\(\int (1-2 x) (2+3 x)^3 (3+5 x) \, dx\) [1148]

Optimal result

Integrand size = 18, antiderivative size = 34 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-\frac {7}{108} (2+3 x)^4+\frac {37}{135} (2+3 x)^5-\frac {5}{81} (2+3 x)^6 \]

[Out]

-7/108*(2+3*x)^4+37/135*(2+3*x)^5-5/81*(2+3*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} \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-\frac {5}{81} (3 x+2)^6+\frac {37}{135} (3 x+2)^5-\frac {7}{108} (3 x+2)^4 \]

[In]

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

[Out]

(-7*(2 + 3*x)^4)/108 + (37*(2 + 3*x)^5)/135 - (5*(2 + 3*x)^6)/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}{9} (2+3 x)^3+\frac {37}{9} (2+3 x)^4-\frac {10}{9} (2+3 x)^5\right ) \, dx \\ & = -\frac {7}{108} (2+3 x)^4+\frac {37}{135} (2+3 x)^5-\frac {5}{81} (2+3 x)^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=24 x+50 x^2+\frac {46 x^3}{3}-\frac {333 x^4}{4}-\frac {567 x^5}{5}-45 x^6 \]

[In]

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

[Out]

24*x + 50*x^2 + (46*x^3)/3 - (333*x^4)/4 - (567*x^5)/5 - 45*x^6

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {x \left (2700 x^{5}+6804 x^{4}+4995 x^{3}-920 x^{2}-3000 x -1440\right )}{60}\)	\(29\)
default	\(-45 x^{6}-\frac {567}{5} x^{5}-\frac {333}{4} x^{4}+\frac {46}{3} x^{3}+50 x^{2}+24 x\)	\(30\)
norman	\(-45 x^{6}-\frac {567}{5} x^{5}-\frac {333}{4} x^{4}+\frac {46}{3} x^{3}+50 x^{2}+24 x\)	\(30\)
risch	\(-45 x^{6}-\frac {567}{5} x^{5}-\frac {333}{4} x^{4}+\frac {46}{3} x^{3}+50 x^{2}+24 x\)	\(30\)
parallelrisch	\(-45 x^{6}-\frac {567}{5} x^{5}-\frac {333}{4} x^{4}+\frac {46}{3} x^{3}+50 x^{2}+24 x\)	\(30\)

[In]

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

[Out]

-1/60*x*(2700*x^5+6804*x^4+4995*x^3-920*x^2-3000*x-1440)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-45 \, x^{6} - \frac {567}{5} \, x^{5} - \frac {333}{4} \, x^{4} + \frac {46}{3} \, x^{3} + 50 \, x^{2} + 24 \, x \]

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=- 45 x^{6} - \frac {567 x^{5}}{5} - \frac {333 x^{4}}{4} + \frac {46 x^{3}}{3} + 50 x^{2} + 24 x \]

[In]

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

[Out]

-45*x**6 - 567*x**5/5 - 333*x**4/4 + 46*x**3/3 + 50*x**2 + 24*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-45 \, x^{6} - \frac {567}{5} \, x^{5} - \frac {333}{4} \, x^{4} + \frac {46}{3} \, x^{3} + 50 \, x^{2} + 24 \, x \]

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-45 \, x^{6} - \frac {567}{5} \, x^{5} - \frac {333}{4} \, x^{4} + \frac {46}{3} \, x^{3} + 50 \, x^{2} + 24 \, x \]

[In]

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

[Out]

-45*x^6 - 567/5*x^5 - 333/4*x^4 + 46/3*x^3 + 50*x^2 + 24*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (1-2 x) (2+3 x)^3 (3+5 x) \, dx=-45\,x^6-\frac {567\,x^5}{5}-\frac {333\,x^4}{4}+\frac {46\,x^3}{3}+50\,x^2+24\,x \]

[In]

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

[Out]

24*x + 50*x^2 + (46*x^3)/3 - (333*x^4)/4 - (567*x^5)/5 - 45*x^6