\(\int (1-2 x)^3 (2+3 x) (3+5 x) \, dx\) [1345]

Optimal result

Integrand size = 18, antiderivative size = 34 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-\frac {77}{32} (1-2 x)^4+\frac {17}{10} (1-2 x)^5-\frac {5}{16} (1-2 x)^6 \]

[Out]

-77/32*(1-2*x)^4+17/10*(1-2*x)^5-5/16*(1-2*x)^6

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-\frac {5}{16} (1-2 x)^6+\frac {17}{10} (1-2 x)^5-\frac {77}{32} (1-2 x)^4 \]

[In]

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

[Out]

(-77*(1 - 2*x)^4)/32 + (17*(1 - 2*x)^5)/10 - (5*(1 - 2*x)^6)/16

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 {77}{4} (1-2 x)^3-17 (1-2 x)^4+\frac {15}{4} (1-2 x)^5\right ) \, dx \\ & = -\frac {77}{32} (1-2 x)^4+\frac {17}{10} (1-2 x)^5-\frac {5}{16} (1-2 x)^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=6 x-\frac {17 x^2}{2}-9 x^3+\frac {45 x^4}{2}+\frac {28 x^5}{5}-20 x^6 \]

[In]

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

[Out]

6*x - (17*x^2)/2 - 9*x^3 + (45*x^4)/2 + (28*x^5)/5 - 20*x^6

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {x \left (200 x^{5}-56 x^{4}-225 x^{3}+90 x^{2}+85 x -60\right )}{10}\)	\(29\)
default	\(-20 x^{6}+\frac {28}{5} x^{5}+\frac {45}{2} x^{4}-9 x^{3}-\frac {17}{2} x^{2}+6 x\)	\(30\)
norman	\(-20 x^{6}+\frac {28}{5} x^{5}+\frac {45}{2} x^{4}-9 x^{3}-\frac {17}{2} x^{2}+6 x\)	\(30\)
risch	\(-20 x^{6}+\frac {28}{5} x^{5}+\frac {45}{2} x^{4}-9 x^{3}-\frac {17}{2} x^{2}+6 x\)	\(30\)
parallelrisch	\(-20 x^{6}+\frac {28}{5} x^{5}+\frac {45}{2} x^{4}-9 x^{3}-\frac {17}{2} x^{2}+6 x\)	\(30\)

[In]

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

[Out]

-1/10*x*(200*x^5-56*x^4-225*x^3+90*x^2+85*x-60)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-20 \, x^{6} + \frac {28}{5} \, x^{5} + \frac {45}{2} \, x^{4} - 9 \, x^{3} - \frac {17}{2} \, x^{2} + 6 \, x \]

[In]

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

[Out]

-20*x^6 + 28/5*x^5 + 45/2*x^4 - 9*x^3 - 17/2*x^2 + 6*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=- 20 x^{6} + \frac {28 x^{5}}{5} + \frac {45 x^{4}}{2} - 9 x^{3} - \frac {17 x^{2}}{2} + 6 x \]

[In]

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

[Out]

-20*x**6 + 28*x**5/5 + 45*x**4/2 - 9*x**3 - 17*x**2/2 + 6*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-20 \, x^{6} + \frac {28}{5} \, x^{5} + \frac {45}{2} \, x^{4} - 9 \, x^{3} - \frac {17}{2} \, x^{2} + 6 \, x \]

[In]

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

[Out]

-20*x^6 + 28/5*x^5 + 45/2*x^4 - 9*x^3 - 17/2*x^2 + 6*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-20 \, x^{6} + \frac {28}{5} \, x^{5} + \frac {45}{2} \, x^{4} - 9 \, x^{3} - \frac {17}{2} \, x^{2} + 6 \, x \]

[In]

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

[Out]

-20*x^6 + 28/5*x^5 + 45/2*x^4 - 9*x^3 - 17/2*x^2 + 6*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x)^3 (2+3 x) (3+5 x) \, dx=-20\,x^6+\frac {28\,x^5}{5}+\frac {45\,x^4}{2}-9\,x^3-\frac {17\,x^2}{2}+6\,x \]

[In]

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

[Out]

6*x - (17*x^2)/2 - 9*x^3 + (45*x^4)/2 + (28*x^5)/5 - 20*x^6