3.1287 \(\int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=67 \[ -\frac{500}{729 (3 x+2)}+\frac{1900}{729 (3 x+2)^2}-\frac{8285}{2187 (3 x+2)^3}+\frac{4099}{2916 (3 x+2)^4}-\frac{763}{3645 (3 x+2)^5}+\frac{49}{4374 (3 x+2)^6} \]

[Out]

49/(4374*(2 + 3*x)^6) - 763/(3645*(2 + 3*x)^5) + 4099/(2916*(2 + 3*x)^4) - 8285/(2187*(2 + 3*x)^3) + 1900/(729
*(2 + 3*x)^2) - 500/(729*(2 + 3*x))

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Rubi [A]  time = 0.0233289, antiderivative size = 67, 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} \[ -\frac{500}{729 (3 x+2)}+\frac{1900}{729 (3 x+2)^2}-\frac{8285}{2187 (3 x+2)^3}+\frac{4099}{2916 (3 x+2)^4}-\frac{763}{3645 (3 x+2)^5}+\frac{49}{4374 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

49/(4374*(2 + 3*x)^6) - 763/(3645*(2 + 3*x)^5) + 4099/(2916*(2 + 3*x)^4) - 8285/(2187*(2 + 3*x)^3) + 1900/(729
*(2 + 3*x)^2) - 500/(729*(2 + 3*x))

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)^3}{(2+3 x)^7} \, dx &=\int \left (-\frac{49}{243 (2+3 x)^7}+\frac{763}{243 (2+3 x)^6}-\frac{4099}{243 (2+3 x)^5}+\frac{8285}{243 (2+3 x)^4}-\frac{3800}{243 (2+3 x)^3}+\frac{500}{243 (2+3 x)^2}\right ) \, dx\\ &=\frac{49}{4374 (2+3 x)^6}-\frac{763}{3645 (2+3 x)^5}+\frac{4099}{2916 (2+3 x)^4}-\frac{8285}{2187 (2+3 x)^3}+\frac{1900}{729 (2+3 x)^2}-\frac{500}{729 (2+3 x)}\\ \end{align*}

Mathematica [A]  time = 0.0267871, size = 36, normalized size = 0.54 \[ -\frac{7290000 x^5+15066000 x^4+12249900 x^3+5370435 x^2+1510848 x+233482}{43740 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^7,x]

[Out]

-(233482 + 1510848*x + 5370435*x^2 + 12249900*x^3 + 15066000*x^4 + 7290000*x^5)/(43740*(2 + 3*x)^6)

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Maple [A]  time = 0.005, size = 56, normalized size = 0.8 \begin{align*}{\frac{49}{4374\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{763}{3645\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{4099}{2916\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{8285}{2187\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1900}{729\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{500}{1458+2187\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(3+5*x)^3/(2+3*x)^7,x)

[Out]

49/4374/(2+3*x)^6-763/3645/(2+3*x)^5+4099/2916/(2+3*x)^4-8285/2187/(2+3*x)^3+1900/729/(2+3*x)^2-500/729/(2+3*x
)

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Maxima [A]  time = 1.03119, size = 80, normalized size = 1.19 \begin{align*} -\frac{7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Fricas [A]  time = 1.50294, size = 211, normalized size = 3.15 \begin{align*} -\frac{7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Sympy [A]  time = 0.164601, size = 56, normalized size = 0.84 \begin{align*} - \frac{7290000 x^{5} + 15066000 x^{4} + 12249900 x^{3} + 5370435 x^{2} + 1510848 x + 233482}{31886460 x^{6} + 127545840 x^{5} + 212576400 x^{4} + 188956800 x^{3} + 94478400 x^{2} + 25194240 x + 2799360} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)**3/(2+3*x)**7,x)

[Out]

-(7290000*x**5 + 15066000*x**4 + 12249900*x**3 + 5370435*x**2 + 1510848*x + 233482)/(31886460*x**6 + 127545840
*x**5 + 212576400*x**4 + 188956800*x**3 + 94478400*x**2 + 25194240*x + 2799360)

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Giac [A]  time = 2.3144, size = 46, normalized size = 0.69 \begin{align*} -\frac{7290000 \, x^{5} + 15066000 \, x^{4} + 12249900 \, x^{3} + 5370435 \, x^{2} + 1510848 \, x + 233482}{43740 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/43740*(7290000*x^5 + 15066000*x^4 + 12249900*x^3 + 5370435*x^2 + 1510848*x + 233482)/(3*x + 2)^6