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

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

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Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 36, normalized size = 0.54 \begin {gather*} -\frac {3645000 x^5+5994000 x^4+3139425 x^3+652158 x^2+210534 x+76288}{43740 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.43, size = 64, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.90, size = 34, normalized size = 0.51 \begin {gather*} -\frac {3645000 \, x^{5} + 5994000 \, x^{4} + 3139425 \, x^{3} + 652158 \, x^{2} + 210534 \, x + 76288}{43740 \, {\left (3 \, x + 2\right )}^{7}} \end {gather*}

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

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maple [A]  time = 0.01, size = 56, normalized size = 0.84 \begin {gather*} \frac {7}{729 \left (3 x +2\right )^{7}}-\frac {763}{4374 \left (3 x +2\right )^{6}}+\frac {4099}{3645 \left (3 x +2\right )^{5}}-\frac {8285}{2916 \left (3 x +2\right )^{4}}+\frac {3800}{2187 \left (3 x +2\right )^{3}}-\frac {250}{729 \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 64, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.09, size = 55, normalized size = 0.82 \begin {gather*} \frac {3800}{2187\,{\left (3\,x+2\right )}^3}-\frac {250}{729\,{\left (3\,x+2\right )}^2}-\frac {8285}{2916\,{\left (3\,x+2\right )}^4}+\frac {4099}{3645\,{\left (3\,x+2\right )}^5}-\frac {763}{4374\,{\left (3\,x+2\right )}^6}+\frac {7}{729\,{\left (3\,x+2\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3800/(2187*(3*x + 2)^3) - 250/(729*(3*x + 2)^2) - 8285/(2916*(3*x + 2)^4) + 4099/(3645*(3*x + 2)^5) - 763/(437
4*(3*x + 2)^6) + 7/(729*(3*x + 2)^7)

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sympy [A]  time = 0.19, size = 61, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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