\(\int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx\) [1163]

Optimal result

Integrand size = 20, antiderivative size = 44 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=72 x+210 x^2+\frac {638 x^3}{3}-\frac {769 x^4}{4}-\frac {3366 x^5}{5}-\frac {1215 x^6}{2}-\frac {1350 x^7}{7} \]

[Out]

72*x+210*x^2+638/3*x^3-769/4*x^4-3366/5*x^5-1215/2*x^6-1350/7*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, 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} \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {1350 x^7}{7}-\frac {1215 x^6}{2}-\frac {3366 x^5}{5}-\frac {769 x^4}{4}+\frac {638 x^3}{3}+210 x^2+72 x \]

[In]

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

[Out]

72*x + 210*x^2 + (638*x^3)/3 - (769*x^4)/4 - (3366*x^5)/5 - (1215*x^6)/2 - (1350*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 (72+420 x+638 x^2-769 x^3-3366 x^4-3645 x^5-1350 x^6\right ) \, dx \\ & = 72 x+210 x^2+\frac {638 x^3}{3}-\frac {769 x^4}{4}-\frac {3366 x^5}{5}-\frac {1215 x^6}{2}-\frac {1350 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=72 x+210 x^2+\frac {638 x^3}{3}-\frac {769 x^4}{4}-\frac {3366 x^5}{5}-\frac {1215 x^6}{2}-\frac {1350 x^7}{7} \]

[In]

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

[Out]

72*x + 210*x^2 + (638*x^3)/3 - (769*x^4)/4 - (3366*x^5)/5 - (1215*x^6)/2 - (1350*x^7)/7

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77

method	result	size
gosper	\(-\frac {x \left (81000 x^{6}+255150 x^{5}+282744 x^{4}+80745 x^{3}-89320 x^{2}-88200 x -30240\right )}{420}\)	\(34\)
default	\(72 x +210 x^{2}+\frac {638}{3} x^{3}-\frac {769}{4} x^{4}-\frac {3366}{5} x^{5}-\frac {1215}{2} x^{6}-\frac {1350}{7} x^{7}\)	\(35\)
norman	\(72 x +210 x^{2}+\frac {638}{3} x^{3}-\frac {769}{4} x^{4}-\frac {3366}{5} x^{5}-\frac {1215}{2} x^{6}-\frac {1350}{7} x^{7}\)	\(35\)
risch	\(72 x +210 x^{2}+\frac {638}{3} x^{3}-\frac {769}{4} x^{4}-\frac {3366}{5} x^{5}-\frac {1215}{2} x^{6}-\frac {1350}{7} x^{7}\)	\(35\)
parallelrisch	\(72 x +210 x^{2}+\frac {638}{3} x^{3}-\frac {769}{4} x^{4}-\frac {3366}{5} x^{5}-\frac {1215}{2} x^{6}-\frac {1350}{7} x^{7}\)	\(35\)

[In]

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

[Out]

-1/420*x*(81000*x^6+255150*x^5+282744*x^4+80745*x^3-89320*x^2-88200*x-30240)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {1350}{7} \, x^{7} - \frac {1215}{2} \, x^{6} - \frac {3366}{5} \, x^{5} - \frac {769}{4} \, x^{4} + \frac {638}{3} \, x^{3} + 210 \, x^{2} + 72 \, x \]

[In]

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

[Out]

-1350/7*x^7 - 1215/2*x^6 - 3366/5*x^5 - 769/4*x^4 + 638/3*x^3 + 210*x^2 + 72*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=- \frac {1350 x^{7}}{7} - \frac {1215 x^{6}}{2} - \frac {3366 x^{5}}{5} - \frac {769 x^{4}}{4} + \frac {638 x^{3}}{3} + 210 x^{2} + 72 x \]

[In]

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

[Out]

-1350*x**7/7 - 1215*x**6/2 - 3366*x**5/5 - 769*x**4/4 + 638*x**3/3 + 210*x**2 + 72*x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {1350}{7} \, x^{7} - \frac {1215}{2} \, x^{6} - \frac {3366}{5} \, x^{5} - \frac {769}{4} \, x^{4} + \frac {638}{3} \, x^{3} + 210 \, x^{2} + 72 \, x \]

[In]

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

[Out]

-1350/7*x^7 - 1215/2*x^6 - 3366/5*x^5 - 769/4*x^4 + 638/3*x^3 + 210*x^2 + 72*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {1350}{7} \, x^{7} - \frac {1215}{2} \, x^{6} - \frac {3366}{5} \, x^{5} - \frac {769}{4} \, x^{4} + \frac {638}{3} \, x^{3} + 210 \, x^{2} + 72 \, x \]

[In]

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

[Out]

-1350/7*x^7 - 1215/2*x^6 - 3366/5*x^5 - 769/4*x^4 + 638/3*x^3 + 210*x^2 + 72*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int (1-2 x) (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {1350\,x^7}{7}-\frac {1215\,x^6}{2}-\frac {3366\,x^5}{5}-\frac {769\,x^4}{4}+\frac {638\,x^3}{3}+210\,x^2+72\,x \]

[In]

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

[Out]

72*x + 210*x^2 + (638*x^3)/3 - (769*x^4)/4 - (3366*x^5)/5 - (1215*x^6)/2 - (1350*x^7)/7