\(\int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx\) [1160]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=\frac {1}{81} (2+3 x)^7-\frac {1}{9} (2+3 x)^8+\frac {65}{243} (2+3 x)^9-\frac {5}{81} (2+3 x)^{10} \]

[Out]

1/81*(2+3*x)^7-1/9*(2+3*x)^8+65/243*(2+3*x)^9-5/81*(2+3*x)^10

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} \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=-\frac {5}{81} (3 x+2)^{10}+\frac {65}{243} (3 x+2)^9-\frac {1}{9} (3 x+2)^8+\frac {1}{81} (3 x+2)^7 \]

[In]

Int[(1 - 2*x)*(2 + 3*x)^6*(3 + 5*x)^2,x]

[Out]

(2 + 3*x)^7/81 - (2 + 3*x)^8/9 + (65*(2 + 3*x)^9)/243 - (5*(2 + 3*x)^10)/81

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)^6-\frac {8}{3} (2+3 x)^7+\frac {65}{9} (2+3 x)^8-\frac {50}{27} (2+3 x)^9\right ) \, dx \\ & = \frac {1}{81} (2+3 x)^7-\frac {1}{9} (2+3 x)^8+\frac {65}{243} (2+3 x)^9-\frac {5}{81} (2+3 x)^{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=576 x+2976 x^2+\frac {24112 x^3}{3}+10360 x^4-1764 x^5-29106 x^6-49221 x^7-42039 x^8-19035 x^9-3645 x^{10} \]

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^6*(3 + 5*x)^2,x]

[Out]

576*x + 2976*x^2 + (24112*x^3)/3 + 10360*x^4 - 1764*x^5 - 29106*x^6 - 49221*x^7 - 42039*x^8 - 19035*x^9 - 3645
*x^10

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {x \left (10935 x^{9}+57105 x^{8}+126117 x^{7}+147663 x^{6}+87318 x^{5}+5292 x^{4}-31080 x^{3}-24112 x^{2}-8928 x -1728\right )}{3}\)	\(49\)
default	\(-3645 x^{10}-19035 x^{9}-42039 x^{8}-49221 x^{7}-29106 x^{6}-1764 x^{5}+10360 x^{4}+\frac {24112}{3} x^{3}+2976 x^{2}+576 x\)	\(50\)
norman	\(-3645 x^{10}-19035 x^{9}-42039 x^{8}-49221 x^{7}-29106 x^{6}-1764 x^{5}+10360 x^{4}+\frac {24112}{3} x^{3}+2976 x^{2}+576 x\)	\(50\)
risch	\(-3645 x^{10}-19035 x^{9}-42039 x^{8}-49221 x^{7}-29106 x^{6}-1764 x^{5}+10360 x^{4}+\frac {24112}{3} x^{3}+2976 x^{2}+576 x\)	\(50\)
parallelrisch	\(-3645 x^{10}-19035 x^{9}-42039 x^{8}-49221 x^{7}-29106 x^{6}-1764 x^{5}+10360 x^{4}+\frac {24112}{3} x^{3}+2976 x^{2}+576 x\)	\(50\)

[In]

int((1-2*x)*(2+3*x)^6*(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/3*x*(10935*x^9+57105*x^8+126117*x^7+147663*x^6+87318*x^5+5292*x^4-31080*x^3-24112*x^2-8928*x-1728)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=-3645 \, x^{10} - 19035 \, x^{9} - 42039 \, x^{8} - 49221 \, x^{7} - 29106 \, x^{6} - 1764 \, x^{5} + 10360 \, x^{4} + \frac {24112}{3} \, x^{3} + 2976 \, x^{2} + 576 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x)^2,x, algorithm="fricas")

[Out]

-3645*x^10 - 19035*x^9 - 42039*x^8 - 49221*x^7 - 29106*x^6 - 1764*x^5 + 10360*x^4 + 24112/3*x^3 + 2976*x^2 + 5
76*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=- 3645 x^{10} - 19035 x^{9} - 42039 x^{8} - 49221 x^{7} - 29106 x^{6} - 1764 x^{5} + 10360 x^{4} + \frac {24112 x^{3}}{3} + 2976 x^{2} + 576 x \]

[In]

integrate((1-2*x)*(2+3*x)**6*(3+5*x)**2,x)

[Out]

-3645*x**10 - 19035*x**9 - 42039*x**8 - 49221*x**7 - 29106*x**6 - 1764*x**5 + 10360*x**4 + 24112*x**3/3 + 2976
*x**2 + 576*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=-3645 \, x^{10} - 19035 \, x^{9} - 42039 \, x^{8} - 49221 \, x^{7} - 29106 \, x^{6} - 1764 \, x^{5} + 10360 \, x^{4} + \frac {24112}{3} \, x^{3} + 2976 \, x^{2} + 576 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x)^2,x, algorithm="maxima")

[Out]

-3645*x^10 - 19035*x^9 - 42039*x^8 - 49221*x^7 - 29106*x^6 - 1764*x^5 + 10360*x^4 + 24112/3*x^3 + 2976*x^2 + 5
76*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=-3645 \, x^{10} - 19035 \, x^{9} - 42039 \, x^{8} - 49221 \, x^{7} - 29106 \, x^{6} - 1764 \, x^{5} + 10360 \, x^{4} + \frac {24112}{3} \, x^{3} + 2976 \, x^{2} + 576 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x)^2,x, algorithm="giac")

[Out]

-3645*x^10 - 19035*x^9 - 42039*x^8 - 49221*x^7 - 29106*x^6 - 1764*x^5 + 10360*x^4 + 24112/3*x^3 + 2976*x^2 + 5
76*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^6 (3+5 x)^2 \, dx=-3645\,x^{10}-19035\,x^9-42039\,x^8-49221\,x^7-29106\,x^6-1764\,x^5+10360\,x^4+\frac {24112\,x^3}{3}+2976\,x^2+576\,x \]

[In]

int(-(2*x - 1)*(3*x + 2)^6*(5*x + 3)^2,x)

[Out]

576*x + 2976*x^2 + (24112*x^3)/3 + 10360*x^4 - 1764*x^5 - 29106*x^6 - 49221*x^7 - 42039*x^8 - 19035*x^9 - 3645
*x^10