3.14.71 \(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^8} \, dx\) [1371]

Optimal. Leaf size=67 \[ -\frac {49}{729 (2+3 x)^7}+\frac {1862}{2187 (2+3 x)^6}-\frac {11599}{3645 (2+3 x)^5}+\frac {4099}{1458 (2+3 x)^4}-\frac {2180}{2187 (2+3 x)^3}+\frac {100}{729 (2+3 x)^2} \]

[Out]

-49/729/(2+3*x)^7+1862/2187/(2+3*x)^6-11599/3645/(2+3*x)^5+4099/1458/(2+3*x)^4-2180/2187/(2+3*x)^3+100/729/(2+
3*x)^2

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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 = {90} \begin {gather*} \frac {100}{729 (3 x+2)^2}-\frac {2180}{2187 (3 x+2)^3}+\frac {4099}{1458 (3 x+2)^4}-\frac {11599}{3645 (3 x+2)^5}+\frac {1862}{2187 (3 x+2)^6}-\frac {49}{729 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(729*(2 + 3*x)^7) + 1862/(2187*(2 + 3*x)^6) - 11599/(3645*(2 + 3*x)^5) + 4099/(1458*(2 + 3*x)^4) - 2180/(2
187*(2 + 3*x)^3) + 100/(729*(2 + 3*x)^2)

Rule 90

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)^8} \, dx &=\int \left (\frac {343}{243 (2+3 x)^8}-\frac {3724}{243 (2+3 x)^7}+\frac {11599}{243 (2+3 x)^6}-\frac {8198}{243 (2+3 x)^5}+\frac {2180}{243 (2+3 x)^4}-\frac {200}{243 (2+3 x)^3}\right ) \, dx\\ &=-\frac {49}{729 (2+3 x)^7}+\frac {1862}{2187 (2+3 x)^6}-\frac {11599}{3645 (2+3 x)^5}+\frac {4099}{1458 (2+3 x)^4}-\frac {2180}{2187 (2+3 x)^3}+\frac {100}{729 (2+3 x)^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.54 \begin {gather*} \frac {-3526+61392 x+145044 x^2+191295 x^3+664200 x^4+729000 x^5}{21870 (2+3 x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3526 + 61392*x + 145044*x^2 + 191295*x^3 + 664200*x^4 + 729000*x^5)/(21870*(2 + 3*x)^7)

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Maple [A]
time = 0.10, size = 56, normalized size = 0.84

method	result	size
gosper	\(\frac {729000 x^{5}+664200 x^{4}+191295 x^{3}+145044 x^{2}+61392 x -3526}{21870 \left (2+3 x \right )^{7}}\)	\(35\)
risch	\(\frac {\frac {100}{3} x^{5}+\frac {820}{27} x^{4}+\frac {1417}{162} x^{3}+\frac {2686}{405} x^{2}+\frac {10232}{3645} x -\frac {1763}{10935}}{\left (2+3 x \right )^{7}}\)	\(35\)
norman	\(\frac {\frac {943}{16} x^{4}+\frac {9447}{160} x^{5}+\frac {9}{2} x +\frac {57}{4} x^{2}+\frac {667}{24} x^{3}+\frac {1763}{640} x^{7}+\frac {12341}{960} x^{6}}{\left (2+3 x \right )^{7}}\)	\(43\)
default	\(-\frac {49}{729 \left (2+3 x \right )^{7}}+\frac {1862}{2187 \left (2+3 x \right )^{6}}-\frac {11599}{3645 \left (2+3 x \right )^{5}}+\frac {4099}{1458 \left (2+3 x \right )^{4}}-\frac {2180}{2187 \left (2+3 x \right )^{3}}+\frac {100}{729 \left (2+3 x \right )^{2}}\)	\(56\)
meijerg	\(\frac {9 x \left (\frac {729}{64} x^{6}+\frac {1701}{32} x^{5}+\frac {1701}{16} x^{4}+\frac {945}{8} x^{3}+\frac {315}{4} x^{2}+\frac {63}{2} x +7\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {x^{2} \left (\frac {243}{32} x^{5}+\frac {567}{16} x^{4}+\frac {567}{8} x^{3}+\frac {315}{4} x^{2}+\frac {105}{2} x +21\right )}{448 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {47 x^{3} \left (\frac {81}{16} x^{4}+\frac {189}{8} x^{3}+\frac {189}{4} x^{2}+\frac {105}{2} x +35\right )}{26880 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {69 x^{4} \left (\frac {27}{8} x^{3}+\frac {63}{4} x^{2}+\frac {63}{2} x +35\right )}{17920 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {x^{5} \left (\frac {9}{4} x^{2}+\frac {21}{2} x +21\right )}{448 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {25 x^{6} \left (\frac {3 x}{2}+7\right )}{1344 \left (1+\frac {3 x}{2}\right )^{7}}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/729/(2+3*x)^7+1862/2187/(2+3*x)^6-11599/3645/(2+3*x)^5+4099/1458/(2+3*x)^4-2180/2187/(2+3*x)^3+100/729/(2+
3*x)^2

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Maxima [A]
time = 0.48, size = 64, normalized size = 0.96 \begin {gather*} \frac {729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \, {\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)^2/(2+3*x)^8,x, algorithm="maxima")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(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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Fricas [A]
time = 0.45, size = 64, normalized size = 0.96 \begin {gather*} \frac {729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \, {\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)^2/(2+3*x)^8,x, algorithm="fricas")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(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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Sympy [A]
time = 0.08, size = 61, normalized size = 0.91 \begin {gather*} - \frac {- 729000 x^{5} - 664200 x^{4} - 191295 x^{3} - 145044 x^{2} - 61392 x + 3526}{47829690 x^{7} + 223205220 x^{6} + 446410440 x^{5} + 496011600 x^{4} + 330674400 x^{3} + 132269760 x^{2} + 29393280 x + 2799360} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-729000*x**5 - 664200*x**4 - 191295*x**3 - 145044*x**2 - 61392*x + 3526)/(47829690*x**7 + 223205220*x**6 + 4
46410440*x**5 + 496011600*x**4 + 330674400*x**3 + 132269760*x**2 + 29393280*x + 2799360)

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Giac [A]
time = 1.26, size = 34, normalized size = 0.51 \begin {gather*} \frac {729000 \, x^{5} + 664200 \, x^{4} + 191295 \, x^{3} + 145044 \, x^{2} + 61392 \, x - 3526}{21870 \, {\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)^2/(2+3*x)^8,x, algorithm="giac")

[Out]

1/21870*(729000*x^5 + 664200*x^4 + 191295*x^3 + 145044*x^2 + 61392*x - 3526)/(3*x + 2)^7

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.82 \begin {gather*} \frac {100}{729\,{\left (3\,x+2\right )}^2}-\frac {2180}{2187\,{\left (3\,x+2\right )}^3}+\frac {4099}{1458\,{\left (3\,x+2\right )}^4}-\frac {11599}{3645\,{\left (3\,x+2\right )}^5}+\frac {1862}{2187\,{\left (3\,x+2\right )}^6}-\frac {49}{729\,{\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)^2)/(3*x + 2)^8,x)

[Out]

100/(729*(3*x + 2)^2) - 2180/(2187*(3*x + 2)^3) + 4099/(1458*(3*x + 2)^4) - 11599/(3645*(3*x + 2)^5) + 1862/(2
187*(3*x + 2)^6) - 49/(729*(3*x + 2)^7)

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