 0.004, size = 47, normalized size = 0.8 \begin{align*} -{\frac{7}{243\, \left ( 2+3\,x \right ) ^{7}}}+{\frac{259}{729\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{503}{405\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{185}{243\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{100}{729\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/243/(2+3*x)^7+259/729/(2+3*x)^6-503/405/(2+3*x)^5+185/243/(2+3*x)^4-100/729/(2+3*x)^3

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Maxima [A]  time = 1.07262, size = 80, normalized size = 1.43 \begin{align*} -\frac{40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \,{\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/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

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Fricas [A]  time = 1.70604, size = 196, normalized size = 3.5 \begin{align*} -\frac{40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \,{\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/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

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Sympy [A]  time = 0.169625, size = 56, normalized size = 1. \begin{align*} - \frac{40500 x^{4} + 33075 x^{3} - 1107 x^{2} - 1461 x + 1423}{7971615 x^{7} + 37200870 x^{6} + 74401740 x^{5} + 82668600 x^{4} + 55112400 x^{3} + 22044960 x^{2} + 4898880 x + 466560} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(40500*x**4 + 33075*x**3 - 1107*x**2 - 1461*x + 1423)/(7971615*x**7 + 37200870*x**6 + 74401740*x**5 + 8266860
0*x**4 + 55112400*x**3 + 22044960*x**2 + 4898880*x + 466560)

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Giac [A]  time = 1.29221, size = 39, normalized size = 0.7 \begin{align*} -\frac{40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \,{\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/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(3*x + 2)^7