\(\int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx\) [1180]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=216 x+810 x^2+1338 x^3+\frac {883 x^4}{4}-\frac {13943 x^5}{5}-\frac {9255 x^6}{2}-\frac {22275 x^7}{7}-\frac {3375 x^8}{4} \]

[Out]

216*x+810*x^2+1338*x^3+883/4*x^4-13943/5*x^5-9255/2*x^6-22275/7*x^7-3375/4*x^8

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {3375 x^8}{4}-\frac {22275 x^7}{7}-\frac {9255 x^6}{2}-\frac {13943 x^5}{5}+\frac {883 x^4}{4}+1338 x^3+810 x^2+216 x \]

[In]

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

[Out]

216*x + 810*x^2 + 1338*x^3 + (883*x^4)/4 - (13943*x^5)/5 - (9255*x^6)/2 - (22275*x^7)/7 - (3375*x^8)/4

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 (216+1620 x+4014 x^2+883 x^3-13943 x^4-27765 x^5-22275 x^6-6750 x^7\right ) \, dx \\ & = 216 x+810 x^2+1338 x^3+\frac {883 x^4}{4}-\frac {13943 x^5}{5}-\frac {9255 x^6}{2}-\frac {22275 x^7}{7}-\frac {3375 x^8}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=216 x+810 x^2+1338 x^3+\frac {883 x^4}{4}-\frac {13943 x^5}{5}-\frac {9255 x^6}{2}-\frac {22275 x^7}{7}-\frac {3375 x^8}{4} \]

[In]

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

[Out]

216*x + 810*x^2 + 1338*x^3 + (883*x^4)/4 - (13943*x^5)/5 - (9255*x^6)/2 - (22275*x^7)/7 - (3375*x^8)/4

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80

method	result	size
gosper	\(-\frac {x \left (118125 x^{7}+445500 x^{6}+647850 x^{5}+390404 x^{4}-30905 x^{3}-187320 x^{2}-113400 x -30240\right )}{140}\)	\(39\)
default	\(216 x +810 x^{2}+1338 x^{3}+\frac {883}{4} x^{4}-\frac {13943}{5} x^{5}-\frac {9255}{2} x^{6}-\frac {22275}{7} x^{7}-\frac {3375}{4} x^{8}\)	\(40\)
norman	\(216 x +810 x^{2}+1338 x^{3}+\frac {883}{4} x^{4}-\frac {13943}{5} x^{5}-\frac {9255}{2} x^{6}-\frac {22275}{7} x^{7}-\frac {3375}{4} x^{8}\)	\(40\)
risch	\(216 x +810 x^{2}+1338 x^{3}+\frac {883}{4} x^{4}-\frac {13943}{5} x^{5}-\frac {9255}{2} x^{6}-\frac {22275}{7} x^{7}-\frac {3375}{4} x^{8}\)	\(40\)
parallelrisch	\(216 x +810 x^{2}+1338 x^{3}+\frac {883}{4} x^{4}-\frac {13943}{5} x^{5}-\frac {9255}{2} x^{6}-\frac {22275}{7} x^{7}-\frac {3375}{4} x^{8}\)	\(40\)

[In]

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

[Out]

-1/140*x*(118125*x^7+445500*x^6+647850*x^5+390404*x^4-30905*x^3-187320*x^2-113400*x-30240)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {3375}{4} \, x^{8} - \frac {22275}{7} \, x^{7} - \frac {9255}{2} \, x^{6} - \frac {13943}{5} \, x^{5} + \frac {883}{4} \, x^{4} + 1338 \, x^{3} + 810 \, x^{2} + 216 \, x \]

[In]

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

[Out]

-3375/4*x^8 - 22275/7*x^7 - 9255/2*x^6 - 13943/5*x^5 + 883/4*x^4 + 1338*x^3 + 810*x^2 + 216*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=- \frac {3375 x^{8}}{4} - \frac {22275 x^{7}}{7} - \frac {9255 x^{6}}{2} - \frac {13943 x^{5}}{5} + \frac {883 x^{4}}{4} + 1338 x^{3} + 810 x^{2} + 216 x \]

[In]

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

[Out]

-3375*x**8/4 - 22275*x**7/7 - 9255*x**6/2 - 13943*x**5/5 + 883*x**4/4 + 1338*x**3 + 810*x**2 + 216*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {3375}{4} \, x^{8} - \frac {22275}{7} \, x^{7} - \frac {9255}{2} \, x^{6} - \frac {13943}{5} \, x^{5} + \frac {883}{4} \, x^{4} + 1338 \, x^{3} + 810 \, x^{2} + 216 \, x \]

[In]

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

[Out]

-3375/4*x^8 - 22275/7*x^7 - 9255/2*x^6 - 13943/5*x^5 + 883/4*x^4 + 1338*x^3 + 810*x^2 + 216*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {3375}{4} \, x^{8} - \frac {22275}{7} \, x^{7} - \frac {9255}{2} \, x^{6} - \frac {13943}{5} \, x^{5} + \frac {883}{4} \, x^{4} + 1338 \, x^{3} + 810 \, x^{2} + 216 \, x \]

[In]

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

[Out]

-3375/4*x^8 - 22275/7*x^7 - 9255/2*x^6 - 13943/5*x^5 + 883/4*x^4 + 1338*x^3 + 810*x^2 + 216*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {3375\,x^8}{4}-\frac {22275\,x^7}{7}-\frac {9255\,x^6}{2}-\frac {13943\,x^5}{5}+\frac {883\,x^4}{4}+1338\,x^3+810\,x^2+216\,x \]

[In]

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

[Out]

216*x + 810*x^2 + 1338*x^3 + (883*x^4)/4 - (13943*x^5)/5 - (9255*x^6)/2 - (22275*x^7)/7 - (3375*x^8)/4