\(\int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx\) [1387]

Optimal result

Integrand size = 22, antiderivative size = 78 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {49}{2187 (2+3 x)^7}-\frac {1813}{4374 (2+3 x)^6}+\frac {10073}{3645 (2+3 x)^5}-\frac {66193}{8748 (2+3 x)^4}+\frac {14390}{2187 (2+3 x)^3}-\frac {1850}{729 (2+3 x)^2}+\frac {1000}{2187 (2+3 x)} \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {1000}{2187 (3 x+2)}-\frac {1850}{729 (3 x+2)^2}+\frac {14390}{2187 (3 x+2)^3}-\frac {66193}{8748 (3 x+2)^4}+\frac {10073}{3645 (3 x+2)^5}-\frac {1813}{4374 (3 x+2)^6}+\frac {49}{2187 (3 x+2)^7} \]

[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 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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {133304+1990182 x+8660574 x^2+19748745 x^3+30601800 x^4+31347000 x^5+14580000 x^6}{43740 (2+3 x)^7} \]

[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)

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50

method	result	size
norman	\(\frac {\frac {1000}{3} x^{6}+\frac {2150}{3} x^{5}+\frac {18890}{27} x^{4}+\frac {146287}{324} x^{3}+\frac {160381}{810} x^{2}+\frac {331697}{7290} x +\frac {33326}{10935}}{\left (2+3 x \right )^{7}}\)	\(39\)
gosper	\(\frac {14580000 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}+8660574 x^{2}+1990182 x +133304}{43740 \left (2+3 x \right )^{7}}\)	\(40\)
risch	\(\frac {\frac {1000}{3} x^{6}+\frac {2150}{3} x^{5}+\frac {18890}{27} x^{4}+\frac {146287}{324} x^{3}+\frac {160381}{810} x^{2}+\frac {331697}{7290} x +\frac {33326}{10935}}{\left (2+3 x \right )^{7}}\)	\(40\)
parallelrisch	\(\frac {-33326 x^{7}+57812 x^{6}+147624 x^{5}+102160 x^{4}+58560 x^{3}+34560 x^{2}+8640 x}{640 \left (2+3 x \right )^{7}}\)	\(44\)
default	\(\frac {49}{2187 \left (2+3 x \right )^{7}}-\frac {1813}{4374 \left (2+3 x \right )^{6}}+\frac {10073}{3645 \left (2+3 x \right )^{5}}-\frac {66193}{8748 \left (2+3 x \right )^{4}}+\frac {14390}{2187 \left (2+3 x \right )^{3}}-\frac {1850}{729 \left (2+3 x \right )^{2}}+\frac {1000}{2187 \left (2+3 x \right )}\)	\(65\)
meijerg	\(\frac {27 x \left (\frac {729}{64} x^{6}+\frac {1701}{32} x^{5}+\frac {1701}{16} x^{4}+\frac {945}{8} x^{3}+\frac {315}{4} x^{2}+\frac {63}{2} x +7\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {9 x^{2} \left (\frac {243}{32} x^{5}+\frac {567}{16} x^{4}+\frac {567}{8} x^{3}+\frac {315}{4} x^{2}+\frac {105}{2} x +21\right )}{3584 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {87 x^{3} \left (\frac {81}{16} x^{4}+\frac {189}{8} x^{3}+\frac {189}{4} x^{2}+\frac {105}{2} x +35\right )}{8960 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {179 x^{4} \left (\frac {27}{8} x^{3}+\frac {63}{4} x^{2}+\frac {63}{2} x +35\right )}{35840 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {29 x^{5} \left (\frac {9}{4} x^{2}+\frac {21}{2} x +21\right )}{896 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {25 x^{6} \left (\frac {3 x}{2}+7\right )}{896 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {125 x^{7}}{224 \left (1+\frac {3 x}{2}\right )^{7}}\)	\(189\)

[In]

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

[Out]

(1000/3*x^6+2150/3*x^5+18890/27*x^4+146287/324*x^3+160381/810*x^2+331697/7290*x+33326/10935)/(2+3*x)^7

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{43740 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=- \frac {- 14580000 x^{6} - 31347000 x^{5} - 30601800 x^{4} - 19748745 x^{3} - 8660574 x^{2} - 1990182 x - 133304}{95659380 x^{7} + 446410440 x^{6} + 892820880 x^{5} + 992023200 x^{4} + 661348800 x^{3} + 264539520 x^{2} + 58786560 x + 5598720} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{43740 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {14580000 \, x^{6} + 31347000 \, x^{5} + 30601800 \, x^{4} + 19748745 \, x^{3} + 8660574 \, x^{2} + 1990182 \, x + 133304}{43740 \, {\left (3 \, x + 2\right )}^{7}} \]

[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

Mupad [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {1000}{2187\,\left (3\,x+2\right )}-\frac {1850}{729\,{\left (3\,x+2\right )}^2}+\frac {14390}{2187\,{\left (3\,x+2\right )}^3}-\frac {66193}{8748\,{\left (3\,x+2\right )}^4}+\frac {10073}{3645\,{\left (3\,x+2\right )}^5}-\frac {1813}{4374\,{\left (3\,x+2\right )}^6}+\frac {49}{2187\,{\left (3\,x+2\right )}^7} \]

[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)