\(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx\) [1191]

Optimal result

Integrand size = 20, antiderivative size = 56 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {1}{243 (2+3 x)^7}-\frac {107}{1458 (2+3 x)^6}+\frac {37}{81 (2+3 x)^5}-\frac {1025}{972 (2+3 x)^4}+\frac {250}{729 (2+3 x)^3} \]

[Out]

1/243/(2+3*x)^7-107/1458/(2+3*x)^6+37/81/(2+3*x)^5-1025/972/(2+3*x)^4+250/729/(2+3*x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {250}{729 (3 x+2)^3}-\frac {1025}{972 (3 x+2)^4}+\frac {37}{81 (3 x+2)^5}-\frac {107}{1458 (3 x+2)^6}+\frac {1}{243 (3 x+2)^7} \]

[In]

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

[Out]

1/(243*(2 + 3*x)^7) - 107/(1458*(2 + 3*x)^6) + 37/(81*(2 + 3*x)^5) - 1025/(972*(2 + 3*x)^4) + 250/(729*(2 + 3*
x)^3)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {7}{81 (2+3 x)^8}+\frac {107}{81 (2+3 x)^7}-\frac {185}{27 (2+3 x)^6}+\frac {1025}{81 (2+3 x)^5}-\frac {250}{81 (2+3 x)^4}\right ) \, dx \\ & = \frac {1}{243 (2+3 x)^7}-\frac {107}{1458 (2+3 x)^6}+\frac {37}{81 (2+3 x)^5}-\frac {1025}{972 (2+3 x)^4}+\frac {250}{729 (2+3 x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {-3688+642 x+61938 x^2+132975 x^3+81000 x^4}{2916 (2+3 x)^7} \]

[In]

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

[Out]

(-3688 + 642*x + 61938*x^2 + 132975*x^3 + 81000*x^4)/(2916*(2 + 3*x)^7)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.52

method	result	size
norman	\(\frac {\frac {107}{486} x +\frac {250}{9} x^{4}+\frac {1147}{54} x^{2}+\frac {4925}{108} x^{3}-\frac {922}{729}}{\left (2+3 x \right )^{7}}\)	\(29\)
gosper	\(\frac {81000 x^{4}+132975 x^{3}+61938 x^{2}+642 x -3688}{2916 \left (2+3 x \right )^{7}}\)	\(30\)
risch	\(\frac {\frac {107}{486} x +\frac {250}{9} x^{4}+\frac {1147}{54} x^{2}+\frac {4925}{108} x^{3}-\frac {922}{729}}{\left (2+3 x \right )^{7}}\)	\(30\)
parallelrisch	\(\frac {2766 x^{7}+12908 x^{6}+25816 x^{5}+32240 x^{4}+24960 x^{3}+10368 x^{2}+1728 x}{128 \left (2+3 x \right )^{7}}\)	\(44\)
default	\(\frac {1}{243 \left (2+3 x \right )^{7}}-\frac {107}{1458 \left (2+3 x \right )^{6}}+\frac {37}{81 \left (2+3 x \right )^{5}}-\frac {1025}{972 \left (2+3 x \right )^{4}}+\frac {250}{729 \left (2+3 x \right )^{3}}\)	\(47\)
meijerg	\(\frac {27 x \left (\frac {729}{64} x^{6}+\frac {1701}{32} x^{5}+\frac {1701}{16} x^{4}+\frac {945}{8} x^{3}+\frac {315}{4} x^{2}+\frac {63}{2} x +7\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {27 x^{2} \left (\frac {243}{32} x^{5}+\frac {567}{16} x^{4}+\frac {567}{8} x^{3}+\frac {315}{4} x^{2}+\frac {105}{2} x +21\right )}{3584 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {3 x^{3} \left (\frac {81}{16} x^{4}+\frac {189}{8} x^{3}+\frac {189}{4} x^{2}+\frac {105}{2} x +35\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {65 x^{4} \left (\frac {27}{8} x^{3}+\frac {63}{4} x^{2}+\frac {63}{2} x +35\right )}{7168 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {25 x^{5} \left (\frac {9}{4} x^{2}+\frac {21}{2} x +21\right )}{2688 \left (1+\frac {3 x}{2}\right )^{7}}\)	\(160\)

[In]

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

[Out]

(107/486*x+250/9*x^4+1147/54*x^2+4925/108*x^3-922/729)/(2+3*x)^7

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {81000 \, x^{4} + 132975 \, x^{3} + 61938 \, x^{2} + 642 \, x - 3688}{2916 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

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

[Out]

1/2916*(81000*x^4 + 132975*x^3 + 61938*x^2 + 642*x - 3688)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=- \frac {- 81000 x^{4} - 132975 x^{3} - 61938 x^{2} - 642 x + 3688}{6377292 x^{7} + 29760696 x^{6} + 59521392 x^{5} + 66134880 x^{4} + 44089920 x^{3} + 17635968 x^{2} + 3919104 x + 373248} \]

[In]

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

[Out]

-(-81000*x**4 - 132975*x**3 - 61938*x**2 - 642*x + 3688)/(6377292*x**7 + 29760696*x**6 + 59521392*x**5 + 66134
880*x**4 + 44089920*x**3 + 17635968*x**2 + 3919104*x + 373248)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {81000 \, x^{4} + 132975 \, x^{3} + 61938 \, x^{2} + 642 \, x - 3688}{2916 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

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

[Out]

1/2916*(81000*x^4 + 132975*x^3 + 61938*x^2 + 642*x - 3688)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {81000 \, x^{4} + 132975 \, x^{3} + 61938 \, x^{2} + 642 \, x - 3688}{2916 \, {\left (3 \, x + 2\right )}^{7}} \]

[In]

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

[Out]

1/2916*(81000*x^4 + 132975*x^3 + 61938*x^2 + 642*x - 3688)/(3*x + 2)^7

Mupad [B] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^8} \, dx=\frac {250}{729\,{\left (3\,x+2\right )}^3}-\frac {1025}{972\,{\left (3\,x+2\right )}^4}+\frac {37}{81\,{\left (3\,x+2\right )}^5}-\frac {107}{1458\,{\left (3\,x+2\right )}^6}+\frac {1}{243\,{\left (3\,x+2\right )}^7} \]

[In]

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

[Out]

250/(729*(3*x + 2)^3) - 1025/(972*(3*x + 2)^4) + 37/(81*(3*x + 2)^5) - 107/(1458*(3*x + 2)^6) + 1/(243*(3*x +
2)^7)