\(\int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx\) [1178]

Optimal result

Integrand size = 20, antiderivative size = 56 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-\frac {7 (2+3 x)^6}{1458}+\frac {107 (2+3 x)^7}{1701}-\frac {185}{648} (2+3 x)^8+\frac {1025 (2+3 x)^9}{2187}-\frac {25}{243} (2+3 x)^{10} \]

[Out]

-7/1458*(2+3*x)^6+107/1701*(2+3*x)^7-185/648*(2+3*x)^8+1025/2187*(2+3*x)^9-25/243*(2+3*x)^10

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, 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} \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-\frac {25}{243} (3 x+2)^{10}+\frac {1025 (3 x+2)^9}{2187}-\frac {185}{648} (3 x+2)^8+\frac {107 (3 x+2)^7}{1701}-\frac {7 (3 x+2)^6}{1458} \]

[In]

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

[Out]

(-7*(2 + 3*x)^6)/1458 + (107*(2 + 3*x)^7)/1701 - (185*(2 + 3*x)^8)/648 + (1025*(2 + 3*x)^9)/2187 - (25*(2 + 3*
x)^10)/243

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}{81} (2+3 x)^5+\frac {107}{81} (2+3 x)^6-\frac {185}{27} (2+3 x)^7+\frac {1025}{81} (2+3 x)^8-\frac {250}{81} (2+3 x)^9\right ) \, dx \\ & = -\frac {7 (2+3 x)^6}{1458}+\frac {107 (2+3 x)^7}{1701}-\frac {185}{648} (2+3 x)^8+\frac {1025 (2+3 x)^9}{2187}-\frac {25}{243} (2+3 x)^{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=864 x+4536 x^2+12480 x^3+16570 x^4-1810 x^5-\frac {90143 x^6}{2}-\frac {547767 x^7}{7}-\frac {544185 x^8}{8}-31275 x^9-6075 x^{10} \]

[In]

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

[Out]

864*x + 4536*x^2 + 12480*x^3 + 16570*x^4 - 1810*x^5 - (90143*x^6)/2 - (547767*x^7)/7 - (544185*x^8)/8 - 31275*
x^9 - 6075*x^10

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88

method	result	size
gosper	\(-\frac {x \left (340200 x^{9}+1751400 x^{8}+3809295 x^{7}+4382136 x^{6}+2524004 x^{5}+101360 x^{4}-927920 x^{3}-698880 x^{2}-254016 x -48384\right )}{56}\)	\(49\)
default	\(-6075 x^{10}-31275 x^{9}-\frac {544185}{8} x^{8}-\frac {547767}{7} x^{7}-\frac {90143}{2} x^{6}-1810 x^{5}+16570 x^{4}+12480 x^{3}+4536 x^{2}+864 x\)	\(50\)
norman	\(-6075 x^{10}-31275 x^{9}-\frac {544185}{8} x^{8}-\frac {547767}{7} x^{7}-\frac {90143}{2} x^{6}-1810 x^{5}+16570 x^{4}+12480 x^{3}+4536 x^{2}+864 x\)	\(50\)
risch	\(-6075 x^{10}-31275 x^{9}-\frac {544185}{8} x^{8}-\frac {547767}{7} x^{7}-\frac {90143}{2} x^{6}-1810 x^{5}+16570 x^{4}+12480 x^{3}+4536 x^{2}+864 x\)	\(50\)
parallelrisch	\(-6075 x^{10}-31275 x^{9}-\frac {544185}{8} x^{8}-\frac {547767}{7} x^{7}-\frac {90143}{2} x^{6}-1810 x^{5}+16570 x^{4}+12480 x^{3}+4536 x^{2}+864 x\)	\(50\)

[In]

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

[Out]

-1/56*x*(340200*x^9+1751400*x^8+3809295*x^7+4382136*x^6+2524004*x^5+101360*x^4-927920*x^3-698880*x^2-254016*x-
48384)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-6075 \, x^{10} - 31275 \, x^{9} - \frac {544185}{8} \, x^{8} - \frac {547767}{7} \, x^{7} - \frac {90143}{2} \, x^{6} - 1810 \, x^{5} + 16570 \, x^{4} + 12480 \, x^{3} + 4536 \, x^{2} + 864 \, x \]

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=- 6075 x^{10} - 31275 x^{9} - \frac {544185 x^{8}}{8} - \frac {547767 x^{7}}{7} - \frac {90143 x^{6}}{2} - 1810 x^{5} + 16570 x^{4} + 12480 x^{3} + 4536 x^{2} + 864 x \]

[In]

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

[Out]

-6075*x**10 - 31275*x**9 - 544185*x**8/8 - 547767*x**7/7 - 90143*x**6/2 - 1810*x**5 + 16570*x**4 + 12480*x**3
+ 4536*x**2 + 864*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-6075 \, x^{10} - 31275 \, x^{9} - \frac {544185}{8} \, x^{8} - \frac {547767}{7} \, x^{7} - \frac {90143}{2} \, x^{6} - 1810 \, x^{5} + 16570 \, x^{4} + 12480 \, x^{3} + 4536 \, x^{2} + 864 \, x \]

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-6075 \, x^{10} - 31275 \, x^{9} - \frac {544185}{8} \, x^{8} - \frac {547767}{7} \, x^{7} - \frac {90143}{2} \, x^{6} - 1810 \, x^{5} + 16570 \, x^{4} + 12480 \, x^{3} + 4536 \, x^{2} + 864 \, x \]

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx=-6075\,x^{10}-31275\,x^9-\frac {544185\,x^8}{8}-\frac {547767\,x^7}{7}-\frac {90143\,x^6}{2}-1810\,x^5+16570\,x^4+12480\,x^3+4536\,x^2+864\,x \]

[In]

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

[Out]

864*x + 4536*x^2 + 12480*x^3 + 16570*x^4 - 1810*x^5 - (90143*x^6)/2 - (547767*x^7)/7 - (544185*x^8)/8 - 31275*
x^9 - 6075*x^10