\(\int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx\) [1269]

Optimal result

Integrand size = 22, antiderivative size = 56 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {7}{243 (2+3 x)^7}+\frac {259}{729 (2+3 x)^6}-\frac {503}{405 (2+3 x)^5}+\frac {185}{243 (2+3 x)^4}-\frac {100}{729 (2+3 x)^3} \]

[Out]

-7/243/(2+3*x)^7+259/729/(2+3*x)^6-503/405/(2+3*x)^5+185/243/(2+3*x)^4-100/729/(2+3*x)^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {100}{729 (3 x+2)^3}+\frac {185}{243 (3 x+2)^4}-\frac {503}{405 (3 x+2)^5}+\frac {259}{729 (3 x+2)^6}-\frac {7}{243 (3 x+2)^7} \]

[In]

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

[Out]

-7/(243*(2 + 3*x)^7) + 259/(729*(2 + 3*x)^6) - 503/(405*(2 + 3*x)^5) + 185/(243*(2 + 3*x)^4) - 100/(729*(2 + 3
*x)^3)

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 {49}{81 (2+3 x)^8}-\frac {518}{81 (2+3 x)^7}+\frac {503}{27 (2+3 x)^6}-\frac {740}{81 (2+3 x)^5}+\frac {100}{81 (2+3 x)^4}\right ) \, dx \\ & = -\frac {7}{243 (2+3 x)^7}+\frac {259}{729 (2+3 x)^6}-\frac {503}{405 (2+3 x)^5}+\frac {185}{243 (2+3 x)^4}-\frac {100}{729 (2+3 x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {-1423+1461 x+1107 x^2-33075 x^3-40500 x^4}{3645 (2+3 x)^7} \]

[In]

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

[Out]

(-1423 + 1461*x + 1107*x^2 - 33075*x^3 - 40500*x^4)/(3645*(2 + 3*x)^7)

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.52

method	result	size
norman	\(\frac {-\frac {100}{9} x^{4}-\frac {245}{27} x^{3}+\frac {41}{135} x^{2}+\frac {487}{1215} x -\frac {1423}{3645}}{\left (2+3 x \right )^{7}}\)	\(29\)
gosper	\(-\frac {40500 x^{4}+33075 x^{3}-1107 x^{2}-1461 x +1423}{3645 \left (2+3 x \right )^{7}}\)	\(30\)
risch	\(\frac {-\frac {100}{9} x^{4}-\frac {245}{27} x^{3}+\frac {41}{135} x^{2}+\frac {487}{1215} x -\frac {1423}{3645}}{\left (2+3 x \right )^{7}}\)	\(30\)
parallelrisch	\(\frac {12807 x^{7}+59766 x^{6}+119532 x^{5}+111480 x^{4}+71120 x^{3}+36000 x^{2}+8640 x}{1920 \left (2+3 x \right )^{7}}\)	\(44\)
default	\(-\frac {7}{243 \left (2+3 x \right )^{7}}+\frac {259}{729 \left (2+3 x \right )^{6}}-\frac {503}{405 \left (2+3 x \right )^{5}}+\frac {185}{243 \left (2+3 x \right )^{4}}-\frac {100}{729 \left (2+3 x \right )^{3}}\)	\(47\)
meijerg	\(\frac {9 x \left (\frac {729}{64} x^{6}+\frac {1701}{32} x^{5}+\frac {1701}{16} x^{4}+\frac {945}{8} x^{3}+\frac {315}{4} x^{2}+\frac {63}{2} x +7\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {x^{2} \left (\frac {243}{32} x^{5}+\frac {567}{16} x^{4}+\frac {567}{8} x^{3}+\frac {315}{4} x^{2}+\frac {105}{2} x +21\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}-\frac {59 x^{3} \left (\frac {81}{16} x^{4}+\frac {189}{8} x^{3}+\frac {189}{4} x^{2}+\frac {105}{2} x +35\right )}{26880 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {x^{4} \left (\frac {27}{8} x^{3}+\frac {63}{4} x^{2}+\frac {63}{2} x +35\right )}{1792 \left (1+\frac {3 x}{2}\right )^{7}}+\frac {5 x^{5} \left (\frac {9}{4} x^{2}+\frac {21}{2} x +21\right )}{1344 \left (1+\frac {3 x}{2}\right )^{7}}\)	\(160\)

[In]

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

[Out]

(-100/9*x^4-245/27*x^3+41/135*x^2+487/1215*x-1423/3645)/(2+3*x)^7

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

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

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(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 = 54, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {- 40500 x^{4} - 33075 x^{3} + 1107 x^{2} + 1461 x - 1423}{7971615 x^{7} + 37200870 x^{6} + 74401740 x^{5} + 82668600 x^{4} + 55112400 x^{3} + 22044960 x^{2} + 4898880 x + 466560} \]

[In]

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

[Out]

(-40500*x**4 - 33075*x**3 + 1107*x**2 + 1461*x - 1423)/(7971615*x**7 + 37200870*x**6 + 74401740*x**5 + 8266860
0*x**4 + 55112400*x**3 + 22044960*x**2 + 4898880*x + 466560)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

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

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(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 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (3 \, x + 2\right )}^{7}} \]

[In]

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

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(3*x + 2)^7

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {185}{243\,{\left (3\,x+2\right )}^4}-\frac {100}{729\,{\left (3\,x+2\right )}^3}-\frac {503}{405\,{\left (3\,x+2\right )}^5}+\frac {259}{729\,{\left (3\,x+2\right )}^6}-\frac {7}{243\,{\left (3\,x+2\right )}^7} \]

[In]

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

[Out]

185/(243*(3*x + 2)^4) - 100/(729*(3*x + 2)^3) - 503/(405*(3*x + 2)^5) + 259/(729*(3*x + 2)^6) - 7/(243*(3*x +
2)^7)