\(\int (8+24 x+8 x^2-15 x^3+8 x^4)^2 \, dx\) [59]

Optimal result

Integrand size = 22, antiderivative size = 52 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=64 x+192 x^2+\frac {704 x^3}{3}+36 x^4-\frac {528 x^5}{5}+24 x^6+\frac {353 x^7}{7}-30 x^8+\frac {64 x^9}{9} \]

[Out]

64*x+192*x^2+704/3*x^3+36*x^4-528/5*x^5+24*x^6+353/7*x^7-30*x^8+64/9*x^9

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2086} \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64 x^9}{9}-30 x^8+\frac {353 x^7}{7}+24 x^6-\frac {528 x^5}{5}+36 x^4+\frac {704 x^3}{3}+192 x^2+64 x \]

[In]

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^2,x]

[Out]

64*x + 192*x^2 + (704*x^3)/3 + 36*x^4 - (528*x^5)/5 + 24*x^6 + (353*x^7)/7 - 30*x^8 + (64*x^9)/9

Rule 2086

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (64+384 x+704 x^2+144 x^3-528 x^4+144 x^5+353 x^6-240 x^7+64 x^8\right ) \, dx \\ & = 64 x+192 x^2+\frac {704 x^3}{3}+36 x^4-\frac {528 x^5}{5}+24 x^6+\frac {353 x^7}{7}-30 x^8+\frac {64 x^9}{9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=64 x+192 x^2+\frac {704 x^3}{3}+36 x^4-\frac {528 x^5}{5}+24 x^6+\frac {353 x^7}{7}-30 x^8+\frac {64 x^9}{9} \]

[In]

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^2,x]

[Out]

64*x + 192*x^2 + (704*x^3)/3 + 36*x^4 - (528*x^5)/5 + 24*x^6 + (353*x^7)/7 - 30*x^8 + (64*x^9)/9

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87

method	result	size
gosper	\(64 x +192 x^{2}+\frac {704}{3} x^{3}+36 x^{4}-\frac {528}{5} x^{5}+24 x^{6}+\frac {353}{7} x^{7}-30 x^{8}+\frac {64}{9} x^{9}\)	\(45\)
default	\(64 x +192 x^{2}+\frac {704}{3} x^{3}+36 x^{4}-\frac {528}{5} x^{5}+24 x^{6}+\frac {353}{7} x^{7}-30 x^{8}+\frac {64}{9} x^{9}\)	\(45\)
norman	\(64 x +192 x^{2}+\frac {704}{3} x^{3}+36 x^{4}-\frac {528}{5} x^{5}+24 x^{6}+\frac {353}{7} x^{7}-30 x^{8}+\frac {64}{9} x^{9}\)	\(45\)
risch	\(64 x +192 x^{2}+\frac {704}{3} x^{3}+36 x^{4}-\frac {528}{5} x^{5}+24 x^{6}+\frac {353}{7} x^{7}-30 x^{8}+\frac {64}{9} x^{9}\)	\(45\)
parallelrisch	\(64 x +192 x^{2}+\frac {704}{3} x^{3}+36 x^{4}-\frac {528}{5} x^{5}+24 x^{6}+\frac {353}{7} x^{7}-30 x^{8}+\frac {64}{9} x^{9}\)	\(45\)

[In]

int((8*x^4-15*x^3+8*x^2+24*x+8)^2,x,method=_RETURNVERBOSE)

[Out]

64*x+192*x^2+704/3*x^3+36*x^4-528/5*x^5+24*x^6+353/7*x^7-30*x^8+64/9*x^9

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64}{9} \, x^{9} - 30 \, x^{8} + \frac {353}{7} \, x^{7} + 24 \, x^{6} - \frac {528}{5} \, x^{5} + 36 \, x^{4} + \frac {704}{3} \, x^{3} + 192 \, x^{2} + 64 \, x \]

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="fricas")

[Out]

64/9*x^9 - 30*x^8 + 353/7*x^7 + 24*x^6 - 528/5*x^5 + 36*x^4 + 704/3*x^3 + 192*x^2 + 64*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64 x^{9}}{9} - 30 x^{8} + \frac {353 x^{7}}{7} + 24 x^{6} - \frac {528 x^{5}}{5} + 36 x^{4} + \frac {704 x^{3}}{3} + 192 x^{2} + 64 x \]

[In]

integrate((8*x**4-15*x**3+8*x**2+24*x+8)**2,x)

[Out]

64*x**9/9 - 30*x**8 + 353*x**7/7 + 24*x**6 - 528*x**5/5 + 36*x**4 + 704*x**3/3 + 192*x**2 + 64*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64}{9} \, x^{9} - 30 \, x^{8} + \frac {353}{7} \, x^{7} + 24 \, x^{6} - \frac {528}{5} \, x^{5} + 36 \, x^{4} + \frac {704}{3} \, x^{3} + 192 \, x^{2} + 64 \, x \]

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="maxima")

[Out]

64/9*x^9 - 30*x^8 + 353/7*x^7 + 24*x^6 - 528/5*x^5 + 36*x^4 + 704/3*x^3 + 192*x^2 + 64*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64}{9} \, x^{9} - 30 \, x^{8} + \frac {353}{7} \, x^{7} + 24 \, x^{6} - \frac {528}{5} \, x^{5} + 36 \, x^{4} + \frac {704}{3} \, x^{3} + 192 \, x^{2} + 64 \, x \]

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="giac")

[Out]

64/9*x^9 - 30*x^8 + 353/7*x^7 + 24*x^6 - 528/5*x^5 + 36*x^4 + 704/3*x^3 + 192*x^2 + 64*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2 \, dx=\frac {64\,x^9}{9}-30\,x^8+\frac {353\,x^7}{7}+24\,x^6-\frac {528\,x^5}{5}+36\,x^4+\frac {704\,x^3}{3}+192\,x^2+64\,x \]

[In]

int((24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8)^2,x)

[Out]

64*x + 192*x^2 + (704*x^3)/3 + 36*x^4 - (528*x^5)/5 + 24*x^6 + (353*x^7)/7 - 30*x^8 + (64*x^9)/9