\(\int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx\) [1159]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=\frac {7}{648} (2+3 x)^8-\frac {8}{81} (2+3 x)^9+\frac {13}{54} (2+3 x)^{10}-\frac {50}{891} (2+3 x)^{11} \]

[Out]

7/648*(2+3*x)^8-8/81*(2+3*x)^9+13/54*(2+3*x)^10-50/891*(2+3*x)^11

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, 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} \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=-\frac {50}{891} (3 x+2)^{11}+\frac {13}{54} (3 x+2)^{10}-\frac {8}{81} (3 x+2)^9+\frac {7}{648} (3 x+2)^8 \]

[In]

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

[Out]

(7*(2 + 3*x)^8)/648 - (8*(2 + 3*x)^9)/81 + (13*(2 + 3*x)^10)/54 - (50*(2 + 3*x)^11)/891

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}{27} (2+3 x)^7-\frac {8}{3} (2+3 x)^8+\frac {65}{9} (2+3 x)^9-\frac {50}{27} (2+3 x)^{10}\right ) \, dx \\ & = \frac {7}{648} (2+3 x)^8-\frac {8}{81} (2+3 x)^9+\frac {13}{54} (2+3 x)^{10}-\frac {50}{891} (2+3 x)^{11} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.38 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=1152 x+6816 x^2+\frac {66080 x^3}{3}+38804 x^4+21336 x^5-62622 x^6-173286 x^7-\frac {1706265 x^8}{8}-150174 x^9-\frac {117369 x^{10}}{2}-\frac {109350 x^{11}}{11} \]

[In]

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

[Out]

1152*x + 6816*x^2 + (66080*x^3)/3 + 38804*x^4 + 21336*x^5 - 62622*x^6 - 173286*x^7 - (1706265*x^8)/8 - 150174*
x^9 - (117369*x^10)/2 - (109350*x^11)/11

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20

method	result	size
gosper	\(-\frac {x \left (2624400 x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-5632704 x^{4}-10244256 x^{3}-5815040 x^{2}-1799424 x -304128\right )}{264}\)	\(54\)
default	\(-\frac {109350}{11} x^{11}-\frac {117369}{2} x^{10}-150174 x^{9}-\frac {1706265}{8} x^{8}-173286 x^{7}-62622 x^{6}+21336 x^{5}+38804 x^{4}+\frac {66080}{3} x^{3}+6816 x^{2}+1152 x\)	\(55\)
norman	\(-\frac {109350}{11} x^{11}-\frac {117369}{2} x^{10}-150174 x^{9}-\frac {1706265}{8} x^{8}-173286 x^{7}-62622 x^{6}+21336 x^{5}+38804 x^{4}+\frac {66080}{3} x^{3}+6816 x^{2}+1152 x\)	\(55\)
risch	\(-\frac {109350}{11} x^{11}-\frac {117369}{2} x^{10}-150174 x^{9}-\frac {1706265}{8} x^{8}-173286 x^{7}-62622 x^{6}+21336 x^{5}+38804 x^{4}+\frac {66080}{3} x^{3}+6816 x^{2}+1152 x\)	\(55\)
parallelrisch	\(-\frac {109350}{11} x^{11}-\frac {117369}{2} x^{10}-150174 x^{9}-\frac {1706265}{8} x^{8}-173286 x^{7}-62622 x^{6}+21336 x^{5}+38804 x^{4}+\frac {66080}{3} x^{3}+6816 x^{2}+1152 x\)	\(55\)

[In]

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

[Out]

-1/264*x*(2624400*x^10+15492708*x^9+39645936*x^8+56306745*x^7+45747504*x^6+16532208*x^5-5632704*x^4-10244256*x
^3-5815040*x^2-1799424*x-304128)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=-\frac {109350}{11} \, x^{11} - \frac {117369}{2} \, x^{10} - 150174 \, x^{9} - \frac {1706265}{8} \, x^{8} - 173286 \, x^{7} - 62622 \, x^{6} + 21336 \, x^{5} + 38804 \, x^{4} + \frac {66080}{3} \, x^{3} + 6816 \, x^{2} + 1152 \, x \]

[In]

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

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.33 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=- \frac {109350 x^{11}}{11} - \frac {117369 x^{10}}{2} - 150174 x^{9} - \frac {1706265 x^{8}}{8} - 173286 x^{7} - 62622 x^{6} + 21336 x^{5} + 38804 x^{4} + \frac {66080 x^{3}}{3} + 6816 x^{2} + 1152 x \]

[In]

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

[Out]

-109350*x**11/11 - 117369*x**10/2 - 150174*x**9 - 1706265*x**8/8 - 173286*x**7 - 62622*x**6 + 21336*x**5 + 388
04*x**4 + 66080*x**3/3 + 6816*x**2 + 1152*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=-\frac {109350}{11} \, x^{11} - \frac {117369}{2} \, x^{10} - 150174 \, x^{9} - \frac {1706265}{8} \, x^{8} - 173286 \, x^{7} - 62622 \, x^{6} + 21336 \, x^{5} + 38804 \, x^{4} + \frac {66080}{3} \, x^{3} + 6816 \, x^{2} + 1152 \, x \]

[In]

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

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=-\frac {109350}{11} \, x^{11} - \frac {117369}{2} \, x^{10} - 150174 \, x^{9} - \frac {1706265}{8} \, x^{8} - 173286 \, x^{7} - 62622 \, x^{6} + 21336 \, x^{5} + 38804 \, x^{4} + \frac {66080}{3} \, x^{3} + 6816 \, x^{2} + 1152 \, x \]

[In]

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

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx=-\frac {109350\,x^{11}}{11}-\frac {117369\,x^{10}}{2}-150174\,x^9-\frac {1706265\,x^8}{8}-173286\,x^7-62622\,x^6+21336\,x^5+38804\,x^4+\frac {66080\,x^3}{3}+6816\,x^2+1152\,x \]

[In]

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

[Out]

1152*x + 6816*x^2 + (66080*x^3)/3 + 38804*x^4 + 21336*x^5 - 62622*x^6 - 173286*x^7 - (1706265*x^8)/8 - 150174*
x^9 - (117369*x^10)/2 - (109350*x^11)/11