\(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx\) [1370]

Optimal result

Integrand size = 22, antiderivative size = 67 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {343}{4374 (2+3 x)^6}+\frac {3724}{3645 (2+3 x)^5}-\frac {11599}{2916 (2+3 x)^4}+\frac {8198}{2187 (2+3 x)^3}-\frac {1090}{729 (2+3 x)^2}+\frac {200}{729 (2+3 x)} \]

[Out]

-343/4374/(2+3*x)^6+3724/3645/(2+3*x)^5-11599/2916/(2+3*x)^4+8198/2187/(2+3*x)^3-1090/729/(2+3*x)^2+200/729/(2
+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, 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} \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {200}{729 (3 x+2)}-\frac {1090}{729 (3 x+2)^2}+\frac {8198}{2187 (3 x+2)^3}-\frac {11599}{2916 (3 x+2)^4}+\frac {3724}{3645 (3 x+2)^5}-\frac {343}{4374 (3 x+2)^6} \]

[In]

Int[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-343/(4374*(2 + 3*x)^6) + 3724/(3645*(2 + 3*x)^5) - 11599/(2916*(2 + 3*x)^4) + 8198/(2187*(2 + 3*x)^3) - 1090/
(729*(2 + 3*x)^2) + 200/(729*(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}{243 (2+3 x)^7}-\frac {3724}{243 (2+3 x)^6}+\frac {11599}{243 (2+3 x)^5}-\frac {8198}{243 (2+3 x)^4}+\frac {2180}{243 (2+3 x)^3}-\frac {200}{243 (2+3 x)^2}\right ) \, dx \\ & = -\frac {343}{4374 (2+3 x)^6}+\frac {3724}{3645 (2+3 x)^5}-\frac {11599}{2916 (2+3 x)^4}+\frac {8198}{2187 (2+3 x)^3}-\frac {1090}{729 (2+3 x)^2}+\frac {200}{729 (2+3 x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {39286+550404 x+1801575 x^2+3260520 x^3+4422600 x^4+2916000 x^5}{43740 (2+3 x)^6} \]

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

(39286 + 550404*x + 1801575*x^2 + 3260520*x^3 + 4422600*x^4 + 2916000*x^5)/(43740*(2 + 3*x)^6)

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.51

method	result	size
norman	\(\frac {\frac {200}{3} x^{5}+\frac {910}{9} x^{4}+\frac {6038}{81} x^{3}+\frac {13345}{324} x^{2}+\frac {15289}{1215} x +\frac {19643}{21870}}{\left (2+3 x \right )^{6}}\)	\(34\)
gosper	\(\frac {2916000 x^{5}+4422600 x^{4}+3260520 x^{3}+1801575 x^{2}+550404 x +39286}{43740 \left (2+3 x \right )^{6}}\)	\(35\)
risch	\(\frac {\frac {200}{3} x^{5}+\frac {910}{9} x^{4}+\frac {6038}{81} x^{3}+\frac {13345}{324} x^{2}+\frac {15289}{1215} x +\frac {19643}{21870}}{\left (2+3 x \right )^{6}}\)	\(35\)
parallelrisch	\(\frac {-19643 x^{6}+49428 x^{5}+63180 x^{4}+26720 x^{3}+20880 x^{2}+8640 x}{1920 \left (2+3 x \right )^{6}}\)	\(39\)
default	\(-\frac {343}{4374 \left (2+3 x \right )^{6}}+\frac {3724}{3645 \left (2+3 x \right )^{5}}-\frac {11599}{2916 \left (2+3 x \right )^{4}}+\frac {8198}{2187 \left (2+3 x \right )^{3}}-\frac {1090}{729 \left (2+3 x \right )^{2}}+\frac {200}{729 \left (2+3 x \right )}\)	\(56\)
meijerg	\(\frac {3 x \left (\frac {243}{32} x^{5}+\frac {243}{8} x^{4}+\frac {405}{8} x^{3}+45 x^{2}+\frac {45}{2} x +6\right )}{256 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {x^{2} \left (\frac {81}{16} x^{4}+\frac {81}{4} x^{3}+\frac {135}{4} x^{2}+30 x +15\right )}{160 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {47 x^{3} \left (\frac {27}{8} x^{3}+\frac {27}{2} x^{2}+\frac {45}{2} x +20\right )}{7680 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {23 x^{4} \left (\frac {9}{4} x^{2}+9 x +15\right )}{1280 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {x^{5} \left (\frac {3 x}{2}+6\right )}{64 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {25 x^{6}}{96 \left (1+\frac {3 x}{2}\right )^{6}}\)	\(147\)

[In]

int((1-2*x)^3*(3+5*x)^2/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

(200/3*x^5+910/9*x^4+6038/81*x^3+13345/324*x^2+15289/1215*x+19643/21870)/(2+3*x)^6

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{43740 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=- \frac {- 2916000 x^{5} - 4422600 x^{4} - 3260520 x^{3} - 1801575 x^{2} - 550404 x - 39286}{31886460 x^{6} + 127545840 x^{5} + 212576400 x^{4} + 188956800 x^{3} + 94478400 x^{2} + 25194240 x + 2799360} \]

[In]

integrate((1-2*x)**3*(3+5*x)**2/(2+3*x)**7,x)

[Out]

-(-2916000*x**5 - 4422600*x**4 - 3260520*x**3 - 1801575*x**2 - 550404*x - 39286)/(31886460*x**6 + 127545840*x*
*5 + 212576400*x**4 + 188956800*x**3 + 94478400*x**2 + 25194240*x + 2799360)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{43740 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {2916000 \, x^{5} + 4422600 \, x^{4} + 3260520 \, x^{3} + 1801575 \, x^{2} + 550404 \, x + 39286}{43740 \, {\left (3 \, x + 2\right )}^{6}} \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^7,x, algorithm="giac")

[Out]

1/43740*(2916000*x^5 + 4422600*x^4 + 3260520*x^3 + 1801575*x^2 + 550404*x + 39286)/(3*x + 2)^6

Mupad [B] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {200}{729\,\left (3\,x+2\right )}-\frac {1090}{729\,{\left (3\,x+2\right )}^2}+\frac {8198}{2187\,{\left (3\,x+2\right )}^3}-\frac {11599}{2916\,{\left (3\,x+2\right )}^4}+\frac {3724}{3645\,{\left (3\,x+2\right )}^5}-\frac {343}{4374\,{\left (3\,x+2\right )}^6} \]

[In]

int(-((2*x - 1)^3*(5*x + 3)^2)/(3*x + 2)^7,x)

[Out]

200/(729*(3*x + 2)) - 1090/(729*(3*x + 2)^2) + 8198/(2187*(3*x + 2)^3) - 11599/(2916*(3*x + 2)^4) + 3724/(3645
*(3*x + 2)^5) - 343/(4374*(3*x + 2)^6)