\(\int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx\) [1158]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=\frac {7}{729} (2+3 x)^9-\frac {4}{45} (2+3 x)^{10}+\frac {65}{297} (2+3 x)^{11}-\frac {25}{486} (2+3 x)^{12} \]

[Out]

7/729*(2+3*x)^9-4/45*(2+3*x)^10+65/297*(2+3*x)^11-25/486*(2+3*x)^12

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} \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=-\frac {25}{486} (3 x+2)^{12}+\frac {65}{297} (3 x+2)^{11}-\frac {4}{45} (3 x+2)^{10}+\frac {7}{729} (3 x+2)^9 \]

[In]

Int[(1 - 2*x)*(2 + 3*x)^8*(3 + 5*x)^2,x]

[Out]

(7*(2 + 3*x)^9)/729 - (4*(2 + 3*x)^10)/45 + (65*(2 + 3*x)^11)/297 - (25*(2 + 3*x)^12)/486

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)^8-\frac {8}{3} (2+3 x)^9+\frac {65}{9} (2+3 x)^{10}-\frac {50}{27} (2+3 x)^{11}\right ) \, dx \\ & = \frac {7}{729} (2+3 x)^9-\frac {4}{45} (2+3 x)^{10}+\frac {65}{297} (2+3 x)^{11}-\frac {25}{486} (2+3 x)^{12} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.53 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=2304 x+15360 x^2+\frac {173056 x^3}{3}+127168 x^4+\frac {679008 x^5}{5}-71904 x^6-507600 x^7-881442 x^8-869103 x^9-\frac {2614194 x^{10}}{5}-\frac {1979235 x^{11}}{11}-\frac {54675 x^{12}}{2} \]

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^8*(3 + 5*x)^2,x]

[Out]

2304*x + 15360*x^2 + (173056*x^3)/3 + 127168*x^4 + (679008*x^5)/5 - 71904*x^6 - 507600*x^7 - 881442*x^8 - 8691
03*x^9 - (2614194*x^10)/5 - (1979235*x^11)/11 - (54675*x^12)/2

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31

method	result	size
gosper	\(-\frac {x \left (9021375 x^{11}[phone] x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-44814528 x^{4}-41965440 x^{3}-19036160 x^{2}-5068800 x -760320\right )}{330}\)	\(59\)
default	\(-\frac {54675}{2} x^{12}-\frac {1979235}{11} x^{11}-\frac {2614194}{5} x^{10}-869103 x^{9}-881442 x^{8}-507600 x^{7}-71904 x^{6}+\frac {679008}{5} x^{5}+127168 x^{4}+\frac {173056}{3} x^{3}+15360 x^{2}+2304 x\)	\(60\)
norman	\(-\frac {54675}{2} x^{12}-\frac {1979235}{11} x^{11}-\frac {2614194}{5} x^{10}-869103 x^{9}-881442 x^{8}-507600 x^{7}-71904 x^{6}+\frac {679008}{5} x^{5}+127168 x^{4}+\frac {173056}{3} x^{3}+15360 x^{2}+2304 x\)	\(60\)
risch	\(-\frac {54675}{2} x^{12}-\frac {1979235}{11} x^{11}-\frac {2614194}{5} x^{10}-869103 x^{9}-881442 x^{8}-507600 x^{7}-71904 x^{6}+\frac {679008}{5} x^{5}+127168 x^{4}+\frac {173056}{3} x^{3}+15360 x^{2}+2304 x\)	\(60\)
parallelrisch	\(-\frac {54675}{2} x^{12}-\frac {1979235}{11} x^{11}-\frac {2614194}{5} x^{10}-869103 x^{9}-881442 x^{8}-507600 x^{7}-71904 x^{6}+\frac {679008}{5} x^{5}+127168 x^{4}+\frac {173056}{3} x^{3}+15360 x^{2}+2304 x\)	\(60\)

[In]

int((1-2*x)*(2+3*x)^8*(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/330*x*(9021375*x^11+59377050*x^10+172536804*x^9+286803990*x^8+290875860*x^7+167508000*x^6+23728320*x^5-4481
4528*x^4-41965440*x^3-19036160*x^2-5068800*x-760320)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=-\frac {54675}{2} \, x^{12} - \frac {1979235}{11} \, x^{11} - \frac {2614194}{5} \, x^{10} - 869103 \, x^{9} - 881442 \, x^{8} - 507600 \, x^{7} - 71904 \, x^{6} + \frac {679008}{5} \, x^{5} + 127168 \, x^{4} + \frac {173056}{3} \, x^{3} + 15360 \, x^{2} + 2304 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="fricas")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.47 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=- \frac {54675 x^{12}}{2} - \frac {1979235 x^{11}}{11} - \frac {2614194 x^{10}}{5} - 869103 x^{9} - 881442 x^{8} - 507600 x^{7} - 71904 x^{6} + \frac {679008 x^{5}}{5} + 127168 x^{4} + \frac {173056 x^{3}}{3} + 15360 x^{2} + 2304 x \]

[In]

integrate((1-2*x)*(2+3*x)**8*(3+5*x)**2,x)

[Out]

-54675*x**12/2 - 1979235*x**11/11 - 2614194*x**10/5 - 869103*x**9 - 881442*x**8 - 507600*x**7 - 71904*x**6 + 6
79008*x**5/5 + 127168*x**4 + 173056*x**3/3 + 15360*x**2 + 2304*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=-\frac {54675}{2} \, x^{12} - \frac {1979235}{11} \, x^{11} - \frac {2614194}{5} \, x^{10} - 869103 \, x^{9} - 881442 \, x^{8} - 507600 \, x^{7} - 71904 \, x^{6} + \frac {679008}{5} \, x^{5} + 127168 \, x^{4} + \frac {173056}{3} \, x^{3} + 15360 \, x^{2} + 2304 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="maxima")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=-\frac {54675}{2} \, x^{12} - \frac {1979235}{11} \, x^{11} - \frac {2614194}{5} \, x^{10} - 869103 \, x^{9} - 881442 \, x^{8} - 507600 \, x^{7} - 71904 \, x^{6} + \frac {679008}{5} \, x^{5} + 127168 \, x^{4} + \frac {173056}{3} \, x^{3} + 15360 \, x^{2} + 2304 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="giac")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.31 \[ \int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx=-\frac {54675\,x^{12}}{2}-\frac {1979235\,x^{11}}{11}-\frac {2614194\,x^{10}}{5}-869103\,x^9-881442\,x^8-507600\,x^7-71904\,x^6+\frac {679008\,x^5}{5}+127168\,x^4+\frac {173056\,x^3}{3}+15360\,x^2+2304\,x \]

[In]

int(-(2*x - 1)*(3*x + 2)^8*(5*x + 3)^2,x)

[Out]

2304*x + 15360*x^2 + (173056*x^3)/3 + 127168*x^4 + (679008*x^5)/5 - 71904*x^6 - 507600*x^7 - 881442*x^8 - 8691
03*x^9 - (2614194*x^10)/5 - (1979235*x^11)/11 - (54675*x^12)/2