3.1288 \(\int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=67 \[ -\frac{250}{729 (3 x+2)^2}+\frac{3800}{2187 (3 x+2)^3}-\frac{8285}{2916 (3 x+2)^4}+\frac{4099}{3645 (3 x+2)^5}-\frac{763}{4374 (3 x+2)^6}+\frac{7}{729 (3 x+2)^7} \]

[Out]

7/(729*(2 + 3*x)^7) - 763/(4374*(2 + 3*x)^6) + 4099/(3645*(2 + 3*x)^5) - 8285/(2916*(2 + 3*x)^4) + 3800/(2187*
(2 + 3*x)^3) - 250/(729*(2 + 3*x)^2)

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Rubi [A]  time = 0.0247781, 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} \[ -\frac{250}{729 (3 x+2)^2}+\frac{3800}{2187 (3 x+2)^3}-\frac{8285}{2916 (3 x+2)^4}+\frac{4099}{3645 (3 x+2)^5}-\frac{763}{4374 (3 x+2)^6}+\frac{7}{729 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(729*(2 + 3*x)^7) - 763/(4374*(2 + 3*x)^6) + 4099/(3645*(2 + 3*x)^5) - 8285/(2916*(2 + 3*x)^4) + 3800/(2187*
(2 + 3*x)^3) - 250/(729*(2 + 3*x)^2)

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)^8} \, dx &=\int \left (-\frac{49}{243 (2+3 x)^8}+\frac{763}{243 (2+3 x)^7}-\frac{4099}{243 (2+3 x)^6}+\frac{8285}{243 (2+3 x)^5}-\frac{3800}{243 (2+3 x)^4}+\frac{500}{243 (2+3 x)^3}\right ) \, dx\\ &=\frac{7}{729 (2+3 x)^7}-\frac{763}{4374 (2+3 x)^6}+\frac{4099}{3645 (2+3 x)^5}-\frac{8285}{2916 (2+3 x)^4}+\frac{3800}{2187 (2+3 x)^3}-\frac{250}{729 (2+3 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0270073, size = 36, normalized size = 0.54 \[ -\frac{3645000 x^5+5994000 x^4+3139425 x^3+652158 x^2+210534 x+76288}{43740 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(76288 + 210534*x + 652158*x^2 + 3139425*x^3 + 5994000*x^4 + 3645000*x^5)/(43740*(2 + 3*x)^7)

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Maple [A]  time = 0.003, size = 56, normalized size = 0.8 \begin{align*}{\frac{7}{729\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{763}{4374\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{4099}{3645\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{8285}{2916\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{3800}{2187\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{250}{729\, \left ( 2+3\,x \right ) ^{2}}} \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)^8,x)

[Out]

7/729/(2+3*x)^7-763/4374/(2+3*x)^6+4099/3645/(2+3*x)^5-8285/2916/(2+3*x)^4+3800/2187/(2+3*x)^3-250/729/(2+3*x)
^2

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Maxima [A]  time = 1.0129, size = 86, normalized size = 1.28 \begin{align*} -\frac{3645000 \, x^{5} + 5994000 \, x^{4} + 3139425 \, x^{3} + 652158 \, x^{2} + 210534 \, x + 76288}{43740 \,{\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)^3/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/43740*(3645000*x^5 + 5994000*x^4 + 3139425*x^3 + 652158*x^2 + 210534*x + 76288)/(2187*x^7 + 10206*x^6 + 204
12*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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Fricas [A]  time = 1.50586, size = 228, normalized size = 3.4 \begin{align*} -\frac{3645000 \, x^{5} + 5994000 \, x^{4} + 3139425 \, x^{3} + 652158 \, x^{2} + 210534 \, x + 76288}{43740 \,{\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)^3/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/43740*(3645000*x^5 + 5994000*x^4 + 3139425*x^3 + 652158*x^2 + 210534*x + 76288)/(2187*x^7 + 10206*x^6 + 204
12*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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Sympy [A]  time = 0.17609, size = 61, normalized size = 0.91 \begin{align*} - \frac{3645000 x^{5} + 5994000 x^{4} + 3139425 x^{3} + 652158 x^{2} + 210534 x + 76288}{95659380 x^{7} + 446410440 x^{6} + 892820880 x^{5} + 992023200 x^{4} + 661348800 x^{3} + 264539520 x^{2} + 58786560 x + 5598720} \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)**8,x)

[Out]

-(3645000*x**5 + 5994000*x**4 + 3139425*x**3 + 652158*x**2 + 210534*x + 76288)/(95659380*x**7 + 446410440*x**6
+ 892820880*x**5 + 992023200*x**4 + 661348800*x**3 + 264539520*x**2 + 58786560*x + 5598720)

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Giac [A]  time = 1.97805, size = 46, normalized size = 0.69 \begin{align*} -\frac{3645000 \, x^{5} + 5994000 \, x^{4} + 3139425 \, x^{3} + 652158 \, x^{2} + 210534 \, x + 76288}{43740 \,{\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)^3/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/43740*(3645000*x^5 + 5994000*x^4 + 3139425*x^3 + 652158*x^2 + 210534*x + 76288)/(3*x + 2)^7