Integrand size = 22, antiderivative size = 104 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=a^2 x+a b x^2+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {1}{2} b (a+c) x^4+\frac {1}{5} \left (2 a^2+2 b^2+c^2\right ) x^5+\frac {1}{3} b (a+c) x^6+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {1}{4} a b x^8+\frac {a^2 x^9}{9} \] Output:
a^2*x+a*b*x^2+1/3*(2*a*c+b^2)*x^3+1/2*b*(a+c)*x^4+1/5*(2*a^2+2*b^2+c^2)*x^ 5+1/3*b*(a+c)*x^6+1/7*(2*a*c+b^2)*x^7+1/4*a*b*x^8+1/9*a^2*x^9
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=a^2 x+a b x^2+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {1}{2} b (a+c) x^4+\frac {1}{5} \left (2 a^2+2 b^2+c^2\right ) x^5+\frac {1}{3} b (a+c) x^6+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {1}{4} a b x^8+\frac {a^2 x^9}{9} \] Input:
Integrate[(a + b*x + c*x^2 + b*x^3 + a*x^4)^2,x]
Output:
a^2*x + a*b*x^2 + ((b^2 + 2*a*c)*x^3)/3 + (b*(a + c)*x^4)/2 + ((2*a^2 + 2* b^2 + c^2)*x^5)/5 + (b*(a + c)*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (a*b*x^8)/ 4 + (a^2*x^9)/9
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2465, 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.
\(\displaystyle \int \left (a x^4+a+b x^3+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 2465 |
\(\displaystyle \int \left (x^4 \left (2 a^2+2 b^2+c^2\right )+a^2 x^8+a^2+x^6 \left (2 a c+b^2\right )+x^2 \left (2 a c+b^2\right )+2 b x^5 (a+c)+2 b x^3 (a+c)+2 a b x^7+2 a b x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \left (2 a^2+2 b^2+c^2\right )+\frac {a^2 x^9}{9}+a^2 x+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {1}{3} x^3 \left (2 a c+b^2\right )+\frac {1}{3} b x^6 (a+c)+\frac {1}{2} b x^4 (a+c)+\frac {1}{4} a b x^8+a b x^2\) |
Input:
Int[(a + b*x + c*x^2 + b*x^3 + a*x^4)^2,x]
Output:
a^2*x + a*b*x^2 + ((b^2 + 2*a*c)*x^3)/3 + (b*(a + c)*x^4)/2 + ((2*a^2 + 2* b^2 + c^2)*x^5)/5 + (b*(a + c)*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (a*b*x^8)/ 4 + (a^2*x^9)/9
Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ [Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x ] && IGtQ[p, 0]
Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {a^{2} x^{9}}{9}+\frac {a b \,x^{8}}{4}+\frac {\left (2 a c +b^{2}\right ) x^{7}}{7}+\frac {\left (2 a b +2 b c \right ) x^{6}}{6}+\frac {\left (2 a^{2}+2 b^{2}+c^{2}\right ) x^{5}}{5}+\frac {\left (2 a b +2 b c \right ) x^{4}}{4}+\frac {\left (2 a c +b^{2}\right ) x^{3}}{3}+a b \,x^{2}+x \,a^{2}\) | \(101\) |
norman | \(\frac {a^{2} x^{9}}{9}+\frac {a b \,x^{8}}{4}+\left (\frac {2 a c}{7}+\frac {b^{2}}{7}\right ) x^{7}+\left (\frac {1}{3} a b +\frac {1}{3} b c \right ) x^{6}+\left (\frac {2 a^{2}}{5}+\frac {2 b^{2}}{5}+\frac {c^{2}}{5}\right ) x^{5}+\left (\frac {1}{2} a b +\frac {1}{2} b c \right ) x^{4}+\left (\frac {2 a c}{3}+\frac {b^{2}}{3}\right ) x^{3}+a b \,x^{2}+x \,a^{2}\) | \(102\) |
gosper | \(\frac {1}{9} a^{2} x^{9}+\frac {1}{4} a b \,x^{8}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} b c \,x^{6}+\frac {2}{5} a^{2} x^{5}+\frac {2}{5} b^{2} x^{5}+\frac {1}{5} c^{2} x^{5}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} b c \,x^{4}+\frac {2}{3} a \,x^{3} c +\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+x \,a^{2}\) | \(110\) |
risch | \(\frac {1}{9} a^{2} x^{9}+\frac {1}{4} a b \,x^{8}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} b c \,x^{6}+\frac {2}{5} a^{2} x^{5}+\frac {2}{5} b^{2} x^{5}+\frac {1}{5} c^{2} x^{5}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} b c \,x^{4}+\frac {2}{3} a \,x^{3} c +\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+x \,a^{2}\) | \(110\) |
parallelrisch | \(\frac {1}{9} a^{2} x^{9}+\frac {1}{4} a b \,x^{8}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} b c \,x^{6}+\frac {2}{5} a^{2} x^{5}+\frac {2}{5} b^{2} x^{5}+\frac {1}{5} c^{2} x^{5}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} b c \,x^{4}+\frac {2}{3} a \,x^{3} c +\frac {1}{3} b^{2} x^{3}+a b \,x^{2}+x \,a^{2}\) | \(110\) |
orering | \(\frac {x \left (140 a^{2} x^{8}+315 a b \,x^{7}+360 a c \,x^{6}+180 b^{2} x^{6}+420 a b \,x^{5}+420 b c \,x^{5}+504 a^{2} x^{4}+504 b^{2} x^{4}+252 c^{2} x^{4}+630 a \,x^{3} b +630 b c \,x^{3}+840 a c \,x^{2}+420 b^{2} x^{2}+1260 a b x +1260 a^{2}\right )}{1260}\) | \(112\) |
Input:
int((a*x^4+b*x^3+c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/9*a^2*x^9+1/4*a*b*x^8+1/7*(2*a*c+b^2)*x^7+1/6*(2*a*b+2*b*c)*x^6+1/5*(2*a ^2+2*b^2+c^2)*x^5+1/4*(2*a*b+2*b*c)*x^4+1/3*(2*a*c+b^2)*x^3+a*b*x^2+x*a^2
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} x^{9} + \frac {1}{4} \, a b x^{8} + \frac {1}{7} \, {\left (b^{2} + 2 \, a c\right )} x^{7} + \frac {1}{3} \, {\left (a b + b c\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a^{2} + 2 \, b^{2} + c^{2}\right )} x^{5} + \frac {1}{2} \, {\left (a b + b c\right )} x^{4} + a b x^{2} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} x^{3} + a^{2} x \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/9*a^2*x^9 + 1/4*a*b*x^8 + 1/7*(b^2 + 2*a*c)*x^7 + 1/3*(a*b + b*c)*x^6 + 1/5*(2*a^2 + 2*b^2 + c^2)*x^5 + 1/2*(a*b + b*c)*x^4 + a*b*x^2 + 1/3*(b^2 + 2*a*c)*x^3 + a^2*x
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=\frac {a^{2} x^{9}}{9} + a^{2} x + \frac {a b x^{8}}{4} + a b x^{2} + x^{7} \cdot \left (\frac {2 a c}{7} + \frac {b^{2}}{7}\right ) + x^{6} \left (\frac {a b}{3} + \frac {b c}{3}\right ) + x^{5} \cdot \left (\frac {2 a^{2}}{5} + \frac {2 b^{2}}{5} + \frac {c^{2}}{5}\right ) + x^{4} \left (\frac {a b}{2} + \frac {b c}{2}\right ) + x^{3} \cdot \left (\frac {2 a c}{3} + \frac {b^{2}}{3}\right ) \] Input:
integrate((a*x**4+b*x**3+c*x**2+b*x+a)**2,x)
Output:
a**2*x**9/9 + a**2*x + a*b*x**8/4 + a*b*x**2 + x**7*(2*a*c/7 + b**2/7) + x **6*(a*b/3 + b*c/3) + x**5*(2*a**2/5 + 2*b**2/5 + c**2/5) + x**4*(a*b/2 + b*c/2) + x**3*(2*a*c/3 + b**2/3)
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} x^{9} + \frac {1}{4} \, a b x^{8} + \frac {1}{7} \, b^{2} x^{7} + \frac {1}{5} \, c^{2} x^{5} + \frac {1}{3} \, b^{2} x^{3} + a^{2} x + \frac {1}{30} \, {\left (12 \, a x^{5} + 15 \, b x^{4} + 20 \, c x^{3} + 30 \, b x^{2}\right )} a + \frac {1}{30} \, {\left (10 \, a x^{6} + 12 \, b x^{5} + 15 \, c x^{4}\right )} b + \frac {1}{21} \, {\left (6 \, a x^{7} + 7 \, b x^{6}\right )} c \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/9*a^2*x^9 + 1/4*a*b*x^8 + 1/7*b^2*x^7 + 1/5*c^2*x^5 + 1/3*b^2*x^3 + a^2* x + 1/30*(12*a*x^5 + 15*b*x^4 + 20*c*x^3 + 30*b*x^2)*a + 1/30*(10*a*x^6 + 12*b*x^5 + 15*c*x^4)*b + 1/21*(6*a*x^7 + 7*b*x^6)*c
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.05 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=\frac {1}{9} \, a^{2} x^{9} + \frac {1}{4} \, a b x^{8} + \frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a c x^{7} + \frac {1}{3} \, a b x^{6} + \frac {1}{3} \, b c x^{6} + \frac {2}{5} \, a^{2} x^{5} + \frac {2}{5} \, b^{2} x^{5} + \frac {1}{5} \, c^{2} x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, b c x^{4} + \frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a c x^{3} + a b x^{2} + a^{2} x \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/9*a^2*x^9 + 1/4*a*b*x^8 + 1/7*b^2*x^7 + 2/7*a*c*x^7 + 1/3*a*b*x^6 + 1/3* b*c*x^6 + 2/5*a^2*x^5 + 2/5*b^2*x^5 + 1/5*c^2*x^5 + 1/2*a*b*x^4 + 1/2*b*c* x^4 + 1/3*b^2*x^3 + 2/3*a*c*x^3 + a*b*x^2 + a^2*x
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=a^2\,x+x^3\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )+x^7\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {a^2\,x^9}{9}+x^5\,\left (\frac {2\,a^2}{5}+\frac {2\,b^2}{5}+\frac {c^2}{5}\right )+\frac {b\,x^4\,\left (a+c\right )}{2}+\frac {b\,x^6\,\left (a+c\right )}{3}+a\,b\,x^2+\frac {a\,b\,x^8}{4} \] Input:
int((a + b*x + a*x^4 + b*x^3 + c*x^2)^2,x)
Output:
a^2*x + x^3*((2*a*c)/3 + b^2/3) + x^7*((2*a*c)/7 + b^2/7) + (a^2*x^9)/9 + x^5*((2*a^2)/5 + (2*b^2)/5 + c^2/5) + (b*x^4*(a + c))/2 + (b*x^6*(a + c))/ 3 + a*b*x^2 + (a*b*x^8)/4
Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^2 \, dx=\frac {x \left (140 a^{2} x^{8}+315 a b \,x^{7}+360 a c \,x^{6}+180 b^{2} x^{6}+420 a b \,x^{5}+420 b c \,x^{5}+504 a^{2} x^{4}+504 b^{2} x^{4}+252 c^{2} x^{4}+630 a b \,x^{3}+630 b c \,x^{3}+840 a c \,x^{2}+420 b^{2} x^{2}+1260 a b x +1260 a^{2}\right )}{1260} \] Input:
int((a*x^4+b*x^3+c*x^2+b*x+a)^2,x)
Output:
(x*(140*a**2*x**8 + 504*a**2*x**4 + 1260*a**2 + 315*a*b*x**7 + 420*a*b*x** 5 + 630*a*b*x**3 + 1260*a*b*x + 360*a*c*x**6 + 840*a*c*x**2 + 180*b**2*x** 6 + 504*b**2*x**4 + 420*b**2*x**2 + 420*b*c*x**5 + 630*b*c*x**3 + 252*c**2 *x**4))/1260