\(\int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx\) [1279]

Optimal result

Integrand size = 20, antiderivative size = 42 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=54 x+\frac {135 x^2}{2}-111 x^3-\frac {1091 x^4}{4}+19 x^5+\frac {1100 x^6}{3}+\frac {1500 x^7}{7} \]

[Out]

54*x+135/2*x^2-111*x^3-1091/4*x^4+19*x^5+1100/3*x^6+1500/7*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, 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} \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500 x^7}{7}+\frac {1100 x^6}{3}+19 x^5-\frac {1091 x^4}{4}-111 x^3+\frac {135 x^2}{2}+54 x \]

[In]

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

[Out]

54*x + (135*x^2)/2 - 111*x^3 - (1091*x^4)/4 + 19*x^5 + (1100*x^6)/3 + (1500*x^7)/7

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 (54+135 x-333 x^2-1091 x^3+95 x^4+2200 x^5+1500 x^6\right ) \, dx \\ & = 54 x+\frac {135 x^2}{2}-111 x^3-\frac {1091 x^4}{4}+19 x^5+\frac {1100 x^6}{3}+\frac {1500 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=54 x+\frac {135 x^2}{2}-111 x^3-\frac {1091 x^4}{4}+19 x^5+\frac {1100 x^6}{3}+\frac {1500 x^7}{7} \]

[In]

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

[Out]

54*x + (135*x^2)/2 - 111*x^3 - (1091*x^4)/4 + 19*x^5 + (1100*x^6)/3 + (1500*x^7)/7

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {x \left (18000 x^{6}+30800 x^{5}+1596 x^{4}-22911 x^{3}-9324 x^{2}+5670 x +4536\right )}{84}\)	\(34\)
default	\(54 x +\frac {135}{2} x^{2}-111 x^{3}-\frac {1091}{4} x^{4}+19 x^{5}+\frac {1100}{3} x^{6}+\frac {1500}{7} x^{7}\)	\(35\)
norman	\(54 x +\frac {135}{2} x^{2}-111 x^{3}-\frac {1091}{4} x^{4}+19 x^{5}+\frac {1100}{3} x^{6}+\frac {1500}{7} x^{7}\)	\(35\)
risch	\(54 x +\frac {135}{2} x^{2}-111 x^{3}-\frac {1091}{4} x^{4}+19 x^{5}+\frac {1100}{3} x^{6}+\frac {1500}{7} x^{7}\)	\(35\)
parallelrisch	\(54 x +\frac {135}{2} x^{2}-111 x^{3}-\frac {1091}{4} x^{4}+19 x^{5}+\frac {1100}{3} x^{6}+\frac {1500}{7} x^{7}\)	\(35\)

[In]

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

[Out]

1/84*x*(18000*x^6+30800*x^5+1596*x^4-22911*x^3-9324*x^2+5670*x+4536)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500}{7} \, x^{7} + \frac {1100}{3} \, x^{6} + 19 \, x^{5} - \frac {1091}{4} \, x^{4} - 111 \, x^{3} + \frac {135}{2} \, x^{2} + 54 \, x \]

[In]

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

[Out]

1500/7*x^7 + 1100/3*x^6 + 19*x^5 - 1091/4*x^4 - 111*x^3 + 135/2*x^2 + 54*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500 x^{7}}{7} + \frac {1100 x^{6}}{3} + 19 x^{5} - \frac {1091 x^{4}}{4} - 111 x^{3} + \frac {135 x^{2}}{2} + 54 x \]

[In]

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

[Out]

1500*x**7/7 + 1100*x**6/3 + 19*x**5 - 1091*x**4/4 - 111*x**3 + 135*x**2/2 + 54*x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500}{7} \, x^{7} + \frac {1100}{3} \, x^{6} + 19 \, x^{5} - \frac {1091}{4} \, x^{4} - 111 \, x^{3} + \frac {135}{2} \, x^{2} + 54 \, x \]

[In]

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

[Out]

1500/7*x^7 + 1100/3*x^6 + 19*x^5 - 1091/4*x^4 - 111*x^3 + 135/2*x^2 + 54*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500}{7} \, x^{7} + \frac {1100}{3} \, x^{6} + 19 \, x^{5} - \frac {1091}{4} \, x^{4} - 111 \, x^{3} + \frac {135}{2} \, x^{2} + 54 \, x \]

[In]

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

[Out]

1500/7*x^7 + 1100/3*x^6 + 19*x^5 - 1091/4*x^4 - 111*x^3 + 135/2*x^2 + 54*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int (1-2 x)^2 (2+3 x) (3+5 x)^3 \, dx=\frac {1500\,x^7}{7}+\frac {1100\,x^6}{3}+19\,x^5-\frac {1091\,x^4}{4}-111\,x^3+\frac {135\,x^2}{2}+54\,x \]

[In]

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

[Out]

54*x + (135*x^2)/2 - 111*x^3 - (1091*x^4)/4 + 19*x^5 + (1100*x^6)/3 + (1500*x^7)/7