\(\int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx\) [1161]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=\frac {7}{486} (2+3 x)^6-\frac {8}{63} (2+3 x)^7+\frac {65}{216} (2+3 x)^8-\frac {50}{729} (2+3 x)^9 \]

[Out]

7/486*(2+3*x)^6-8/63*(2+3*x)^7+65/216*(2+3*x)^8-50/729*(2+3*x)^9

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, 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} \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=-\frac {50}{729} (3 x+2)^9+\frac {65}{216} (3 x+2)^8-\frac {8}{63} (3 x+2)^7+\frac {7}{486} (3 x+2)^6 \]

[In]

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

[Out]

(7*(2 + 3*x)^6)/486 - (8*(2 + 3*x)^7)/63 + (65*(2 + 3*x)^8)/216 - (50*(2 + 3*x)^9)/729

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}{27} (2+3 x)^5-\frac {8}{3} (2+3 x)^6+\frac {65}{9} (2+3 x)^7-\frac {50}{27} (2+3 x)^8\right ) \, dx \\ & = \frac {7}{486} (2+3 x)^6-\frac {8}{63} (2+3 x)^7+\frac {65}{216} (2+3 x)^8-\frac {50}{729} (2+3 x)^9 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=288 x+1272 x^2+\frac {8240 x^3}{3}+2090 x^4-3390 x^5-\frac {20631 x^6}{2}-\frac {79434 x^7}{7}-\frac {49005 x^8}{8}-1350 x^9 \]

[In]

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

[Out]

288*x + 1272*x^2 + (8240*x^3)/3 + 2090*x^4 - 3390*x^5 - (20631*x^6)/2 - (79434*x^7)/7 - (49005*x^8)/8 - 1350*x
^9

Maple [A] (verified)

Time = 2.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98

method	result	size
gosper	\(-\frac {x \left (226800 x^{8}+1029105 x^{7}+1906416 x^{6}+1733004 x^{5}+569520 x^{4}-351120 x^{3}-461440 x^{2}-213696 x -48384\right )}{168}\)	\(44\)
default	\(-1350 x^{9}-\frac {49005}{8} x^{8}-\frac {79434}{7} x^{7}-\frac {20631}{2} x^{6}-3390 x^{5}+2090 x^{4}+\frac {8240}{3} x^{3}+1272 x^{2}+288 x\)	\(45\)
norman	\(-1350 x^{9}-\frac {49005}{8} x^{8}-\frac {79434}{7} x^{7}-\frac {20631}{2} x^{6}-3390 x^{5}+2090 x^{4}+\frac {8240}{3} x^{3}+1272 x^{2}+288 x\)	\(45\)
risch	\(-1350 x^{9}-\frac {49005}{8} x^{8}-\frac {79434}{7} x^{7}-\frac {20631}{2} x^{6}-3390 x^{5}+2090 x^{4}+\frac {8240}{3} x^{3}+1272 x^{2}+288 x\)	\(45\)
parallelrisch	\(-1350 x^{9}-\frac {49005}{8} x^{8}-\frac {79434}{7} x^{7}-\frac {20631}{2} x^{6}-3390 x^{5}+2090 x^{4}+\frac {8240}{3} x^{3}+1272 x^{2}+288 x\)	\(45\)

[In]

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

[Out]

-1/168*x*(226800*x^8+1029105*x^7+1906416*x^6+1733004*x^5+569520*x^4-351120*x^3-461440*x^2-213696*x-48384)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=-1350 \, x^{9} - \frac {49005}{8} \, x^{8} - \frac {79434}{7} \, x^{7} - \frac {20631}{2} \, x^{6} - 3390 \, x^{5} + 2090 \, x^{4} + \frac {8240}{3} \, x^{3} + 1272 \, x^{2} + 288 \, x \]

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=- 1350 x^{9} - \frac {49005 x^{8}}{8} - \frac {79434 x^{7}}{7} - \frac {20631 x^{6}}{2} - 3390 x^{5} + 2090 x^{4} + \frac {8240 x^{3}}{3} + 1272 x^{2} + 288 x \]

[In]

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

[Out]

-1350*x**9 - 49005*x**8/8 - 79434*x**7/7 - 20631*x**6/2 - 3390*x**5 + 2090*x**4 + 8240*x**3/3 + 1272*x**2 + 28
8*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=-1350 \, x^{9} - \frac {49005}{8} \, x^{8} - \frac {79434}{7} \, x^{7} - \frac {20631}{2} \, x^{6} - 3390 \, x^{5} + 2090 \, x^{4} + \frac {8240}{3} \, x^{3} + 1272 \, x^{2} + 288 \, x \]

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=-1350 \, x^{9} - \frac {49005}{8} \, x^{8} - \frac {79434}{7} \, x^{7} - \frac {20631}{2} \, x^{6} - 3390 \, x^{5} + 2090 \, x^{4} + \frac {8240}{3} \, x^{3} + 1272 \, x^{2} + 288 \, x \]

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx=-1350\,x^9-\frac {49005\,x^8}{8}-\frac {79434\,x^7}{7}-\frac {20631\,x^6}{2}-3390\,x^5+2090\,x^4+\frac {8240\,x^3}{3}+1272\,x^2+288\,x \]

[In]

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

[Out]

288*x + 1272*x^2 + (8240*x^3)/3 + 2090*x^4 - 3390*x^5 - (20631*x^6)/2 - (79434*x^7)/7 - (49005*x^8)/8 - 1350*x
^9