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

Optimal. Leaf size=78 \[ \frac {1000}{2187 (3 x+2)}-\frac {1850}{729 (3 x+2)^2}+\frac {14390}{2187 (3 x+2)^3}-\frac {66193}{8748 (3 x+2)^4}+\frac {10073}{3645 (3 x+2)^5}-\frac {1813}{4374 (3 x+2)^6}+\frac {49}{2187 (3 x+2)^7} \]

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Rubi [A]  time = 0.03, antiderivative size = 78, 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 {1000}{2187 (3 x+2)}-\frac {1850}{729 (3 x+2)^2}+\frac {14390}{2187 (3 x+2)^3}-\frac {66193}{8748 (3 x+2)^4}+\frac {10073}{3645 (3 x+2)^5}-\frac {1813}{4374 (3 x+2)^6}+\frac {49}{2187 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2187*(2 + 3*x)^7) - 1813/(4374*(2 + 3*x)^6) + 10073/(3645*(2 + 3*x)^5) - 66193/(8748*(2 + 3*x)^4) + 14390/
(2187*(2 + 3*x)^3) - 1850/(729*(2 + 3*x)^2) + 1000/(2187*(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)^3}{(2+3 x)^8} \, dx &=\int \left (-\frac {343}{729 (2+3 x)^8}+\frac {1813}{243 (2+3 x)^7}-\frac {10073}{243 (2+3 x)^6}+\frac {66193}{729 (2+3 x)^5}-\frac {14390}{243 (2+3 x)^4}+\frac {3700}{243 (2+3 x)^3}-\frac {1000}{729 (2+3 x)^2}\right ) \, dx\\ &=\frac {49}{2187 (2+3 x)^7}-\frac {1813}{4374 (2+3 x)^6}+\frac {10073}{3645 (2+3 x)^5}-\frac {66193}{8748 (2+3 x)^4}+\frac {14390}{2187 (2+3 x)^3}-\frac {1850}{729 (2+3 x)^2}+\frac {1000}{2187 (2+3 x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.53 \begin {gather*} \frac {14580000 x^6+31347000 x^5+30601800 x^4+19748745 x^3+8660574 x^2+1990182 x+133304}{43740 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(133304 + 1990182*x + 8660574*x^2 + 19748745*x^3 + 30601800*x^4 + 31347000*x^5 + 14580000*x^6)/(43740*(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)^3 (3+5 x)^3}{(2+3 x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 69, normalized size = 0.88 \begin {gather*} \frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{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)^3*(3+5*x)^3/(2+3*x)^8,x, algorithm="fricas")

[Out]

1/43740*(14580000*x^6 + 31347000*x^5 + 30601800*x^4 + 19748745*x^3 + 8660574*x^2 + 1990182*x + 133304)/(2187*x
^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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giac [A]  time = 0.98, size = 39, normalized size = 0.50 \begin {gather*} \frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{43740 \, {\left (3 \, x + 2\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)^3/(2+3*x)^8,x, algorithm="giac")

[Out]

1/43740*(14580000*x^6 + 31347000*x^5 + 30601800*x^4 + 19748745*x^3 + 8660574*x^2 + 1990182*x + 133304)/(3*x +
2)^7

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maple [A]  time = 0.01, size = 65, normalized size = 0.83 \begin {gather*} \frac {49}{2187 \left (3 x +2\right )^{7}}-\frac {1813}{4374 \left (3 x +2\right )^{6}}+\frac {10073}{3645 \left (3 x +2\right )^{5}}-\frac {66193}{8748 \left (3 x +2\right )^{4}}+\frac {14390}{2187 \left (3 x +2\right )^{3}}-\frac {1850}{729 \left (3 x +2\right )^{2}}+\frac {1000}{2187 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/2187/(3*x+2)^7-1813/4374/(3*x+2)^6+10073/3645/(3*x+2)^5-66193/8748/(3*x+2)^4+14390/2187/(3*x+2)^3-1850/729/
(3*x+2)^2+1000/2187/(3*x+2)

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maxima [A]  time = 0.51, size = 69, normalized size = 0.88 \begin {gather*} \frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{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)^3*(3+5*x)^3/(2+3*x)^8,x, algorithm="maxima")

[Out]

1/43740*(14580000*x^6 + 31347000*x^5 + 30601800*x^4 + 19748745*x^3 + 8660574*x^2 + 1990182*x + 133304)/(2187*x
^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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mupad [B]  time = 1.12, size = 64, normalized size = 0.82 \begin {gather*} \frac {1000}{2187\,\left (3\,x+2\right )}-\frac {1850}{729\,{\left (3\,x+2\right )}^2}+\frac {14390}{2187\,{\left (3\,x+2\right )}^3}-\frac {66193}{8748\,{\left (3\,x+2\right )}^4}+\frac {10073}{3645\,{\left (3\,x+2\right )}^5}-\frac {1813}{4374\,{\left (3\,x+2\right )}^6}+\frac {49}{2187\,{\left (3\,x+2\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000/(2187*(3*x + 2)) - 1850/(729*(3*x + 2)^2) + 14390/(2187*(3*x + 2)^3) - 66193/(8748*(3*x + 2)^4) + 10073/(
3645*(3*x + 2)^5) - 1813/(4374*(3*x + 2)^6) + 49/(2187*(3*x + 2)^7)

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sympy [A]  time = 0.19, size = 68, normalized size = 0.87 \begin {gather*} - \frac {- 14580000 x^{6} - 31347000 x^{5} - 30601800 x^{4} - 19748745 x^{3} - 8660574 x^{2} - 1990182 x - 133304}{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)**3*(3+5*x)**3/(2+3*x)**8,x)

[Out]

-(-14580000*x**6 - 31347000*x**5 - 30601800*x**4 - 19748745*x**3 - 8660574*x**2 - 1990182*x - 133304)/(9565938
0*x**7 + 446410440*x**6 + 892820880*x**5 + 992023200*x**4 + 661348800*x**3 + 264539520*x**2 + 58786560*x + 559
8720)

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