3.1370 \(\int \frac{(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=67 \[ \frac{200}{729 (3 x+2)}-\frac{1090}{729 (3 x+2)^2}+\frac{8198}{2187 (3 x+2)^3}-\frac{11599}{2916 (3 x+2)^4}+\frac{3724}{3645 (3 x+2)^5}-\frac{343}{4374 (3 x+2)^6} \]

[Out]

-343/(4374*(2 + 3*x)^6) + 3724/(3645*(2 + 3*x)^5) - 11599/(2916*(2 + 3*x)^4) + 8198/(2187*(2 + 3*x)^3) - 1090/
(729*(2 + 3*x)^2) + 200/(729*(2 + 3*x))

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Rubi [A]  time = 0.0233426, 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} \[ \frac{200}{729 (3 x+2)}-\frac{1090}{729 (3 x+2)^2}+\frac{8198}{2187 (3 x+2)^3}-\frac{11599}{2916 (3 x+2)^4}+\frac{3724}{3645 (3 x+2)^5}-\frac{343}{4374 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-343/(4374*(2 + 3*x)^6) + 3724/(3645*(2 + 3*x)^5) - 11599/(2916*(2 + 3*x)^4) + 8198/(2187*(2 + 3*x)^3) - 1090/
(729*(2 + 3*x)^2) + 200/(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)^3 (3+5 x)^2}{(2+3 x)^7} \, dx &=\int \left (\frac{343}{243 (2+3 x)^7}-\frac{3724}{243 (2+3 x)^6}+\frac{11599}{243 (2+3 x)^5}-\frac{8198}{243 (2+3 x)^4}+\frac{2180}{243 (2+3 x)^3}-\frac{200}{243 (2+3 x)^2}\right ) \, dx\\ &=-\frac{343}{4374 (2+3 x)^6}+\frac{3724}{3645 (2+3 x)^5}-\frac{11599}{2916 (2+3 x)^4}+\frac{8198}{2187 (2+3 x)^3}-\frac{1090}{729 (2+3 x)^2}+\frac{200}{729 (2+3 x)}\\ \end{align*}

Mathematica [A]  time = 0.0263139, size = 36, normalized size = 0.54 \[ \frac{2916000 x^5+4422600 x^4+3260520 x^3+1801575 x^2+550404 x+39286}{43740 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(39286 + 550404*x + 1801575*x^2 + 3260520*x^3 + 4422600*x^4 + 2916000*x^5)/(43740*(2 + 3*x)^6)

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Maple [A]  time = 0.006, size = 56, normalized size = 0.8 \begin{align*} -{\frac{343}{4374\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{3724}{3645\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{11599}{2916\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{8198}{2187\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{1090}{729\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{200}{1458+2187\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/4374/(2+3*x)^6+3724/3645/(2+3*x)^5-11599/2916/(2+3*x)^4+8198/2187/(2+3*x)^3-1090/729/(2+3*x)^2+200/729/(2
+3*x)

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Maxima [A]  time = 2.30597, size = 80, normalized size = 1.19 \begin{align*} \frac{2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{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)^3*(3+5*x)^2/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(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 = 1.42036, size = 204, normalized size = 3.04 \begin{align*} \frac{2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{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)^3*(3+5*x)^2/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(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 = 0.169394, size = 54, normalized size = 0.81 \begin{align*} \frac{2916000 x^{5} + 4422600 x^{4} + 3260520 x^{3} + 1801575 x^{2} + 550404 x + 39286}{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)**3*(3+5*x)**2/(2+3*x)**7,x)

[Out]

(2916000*x**5 + 4422600*x**4 + 3260520*x**3 + 1801575*x**2 + 550404*x + 39286)/(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 = 2.83996, size = 46, normalized size = 0.69 \begin{align*} \frac{2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{43740 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(3*x + 2)^6