\(\int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx\) [1176]

Optimal result

Integrand size = 20, antiderivative size = 56 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {7 (2+3 x)^8}{1944}+\frac {107 (2+3 x)^9}{2187}-\frac {37}{162} (2+3 x)^{10}+\frac {1025 (2+3 x)^{11}}{2673}-\frac {125 (2+3 x)^{12}}{1458} \]

[Out]

-7/1944*(2+3*x)^8+107/2187*(2+3*x)^9-37/162*(2+3*x)^10+1025/2673*(2+3*x)^11-125/1458*(2+3*x)^12

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} \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {125 (3 x+2)^{12}}{1458}+\frac {1025 (3 x+2)^{11}}{2673}-\frac {37}{162} (3 x+2)^{10}+\frac {107 (3 x+2)^9}{2187}-\frac {7 (3 x+2)^8}{1944} \]

[In]

Int[(1 - 2*x)*(2 + 3*x)^7*(3 + 5*x)^3,x]

[Out]

(-7*(2 + 3*x)^8)/1944 + (107*(2 + 3*x)^9)/2187 - (37*(2 + 3*x)^10)/162 + (1025*(2 + 3*x)^11)/2673 - (125*(2 +
3*x)^12)/1458

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)^7+\frac {107}{81} (2+3 x)^8-\frac {185}{27} (2+3 x)^9+\frac {1025}{81} (2+3 x)^{10}-\frac {250}{81} (2+3 x)^{11}\right ) \, dx \\ & = -\frac {7 (2+3 x)^8}{1944}+\frac {107 (2+3 x)^9}{2187}-\frac {37}{162} (2+3 x)^{10}+\frac {1025 (2+3 x)^{11}}{2673}-\frac {125 (2+3 x)^{12}}{1458} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=3456 x+23328 x^2+88800 x^3+199012 x^4+219224 x^5-98966 x^6-788238 x^7-\frac {11183805 x^8}{8}-1398447 x^9-\frac {1703673 x^{10}}{2}-\frac {3262275 x^{11}}{11}-\frac {91125 x^{12}}{2} \]

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^7*(3 + 5*x)^3,x]

[Out]

3456*x + 23328*x^2 + 88800*x^3 + 199012*x^4 + 219224*x^5 - 98966*x^6 - 788238*x^7 - (11183805*x^8)/8 - 1398447
*x^9 - (1703673*x^10)/2 - (3262275*x^11)/11 - (91125*x^12)/2

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05

method	result	size
gosper	\(-\frac {x \left (4009500 x^{11}[phone] x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}+8709008 x^{5}-19291712 x^{4}-17513056 x^{3}-7814400 x^{2}-2052864 x -304128\right )}{88}\)	\(59\)
default	\(-\frac {91125}{2} x^{12}-\frac {3262275}{11} x^{11}-\frac {1703673}{2} x^{10}-1398447 x^{9}-\frac {11183805}{8} x^{8}-788238 x^{7}-98966 x^{6}+219224 x^{5}+199012 x^{4}+88800 x^{3}+23328 x^{2}+3456 x\)	\(60\)
norman	\(-\frac {91125}{2} x^{12}-\frac {3262275}{11} x^{11}-\frac {1703673}{2} x^{10}-1398447 x^{9}-\frac {11183805}{8} x^{8}-788238 x^{7}-98966 x^{6}+219224 x^{5}+199012 x^{4}+88800 x^{3}+23328 x^{2}+3456 x\)	\(60\)
risch	\(-\frac {91125}{2} x^{12}-\frac {3262275}{11} x^{11}-\frac {1703673}{2} x^{10}-1398447 x^{9}-\frac {11183805}{8} x^{8}-788238 x^{7}-98966 x^{6}+219224 x^{5}+199012 x^{4}+88800 x^{3}+23328 x^{2}+3456 x\)	\(60\)
parallelrisch	\(-\frac {91125}{2} x^{12}-\frac {3262275}{11} x^{11}-\frac {1703673}{2} x^{10}-1398447 x^{9}-\frac {11183805}{8} x^{8}-788238 x^{7}-98966 x^{6}+219224 x^{5}+199012 x^{4}+88800 x^{3}+23328 x^{2}+3456 x\)	\(60\)

[In]

int((1-2*x)*(2+3*x)^7*(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/88*x*(4009500*x^11+26098200*x^10+74961612*x^9+123063336*x^8+123021855*x^7+69364944*x^6+8709008*x^5-19291712
*x^4-17513056*x^3-7814400*x^2-2052864*x-304128)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {91125}{2} \, x^{12} - \frac {3262275}{11} \, x^{11} - \frac {1703673}{2} \, x^{10} - 1398447 \, x^{9} - \frac {11183805}{8} \, x^{8} - 788238 \, x^{7} - 98966 \, x^{6} + 219224 \, x^{5} + 199012 \, x^{4} + 88800 \, x^{3} + 23328 \, x^{2} + 3456 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x)^3,x, algorithm="fricas")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=- \frac {91125 x^{12}}{2} - \frac {3262275 x^{11}}{11} - \frac {1703673 x^{10}}{2} - 1398447 x^{9} - \frac {11183805 x^{8}}{8} - 788238 x^{7} - 98966 x^{6} + 219224 x^{5} + 199012 x^{4} + 88800 x^{3} + 23328 x^{2} + 3456 x \]

[In]

integrate((1-2*x)*(2+3*x)**7*(3+5*x)**3,x)

[Out]

-91125*x**12/2 - 3262275*x**11/11 - 1703673*x**10/2 - 1398447*x**9 - 11183805*x**8/8 - 788238*x**7 - 98966*x**
6 + 219224*x**5 + 199012*x**4 + 88800*x**3 + 23328*x**2 + 3456*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {91125}{2} \, x^{12} - \frac {3262275}{11} \, x^{11} - \frac {1703673}{2} \, x^{10} - 1398447 \, x^{9} - \frac {11183805}{8} \, x^{8} - 788238 \, x^{7} - 98966 \, x^{6} + 219224 \, x^{5} + 199012 \, x^{4} + 88800 \, x^{3} + 23328 \, x^{2} + 3456 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x)^3,x, algorithm="maxima")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {91125}{2} \, x^{12} - \frac {3262275}{11} \, x^{11} - \frac {1703673}{2} \, x^{10} - 1398447 \, x^{9} - \frac {11183805}{8} \, x^{8} - 788238 \, x^{7} - 98966 \, x^{6} + 219224 \, x^{5} + 199012 \, x^{4} + 88800 \, x^{3} + 23328 \, x^{2} + 3456 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x)^3,x, algorithm="giac")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx=-\frac {91125\,x^{12}}{2}-\frac {3262275\,x^{11}}{11}-\frac {1703673\,x^{10}}{2}-1398447\,x^9-\frac {11183805\,x^8}{8}-788238\,x^7-98966\,x^6+219224\,x^5+199012\,x^4+88800\,x^3+23328\,x^2+3456\,x \]

[In]

int(-(2*x - 1)*(3*x + 2)^7*(5*x + 3)^3,x)

[Out]

3456*x + 23328*x^2 + 88800*x^3 + 199012*x^4 + 219224*x^5 - 98966*x^6 - 788238*x^7 - (11183805*x^8)/8 - 1398447
*x^9 - (1703673*x^10)/2 - (3262275*x^11)/11 - (91125*x^12)/2