Integrand size = 32, antiderivative size = 113 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {\left (5 d^4+256 a e^3\right )^2 x}{1024 e^2}-\frac {d^2 \left (5 d^4+256 a e^3\right ) (d+4 e x)^3}{1024 e^3}+\frac {\left (23 d^4+256 a e^3\right ) (d+4 e x)^5}{10240 e^3}-\frac {3 d^2 (d+4 e x)^7}{7168 e^3}+\frac {(d+4 e x)^9}{36864 e^3} \] Output:
1/1024*(256*a*e^3+5*d^4)^2*x/e^2-1/1024*d^2*(256*a*e^3+5*d^4)*(4*e*x+d)^3/ e^3+1/10240*(256*a*e^3+23*d^4)*(4*e*x+d)^5/e^3-3/7168*d^2*(4*e*x+d)^7/e^3+ 1/36864*(4*e*x+d)^9/e^3
Time = 0.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^2 e^4 x-8 a d^3 e^2 x^2+\frac {d^6 x^3}{3}+32 a d e^4 x^4+\frac {16}{5} e^2 \left (-d^4+8 a e^3\right ) x^5-\frac {8}{3} d^3 e^3 x^6+\frac {64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac {64 e^6 x^9}{9} \] Input:
Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]
Output:
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
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2458, 1403, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \left (\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^2d\left (\frac {d}{4 e}+x\right )\) |
\(\Big \downarrow \) 1403 |
\(\displaystyle \int \left (9 d^4 e^2 \left (\frac {128 a e^3}{9 d^4}+\frac {23}{18}\right ) \left (\frac {d}{4 e}+x\right )^4+\frac {\left (256 a e^3+5 d^4\right )^2}{1024 e^2}-\frac {3}{16} d^2 \left (256 a e^3+5 d^4\right ) \left (\frac {d}{4 e}+x\right )^2-48 d^2 e^4 \left (\frac {d}{4 e}+x\right )^6+64 e^6 \left (\frac {d}{4 e}+x\right )^8\right )d\left (\frac {d}{4 e}+x\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} e^2 \left (256 a e^3+23 d^4\right ) \left (\frac {d}{4 e}+x\right )^5+\frac {\left (256 a e^3+5 d^4\right )^2 \left (\frac {d}{4 e}+x\right )}{1024 e^2}-\frac {1}{16} d^2 \left (256 a e^3+5 d^4\right ) \left (\frac {d}{4 e}+x\right )^3-\frac {48}{7} d^2 e^4 \left (\frac {d}{4 e}+x\right )^7+\frac {64}{9} e^6 \left (\frac {d}{4 e}+x\right )^9\) |
Input:
Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]
Output:
((5*d^4 + 256*a*e^3)^2*(d/(4*e) + x))/(1024*e^2) - (d^2*(5*d^4 + 256*a*e^3 )*(d/(4*e) + x)^3)/16 + (e^2*(23*d^4 + 256*a*e^3)*(d/(4*e) + x)^5)/10 - (4 8*d^2*e^4*(d/(4*e) + x)^7)/7 + (64*e^6*(d/(4*e) + x)^9)/9
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandInte grand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a *c, 0] && IGtQ[p, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {64 e^{6} x^{9}}{9}+16 d \,e^{5} x^{8}+\frac {64 d^{2} e^{4} x^{7}}{7}-\frac {8 d^{3} e^{3} x^{6}}{3}+\left (\frac {128}{5} a \,e^{5}-\frac {16}{5} d^{4} e^{2}\right ) x^{5}+32 a \,e^{4} d \,x^{4}+\frac {d^{6} x^{3}}{3}-8 a \,e^{2} d^{3} x^{2}+64 a^{2} e^{4} x\) | \(99\) |
gosper | \(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a \,e^{4} d \,x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,e^{2} d^{3} x^{2}+64 a^{2} e^{4} x\) | \(100\) |
default | \(\frac {64 e^{6} x^{9}}{9}+16 d \,e^{5} x^{8}+\frac {64 d^{2} e^{4} x^{7}}{7}-\frac {8 d^{3} e^{3} x^{6}}{3}+\frac {\left (128 a \,e^{5}-16 d^{4} e^{2}\right ) x^{5}}{5}+32 a \,e^{4} d \,x^{4}+\frac {d^{6} x^{3}}{3}-8 a \,e^{2} d^{3} x^{2}+64 a^{2} e^{4} x\) | \(100\) |
risch | \(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a \,e^{4} d \,x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,e^{2} d^{3} x^{2}+64 a^{2} e^{4} x\) | \(100\) |
parallelrisch | \(\frac {64}{9} e^{6} x^{9}+16 d \,e^{5} x^{8}+\frac {64}{7} d^{2} e^{4} x^{7}-\frac {8}{3} d^{3} e^{3} x^{6}+\frac {128}{5} x^{5} a \,e^{5}-\frac {16}{5} x^{5} d^{4} e^{2}+32 a \,e^{4} d \,x^{4}+\frac {1}{3} d^{6} x^{3}-8 a \,e^{2} d^{3} x^{2}+64 a^{2} e^{4} x\) | \(100\) |
orering | \(\frac {x \left (2240 e^{6} x^{8}+5040 d \,e^{5} x^{7}+2880 d^{2} e^{4} x^{6}-840 d^{3} e^{3} x^{5}+8064 a \,e^{5} x^{4}-1008 d^{4} e^{2} x^{4}+10080 a \,e^{4} d \,x^{3}+105 d^{6} x^{2}-2520 a \,e^{2} d^{3} x +20160 a^{2} e^{4}\right )}{315}\) | \(100\) |
Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x,method=_RETURNVERBOSE)
Output:
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*e^4*d*x^4+1/3*d^6*x^3-8*a*e^2*d^3*x^2+64*a^2*e^4*x
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} - \frac {8}{3} \, d^{3} e^{3} x^{6} + 32 \, a d e^{4} x^{4} + \frac {1}{3} \, d^{6} x^{3} - 8 \, a d^{3} e^{2} x^{2} + 64 \, a^{2} e^{4} x - \frac {16}{5} \, {\left (d^{4} e^{2} - 8 \, a e^{5}\right )} x^{5} \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="fricas")
Output:
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
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.99 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=64 a^{2} e^{4} x - 8 a d^{3} e^{2} x^{2} + 32 a d e^{4} x^{4} + \frac {d^{6} x^{3}}{3} - \frac {8 d^{3} e^{3} x^{6}}{3} + \frac {64 d^{2} e^{4} x^{7}}{7} + 16 d e^{5} x^{8} + \frac {64 e^{6} x^{9}}{9} + x^{5} \cdot \left (\frac {128 a e^{5}}{5} - \frac {16 d^{4} e^{2}}{5}\right ) \] Input:
integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**2,x)
Output:
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)
Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} + \frac {1}{3} \, d^{6} x^{3} + 64 \, a^{2} e^{4} x - \frac {8}{15} \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3} + \frac {8}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a e^{2} \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="maxima")
Output:
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
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac {64}{7} \, d^{2} e^{4} x^{7} - \frac {8}{3} \, d^{3} e^{3} x^{6} - \frac {16}{5} \, d^{4} e^{2} x^{5} + \frac {128}{5} \, a e^{5} x^{5} + 32 \, a d e^{4} x^{4} + \frac {1}{3} \, d^{6} x^{3} - 8 \, a d^{3} e^{2} x^{2} + 64 \, a^{2} e^{4} x \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")
Output:
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
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.87 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=x^5\,\left (\frac {128\,a\,e^5}{5}-\frac {16\,d^4\,e^2}{5}\right )+\frac {d^6\,x^3}{3}+\frac {64\,e^6\,x^9}{9}+64\,a^2\,e^4\,x+16\,d\,e^5\,x^8-\frac {8\,d^3\,e^3\,x^6}{3}+\frac {64\,d^2\,e^4\,x^7}{7}-8\,a\,d^3\,e^2\,x^2+32\,a\,d\,e^4\,x^4 \] Input:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^2,x)
Output:
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
Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx=\frac {x \left (2240 e^{6} x^{8}+5040 d \,e^{5} x^{7}+2880 d^{2} e^{4} x^{6}-840 d^{3} e^{3} x^{5}+8064 a \,e^{5} x^{4}-1008 d^{4} e^{2} x^{4}+10080 a d \,e^{4} x^{3}+105 d^{6} x^{2}-2520 a \,d^{3} e^{2} x +20160 a^{2} e^{4}\right )}{315} \] Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x)
Output:
(x*(20160*a**2*e**4 - 2520*a*d**3*e**2*x + 10080*a*d*e**4*x**3 + 8064*a*e* *5*x**4 + 105*d**6*x**2 - 1008*d**4*e**2*x**4 - 840*d**3*e**3*x**5 + 2880* d**2*e**4*x**6 + 5040*d*e**5*x**7 + 2240*e**6*x**8))/315