\(\int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx\) [1238]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=-\frac {7}{81} (2+3 x)^7+\frac {91}{216} (2+3 x)^8-\frac {16}{81} (2+3 x)^9+\frac {2}{81} (2+3 x)^{10} \]

[Out]

-7/81*(2+3*x)^7+91/216*(2+3*x)^8-16/81*(2+3*x)^9+2/81*(2+3*x)^10

Rubi [A] (verified)

Time = 0.01 (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} \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=\frac {2}{81} (3 x+2)^{10}-\frac {16}{81} (3 x+2)^9+\frac {91}{216} (3 x+2)^8-\frac {7}{81} (3 x+2)^7 \]

[In]

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

[Out]

(-7*(2 + 3*x)^7)/81 + (91*(2 + 3*x)^8)/216 - (16*(2 + 3*x)^9)/81 + (2*(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 {49}{27} (2+3 x)^6+\frac {91}{9} (2+3 x)^7-\frac {16}{3} (2+3 x)^8+\frac {20}{27} (2+3 x)^9\right ) \, dx \\ & = -\frac {7}{81} (2+3 x)^7+\frac {91}{216} (2+3 x)^8-\frac {16}{81} (2+3 x)^9+\frac {2}{81} (2+3 x)^{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=192 x+640 x^2+\frac {1936 x^3}{3}-1372 x^4-4284 x^5-2772 x^6+4185 x^7+\frac {68769 x^8}{8}+5832 x^9+1458 x^{10} \]

[In]

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

[Out]

192*x + 640*x^2 + (1936*x^3)/3 - 1372*x^4 - 4284*x^5 - 2772*x^6 + 4185*x^7 + (68769*x^8)/8 + 5832*x^9 + 1458*x
^10

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09

method	result	size
gosper	\(\frac {x \left (34992 x^{9}+139968 x^{8}+206307 x^{7}+100440 x^{6}-66528 x^{5}-102816 x^{4}-32928 x^{3}+15488 x^{2}+15360 x +4608\right )}{24}\)	\(49\)
default	\(1458 x^{10}+5832 x^{9}+\frac {68769}{8} x^{8}+4185 x^{7}-2772 x^{6}-4284 x^{5}-1372 x^{4}+\frac {1936}{3} x^{3}+640 x^{2}+192 x\)	\(50\)
norman	\(1458 x^{10}+5832 x^{9}+\frac {68769}{8} x^{8}+4185 x^{7}-2772 x^{6}-4284 x^{5}-1372 x^{4}+\frac {1936}{3} x^{3}+640 x^{2}+192 x\)	\(50\)
risch	\(1458 x^{10}+5832 x^{9}+\frac {68769}{8} x^{8}+4185 x^{7}-2772 x^{6}-4284 x^{5}-1372 x^{4}+\frac {1936}{3} x^{3}+640 x^{2}+192 x\)	\(50\)
parallelrisch	\(1458 x^{10}+5832 x^{9}+\frac {68769}{8} x^{8}+4185 x^{7}-2772 x^{6}-4284 x^{5}-1372 x^{4}+\frac {1936}{3} x^{3}+640 x^{2}+192 x\)	\(50\)

[In]

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

[Out]

1/24*x*(34992*x^9+139968*x^8+206307*x^7+100440*x^6-66528*x^5-102816*x^4-32928*x^3+15488*x^2+15360*x+4608)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=1458 \, x^{10} + 5832 \, x^{9} + \frac {68769}{8} \, x^{8} + 4185 \, x^{7} - 2772 \, x^{6} - 4284 \, x^{5} - 1372 \, x^{4} + \frac {1936}{3} \, x^{3} + 640 \, x^{2} + 192 \, x \]

[In]

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

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=1458 x^{10} + 5832 x^{9} + \frac {68769 x^{8}}{8} + 4185 x^{7} - 2772 x^{6} - 4284 x^{5} - 1372 x^{4} + \frac {1936 x^{3}}{3} + 640 x^{2} + 192 x \]

[In]

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

[Out]

1458*x**10 + 5832*x**9 + 68769*x**8/8 + 4185*x**7 - 2772*x**6 - 4284*x**5 - 1372*x**4 + 1936*x**3/3 + 640*x**2
+ 192*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=1458 \, x^{10} + 5832 \, x^{9} + \frac {68769}{8} \, x^{8} + 4185 \, x^{7} - 2772 \, x^{6} - 4284 \, x^{5} - 1372 \, x^{4} + \frac {1936}{3} \, x^{3} + 640 \, x^{2} + 192 \, x \]

[In]

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

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=1458 \, x^{10} + 5832 \, x^{9} + \frac {68769}{8} \, x^{8} + 4185 \, x^{7} - 2772 \, x^{6} - 4284 \, x^{5} - 1372 \, x^{4} + \frac {1936}{3} \, x^{3} + 640 \, x^{2} + 192 \, x \]

[In]

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

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx=1458\,x^{10}+5832\,x^9+\frac {68769\,x^8}{8}+4185\,x^7-2772\,x^6-4284\,x^5-1372\,x^4+\frac {1936\,x^3}{3}+640\,x^2+192\,x \]

[In]

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

[Out]

192*x + 640*x^2 + (1936*x^3)/3 - 1372*x^4 - 4284*x^5 - 2772*x^6 + 4185*x^7 + (68769*x^8)/8 + 5832*x^9 + 1458*x
^10