\(\int (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4)^2 \, dx\) [41]

Optimal result

Integrand size = 32, antiderivative size = 107 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^2 e^4 x-8 a d^3 e^2 x^2+\frac {d^6 x^3}{3}+32 a d e^4 x^4-\frac {16}{5} e^2 \left (d^4-8 a e^3\right ) x^5-\frac {8}{3} d^3 e^3 x^6+\frac {64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac {64 e^6 x^9}{9} \]

[Out]

64*a^2*e^4*x-8*a*d^3*e^2*x^2+1/3*d^6*x^3+32*a*d*e^4*x^4-16/5*e^2*(-8*a*e^3+d^4)*x^5-8/3*d^3*e^3*x^6+64/7*d^2*e
^4*x^7+16*d*e^5*x^8+64/9*e^6*x^9

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2086} \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^2 e^4 x-\frac {16}{5} e^2 x^5 \left (d^4-8 a e^3\right )-8 a d^3 e^2 x^2+32 a d e^4 x^4+\frac {d^6 x^3}{3}-\frac {8}{3} d^3 e^3 x^6+\frac {64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac {64 e^6 x^9}{9} \]

[In]

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]

[Out]

64*a^2*e^4*x - 8*a*d^3*e^2*x^2 + (d^6*x^3)/3 + 32*a*d*e^4*x^4 - (16*e^2*(d^4 - 8*a*e^3)*x^5)/5 - (8*d^3*e^3*x^
6)/3 + (64*d^2*e^4*x^7)/7 + 16*d*e^5*x^8 + (64*e^6*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 a^2 e^4-16 a d^3 e^2 x+d^6 x^2+128 a d e^4 x^3-16 e^2 \left (d^4-8 a e^3\right ) x^4-16 d^3 e^3 x^5+64 d^2 e^4 x^6+128 d e^5 x^7+64 e^6 x^8\right ) \, dx \\ & = 64 a^2 e^4 x-8 a d^3 e^2 x^2+\frac {d^6 x^3}{3}+32 a d e^4 x^4-\frac {16}{5} e^2 \left (d^4-8 a e^3\right ) x^5-\frac {8}{3} d^3 e^3 x^6+\frac {64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac {64 e^6 x^9}{9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^2 e^4 x-8 a d^3 e^2 x^2+\frac {d^6 x^3}{3}+32 a d e^4 x^4+\frac {16}{5} e^2 \left (-d^4+8 a e^3\right ) x^5-\frac {8}{3} d^3 e^3 x^6+\frac {64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac {64 e^6 x^9}{9} \]

[In]

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]

[Out]

64*a^2*e^4*x - 8*a*d^3*e^2*x^2 + (d^6*x^3)/3 + 32*a*d*e^4*x^4 + (16*e^2*(-d^4 + 8*a*e^3)*x^5)/5 - (8*d^3*e^3*x
^6)/3 + (64*d^2*e^4*x^7)/7 + 16*d*e^5*x^8 + (64*e^6*x^9)/9

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93

method	result	size
norman	\(\frac {64 e^{6} x^{9}}{9}+16 d \,e^{5} x^{8}+\frac {64 d^{2} e^{4} x^{7}}{7}-\frac {8 d^{3} e^{3} x^{6}}{3}+\left (\frac {128}{5} a \,e^{5}-\frac {16}{5} d^{4} e^{2}\right ) x^{5}+32 a d \,e^{4} x^{4}+\frac {d^{6} x^{3}}{3}-8 a \,d^{3} e^{2} x^{2}+64 a^{2} e^{4} x\)	\(99\)
gosper	\(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a d \,e^{4} x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,d^{3} e^{2} x^{2}+64 a^{2} e^{4} x\)	\(100\)
default	\(\frac {64 e^{6} x^{9}}{9}+16 d \,e^{5} x^{8}+\frac {64 d^{2} e^{4} x^{7}}{7}-\frac {8 d^{3} e^{3} x^{6}}{3}+\frac {\left (128 a \,e^{5}-16 d^{4} e^{2}\right ) x^{5}}{5}+32 a d \,e^{4} x^{4}+\frac {d^{6} x^{3}}{3}-8 a \,d^{3} e^{2} x^{2}+64 a^{2} e^{4} x\)	\(100\)
risch	\(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a d \,e^{4} x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,d^{3} e^{2} x^{2}+64 a^{2} e^{4} x\)	\(100\)
parallelrisch	\(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a d \,e^{4} x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,d^{3} e^{2} x^{2}+64 a^{2} e^{4} x\)	\(100\)

[In]

int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x,method=_RETURNVERBOSE)

[Out]

64/9*e^6*x^9+16*d*e^5*x^8+64/7*d^2*e^4*x^7-8/3*d^3*e^3*x^6+(128/5*a*e^5-16/5*d^4*e^2)*x^5+32*a*d*e^4*x^4+1/3*d
^6*x^3-8*a*d^3*e^2*x^2+64*a^2*e^4*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} - \frac {8}{3} \, d^{3} e^{3} x^{6} + 32 \, a d e^{4} x^{4} + \frac {1}{3} \, d^{6} x^{3} - 8 \, a d^{3} e^{2} x^{2} + 64 \, a^{2} e^{4} x - \frac {16}{5} \, {\left (d^{4} e^{2} - 8 \, a e^{5}\right )} x^{5} \]

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="fricas")

[Out]

64/9*e^6*x^9 + 16*d*e^5*x^8 + 64/7*d^2*e^4*x^7 - 8/3*d^3*e^3*x^6 + 32*a*d*e^4*x^4 + 1/3*d^6*x^3 - 8*a*d^3*e^2*
x^2 + 64*a^2*e^4*x - 16/5*(d^4*e^2 - 8*a*e^5)*x^5

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^{2} e^{4} x - 8 a d^{3} e^{2} x^{2} + 32 a d e^{4} x^{4} + \frac {d^{6} x^{3}}{3} - \frac {8 d^{3} e^{3} x^{6}}{3} + \frac {64 d^{2} e^{4} x^{7}}{7} + 16 d e^{5} x^{8} + \frac {64 e^{6} x^{9}}{9} + x^{5} \cdot \left (\frac {128 a e^{5}}{5} - \frac {16 d^{4} e^{2}}{5}\right ) \]

[In]

integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**2,x)

[Out]

64*a**2*e**4*x - 8*a*d**3*e**2*x**2 + 32*a*d*e**4*x**4 + d**6*x**3/3 - 8*d**3*e**3*x**6/3 + 64*d**2*e**4*x**7/
7 + 16*d*e**5*x**8 + 64*e**6*x**9/9 + x**5*(128*a*e**5/5 - 16*d**4*e**2/5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} + \frac {1}{3} \, d^{6} x^{3} + 64 \, a^{2} e^{4} x - \frac {8}{15} \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3} + \frac {8}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a e^{2} \]

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="maxima")

[Out]

64/9*e^6*x^9 + 16*d*e^5*x^8 + 64/7*d^2*e^4*x^7 + 1/3*d^6*x^3 + 64*a^2*e^4*x - 8/15*(5*e^3*x^6 + 6*d*e^2*x^5)*d
^3 + 8/5*(16*e^3*x^5 + 20*d*e^2*x^4 - 5*d^3*x^2)*a*e^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} - \frac {8}{3} \, d^{3} e^{3} x^{6} - \frac {16}{5} \, d^{4} e^{2} x^{5} + \frac {128}{5} \, a e^{5} x^{5} + 32 \, a d e^{4} x^{4} + \frac {1}{3} \, d^{6} x^{3} - 8 \, a d^{3} e^{2} x^{2} + 64 \, a^{2} e^{4} x \]

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")

[Out]

64/9*e^6*x^9 + 16*d*e^5*x^8 + 64/7*d^2*e^4*x^7 - 8/3*d^3*e^3*x^6 - 16/5*d^4*e^2*x^5 + 128/5*a*e^5*x^5 + 32*a*d
*e^4*x^4 + 1/3*d^6*x^3 - 8*a*d^3*e^2*x^2 + 64*a^2*e^4*x

Mupad [B] (verification not implemented)

Time = 10.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=x^5\,\left (\frac {128\,a\,e^5}{5}-\frac {16\,d^4\,e^2}{5}\right )+\frac {d^6\,x^3}{3}+\frac {64\,e^6\,x^9}{9}+64\,a^2\,e^4\,x+16\,d\,e^5\,x^8-\frac {8\,d^3\,e^3\,x^6}{3}+\frac {64\,d^2\,e^4\,x^7}{7}-8\,a\,d^3\,e^2\,x^2+32\,a\,d\,e^4\,x^4 \]

[In]

int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^2,x)

[Out]

x^5*((128*a*e^5)/5 - (16*d^4*e^2)/5) + (d^6*x^3)/3 + (64*e^6*x^9)/9 + 64*a^2*e^4*x + 16*d*e^5*x^8 - (8*d^3*e^3
*x^6)/3 + (64*d^2*e^4*x^7)/7 - 8*a*d^3*e^2*x^2 + 32*a*d*e^4*x^4