\(\int (1-2 x) (2+3 x)^5 (3+5 x) \, dx\) [1146]

Optimal result

Integrand size = 18, antiderivative size = 34 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {7}{162} (2+3 x)^6+\frac {37}{189} (2+3 x)^7-\frac {5}{108} (2+3 x)^8 \]

[Out]

-7/162*(2+3*x)^6+37/189*(2+3*x)^7-5/108*(2+3*x)^8

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} \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {5}{108} (3 x+2)^8+\frac {37}{189} (3 x+2)^7-\frac {7}{162} (3 x+2)^6 \]

[In]

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

[Out]

(-7*(2 + 3*x)^6)/162 + (37*(2 + 3*x)^7)/189 - (5*(2 + 3*x)^8)/108

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)^5+\frac {37}{9} (2+3 x)^6-\frac {10}{9} (2+3 x)^7\right ) \, dx \\ & = -\frac {7}{162} (2+3 x)^6+\frac {37}{189} (2+3 x)^7-\frac {5}{108} (2+3 x)^8 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=96 x+344 x^2+\frac {1600 x^3}{3}+30 x^4-1170 x^5-\frac {3627 x^6}{2}-\frac {8343 x^7}{7}-\frac {1215 x^8}{4} \]

[In]

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

[Out]

96*x + 344*x^2 + (1600*x^3)/3 + 30*x^4 - 1170*x^5 - (3627*x^6)/2 - (8343*x^7)/7 - (1215*x^8)/4

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15

method	result	size
gosper	\(-\frac {x \left (25515 x^{7}+100116 x^{6}+152334 x^{5}+98280 x^{4}-2520 x^{3}-44800 x^{2}-28896 x -8064\right )}{84}\)	\(39\)
default	\(-\frac {1215}{4} x^{8}-\frac {8343}{7} x^{7}-\frac {3627}{2} x^{6}-1170 x^{5}+30 x^{4}+\frac {1600}{3} x^{3}+344 x^{2}+96 x\)	\(40\)
norman	\(-\frac {1215}{4} x^{8}-\frac {8343}{7} x^{7}-\frac {3627}{2} x^{6}-1170 x^{5}+30 x^{4}+\frac {1600}{3} x^{3}+344 x^{2}+96 x\)	\(40\)
risch	\(-\frac {1215}{4} x^{8}-\frac {8343}{7} x^{7}-\frac {3627}{2} x^{6}-1170 x^{5}+30 x^{4}+\frac {1600}{3} x^{3}+344 x^{2}+96 x\)	\(40\)
parallelrisch	\(-\frac {1215}{4} x^{8}-\frac {8343}{7} x^{7}-\frac {3627}{2} x^{6}-1170 x^{5}+30 x^{4}+\frac {1600}{3} x^{3}+344 x^{2}+96 x\)	\(40\)

[In]

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

[Out]

-1/84*x*(25515*x^7+100116*x^6+152334*x^5+98280*x^4-2520*x^3-44800*x^2-28896*x-8064)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {1215}{4} \, x^{8} - \frac {8343}{7} \, x^{7} - \frac {3627}{2} \, x^{6} - 1170 \, x^{5} + 30 \, x^{4} + \frac {1600}{3} \, x^{3} + 344 \, x^{2} + 96 \, x \]

[In]

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

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=- \frac {1215 x^{8}}{4} - \frac {8343 x^{7}}{7} - \frac {3627 x^{6}}{2} - 1170 x^{5} + 30 x^{4} + \frac {1600 x^{3}}{3} + 344 x^{2} + 96 x \]

[In]

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

[Out]

-1215*x**8/4 - 8343*x**7/7 - 3627*x**6/2 - 1170*x**5 + 30*x**4 + 1600*x**3/3 + 344*x**2 + 96*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {1215}{4} \, x^{8} - \frac {8343}{7} \, x^{7} - \frac {3627}{2} \, x^{6} - 1170 \, x^{5} + 30 \, x^{4} + \frac {1600}{3} \, x^{3} + 344 \, x^{2} + 96 \, x \]

[In]

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

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {1215}{4} \, x^{8} - \frac {8343}{7} \, x^{7} - \frac {3627}{2} \, x^{6} - 1170 \, x^{5} + 30 \, x^{4} + \frac {1600}{3} \, x^{3} + 344 \, x^{2} + 96 \, x \]

[In]

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

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int (1-2 x) (2+3 x)^5 (3+5 x) \, dx=-\frac {1215\,x^8}{4}-\frac {8343\,x^7}{7}-\frac {3627\,x^6}{2}-1170\,x^5+30\,x^4+\frac {1600\,x^3}{3}+344\,x^2+96\,x \]

[In]

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

[Out]

96*x + 344*x^2 + (1600*x^3)/3 + 30*x^4 - 1170*x^5 - (3627*x^6)/2 - (8343*x^7)/7 - (1215*x^8)/4