Integrand size = 25, antiderivative size = 193 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}+\frac {f \left (a+b x^4\right )^5}{20 b} \] Output:
a^4*c*x+1/2*a^4*d*x^2+1/3*a^4*e*x^3+4/5*a^3*b*c*x^5+2/3*a^3*b*d*x^6+4/7*a^ 3*b*e*x^7+2/3*a^2*b^2*c*x^9+3/5*a^2*b^2*d*x^10+6/11*a^2*b^2*e*x^11+4/13*a* b^3*c*x^13+2/7*a*b^3*d*x^14+4/15*a*b^3*e*x^15+1/17*b^4*c*x^17+1/18*b^4*d*x ^18+1/19*b^4*e*x^19+1/20*f*(b*x^4+a)^5/b
Time = 0.01 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.22 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {1}{4} a^4 f x^4+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {1}{2} a^3 b f x^8+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {1}{2} a^2 b^2 f x^{12}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {1}{4} a b^3 f x^{16}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}+\frac {1}{20} b^4 f x^{20} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
Output:
a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + (a^4*f*x^4)/4 + (4*a^3*b*c*x^5)/ 5 + (2*a^3*b*d*x^6)/3 + (4*a^3*b*e*x^7)/7 + (a^3*b*f*x^8)/2 + (2*a^2*b^2*c *x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (a^2*b^2*f*x^12)/ 2 + (4*a*b^3*c*x^13)/13 + (2*a*b^3*d*x^14)/7 + (4*a*b^3*e*x^15)/15 + (a*b^ 3*f*x^16)/4 + (b^4*c*x^17)/17 + (b^4*d*x^18)/18 + (b^4*e*x^19)/19 + (b^4*f *x^20)/20
Time = 0.71 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2017, 2188, 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.
| \(\displaystyle \int \left (a+b x^4\right )^4 \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (e x^2+d x+c\right ) \left (b x^4+a\right )^4dx+\frac {f \left (a+b x^4\right )^5}{20 b}\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \int \left (b^4 e x^{18}+b^4 d x^{17}+b^4 c x^{16}+4 a b^3 e x^{14}+4 a b^3 d x^{13}+4 a b^3 c x^{12}+6 a^2 b^2 e x^{10}+6 a^2 b^2 d x^9+6 a^2 b^2 c x^8+4 a^3 b e x^6+4 a^3 b d x^5+4 a^3 b c x^4+a^4 e x^2+a^4 d x+a^4 c\right )dx+\frac {f \left (a+b x^4\right )^5}{20 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {f \left (a+b x^4\right )^5}{20 b}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
Output:
a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + (4*a^3*b*c*x^5)/5 + (2*a^3*b*d*x ^6)/3 + (4*a^3*b*e*x^7)/7 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + ( 6*a^2*b^2*e*x^11)/11 + (4*a*b^3*c*x^13)/13 + (2*a*b^3*d*x^14)/7 + (4*a*b^3 *e*x^15)/15 + (b^4*c*x^17)/17 + (b^4*d*x^18)/18 + (b^4*e*x^19)/19 + (f*(a + b*x^4)^5)/(20*b)
Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coeff[Px, x, n - 1] *x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p , 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ [{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03
| method | result | size |
| gosper | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} e \,a^{4} x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} c \,a^{2} b^{2} x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
| default | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} e \,a^{4} x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} c \,a^{2} b^{2} x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
| norman | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} e \,a^{4} x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} c \,a^{2} b^{2} x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
| risch | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} e \,a^{4} x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} c \,a^{2} b^{2} x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
| parallelrisch | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} e \,a^{4} x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} c \,a^{2} b^{2} x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
| orering | \(\frac {x \left (2909907 f \,b^{4} x^{19}+3063060 b^{4} e \,x^{18}+3233230 d \,b^{4} x^{17}+3423420 b^{4} c \,x^{16}+14549535 f a \,b^{3} x^{15}+15519504 a \,b^{3} e \,x^{14}+16628040 a \,b^{3} d \,x^{13}+17907120 a \,b^{3} c \,x^{12}+29099070 f \,a^{2} b^{2} x^{11}+31744440 a^{2} b^{2} e \,x^{10}+34918884 a^{2} b^{2} d \,x^{9}+38798760 a^{2} b^{2} c \,x^{8}+29099070 a^{3} b f \,x^{7}+33256080 a^{3} e b \,x^{6}+38798760 a^{3} b d \,x^{5}+46558512 a^{3} b c \,x^{4}+14549535 a^{4} f \,x^{3}+19399380 a^{4} e \,x^{2}+29099070 a^{4} d x +58198140 c \,a^{4}\right )}{58198140}\) | \(200\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x,method=_RETURNVERBOSE)
Output:
a^4*c*x+1/2*a^4*d*x^2+1/3*e*a^4*x^3+1/4*a^4*f*x^4+4/5*a^3*b*c*x^5+2/3*a^3* b*d*x^6+4/7*a^3*b*e*x^7+1/2*a^3*b*f*x^8+2/3*c*a^2*b^2*x^9+3/5*a^2*b^2*d*x^ 10+6/11*a^2*b^2*e*x^11+1/2*f*a^2*b^2*x^12+4/13*a*b^3*c*x^13+2/7*a*b^3*d*x^ 14+4/15*a*b^3*e*x^15+1/4*f*a*b^3*x^16+1/17*b^4*c*x^17+1/18*b^4*d*x^18+1/19 *b^4*e*x^19+1/20*f*b^4*x^20
Time = 0.06 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} e x^{19} + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} e x^{15} + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b e x^{7} + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="fricas")
Output:
1/20*b^4*f*x^20 + 1/19*b^4*e*x^19 + 1/18*b^4*d*x^18 + 1/17*b^4*c*x^17 + 1/ 4*a*b^3*f*x^16 + 4/15*a*b^3*e*x^15 + 2/7*a*b^3*d*x^14 + 4/13*a*b^3*c*x^13 + 1/2*a^2*b^2*f*x^12 + 6/11*a^2*b^2*e*x^11 + 3/5*a^2*b^2*d*x^10 + 2/3*a^2* b^2*c*x^9 + 1/2*a^3*b*f*x^8 + 4/7*a^3*b*e*x^7 + 2/3*a^3*b*d*x^6 + 4/5*a^3* b*c*x^5 + 1/4*a^4*f*x^4 + 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x
Time = 0.03 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.25 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=a^{4} c x + \frac {a^{4} d x^{2}}{2} + \frac {a^{4} e x^{3}}{3} + \frac {a^{4} f x^{4}}{4} + \frac {4 a^{3} b c x^{5}}{5} + \frac {2 a^{3} b d x^{6}}{3} + \frac {4 a^{3} b e x^{7}}{7} + \frac {a^{3} b f x^{8}}{2} + \frac {2 a^{2} b^{2} c x^{9}}{3} + \frac {3 a^{2} b^{2} d x^{10}}{5} + \frac {6 a^{2} b^{2} e x^{11}}{11} + \frac {a^{2} b^{2} f x^{12}}{2} + \frac {4 a b^{3} c x^{13}}{13} + \frac {2 a b^{3} d x^{14}}{7} + \frac {4 a b^{3} e x^{15}}{15} + \frac {a b^{3} f x^{16}}{4} + \frac {b^{4} c x^{17}}{17} + \frac {b^{4} d x^{18}}{18} + \frac {b^{4} e x^{19}}{19} + \frac {b^{4} f x^{20}}{20} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**4,x)
Output:
a**4*c*x + a**4*d*x**2/2 + a**4*e*x**3/3 + a**4*f*x**4/4 + 4*a**3*b*c*x**5 /5 + 2*a**3*b*d*x**6/3 + 4*a**3*b*e*x**7/7 + a**3*b*f*x**8/2 + 2*a**2*b**2 *c*x**9/3 + 3*a**2*b**2*d*x**10/5 + 6*a**2*b**2*e*x**11/11 + a**2*b**2*f*x **12/2 + 4*a*b**3*c*x**13/13 + 2*a*b**3*d*x**14/7 + 4*a*b**3*e*x**15/15 + a*b**3*f*x**16/4 + b**4*c*x**17/17 + b**4*d*x**18/18 + b**4*e*x**19/19 + b **4*f*x**20/20
Time = 0.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} e x^{19} + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} e x^{15} + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b e x^{7} + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="maxima")
Output:
1/20*b^4*f*x^20 + 1/19*b^4*e*x^19 + 1/18*b^4*d*x^18 + 1/17*b^4*c*x^17 + 1/ 4*a*b^3*f*x^16 + 4/15*a*b^3*e*x^15 + 2/7*a*b^3*d*x^14 + 4/13*a*b^3*c*x^13 + 1/2*a^2*b^2*f*x^12 + 6/11*a^2*b^2*e*x^11 + 3/5*a^2*b^2*d*x^10 + 2/3*a^2* b^2*c*x^9 + 1/2*a^3*b*f*x^8 + 4/7*a^3*b*e*x^7 + 2/3*a^3*b*d*x^6 + 4/5*a^3* b*c*x^5 + 1/4*a^4*f*x^4 + 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x
Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} e x^{19} + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} e x^{15} + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b e x^{7} + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="giac")
Output:
1/20*b^4*f*x^20 + 1/19*b^4*e*x^19 + 1/18*b^4*d*x^18 + 1/17*b^4*c*x^17 + 1/ 4*a*b^3*f*x^16 + 4/15*a*b^3*e*x^15 + 2/7*a*b^3*d*x^14 + 4/13*a*b^3*c*x^13 + 1/2*a^2*b^2*f*x^12 + 6/11*a^2*b^2*e*x^11 + 3/5*a^2*b^2*d*x^10 + 2/3*a^2* b^2*c*x^9 + 1/2*a^3*b*f*x^8 + 4/7*a^3*b*e*x^7 + 2/3*a^3*b*d*x^6 + 4/5*a^3* b*c*x^5 + 1/4*a^4*f*x^4 + 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x
Time = 6.80 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {f\,a^4\,x^4}{4}+\frac {e\,a^4\,x^3}{3}+\frac {d\,a^4\,x^2}{2}+c\,a^4\,x+\frac {f\,a^3\,b\,x^8}{2}+\frac {4\,e\,a^3\,b\,x^7}{7}+\frac {2\,d\,a^3\,b\,x^6}{3}+\frac {4\,c\,a^3\,b\,x^5}{5}+\frac {f\,a^2\,b^2\,x^{12}}{2}+\frac {6\,e\,a^2\,b^2\,x^{11}}{11}+\frac {3\,d\,a^2\,b^2\,x^{10}}{5}+\frac {2\,c\,a^2\,b^2\,x^9}{3}+\frac {f\,a\,b^3\,x^{16}}{4}+\frac {4\,e\,a\,b^3\,x^{15}}{15}+\frac {2\,d\,a\,b^3\,x^{14}}{7}+\frac {4\,c\,a\,b^3\,x^{13}}{13}+\frac {f\,b^4\,x^{20}}{20}+\frac {e\,b^4\,x^{19}}{19}+\frac {d\,b^4\,x^{18}}{18}+\frac {c\,b^4\,x^{17}}{17} \] Input:
int((a + b*x^4)^4*(c + d*x + e*x^2 + f*x^3),x)
Output:
(a^4*d*x^2)/2 + (b^4*c*x^17)/17 + (a^4*e*x^3)/3 + (b^4*d*x^18)/18 + (a^4*f *x^4)/4 + (b^4*e*x^19)/19 + (b^4*f*x^20)/20 + a^4*c*x + (2*a^2*b^2*c*x^9)/ 3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (a^2*b^2*f*x^12)/2 + (4 *a^3*b*c*x^5)/5 + (4*a*b^3*c*x^13)/13 + (2*a^3*b*d*x^6)/3 + (2*a*b^3*d*x^1 4)/7 + (4*a^3*b*e*x^7)/7 + (4*a*b^3*e*x^15)/15 + (a^3*b*f*x^8)/2 + (a*b^3* f*x^16)/4
Time = 0.19 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {x \left (2909907 b^{4} f \,x^{19}+3063060 b^{4} e \,x^{18}+3233230 b^{4} d \,x^{17}+3423420 b^{4} c \,x^{16}+14549535 a \,b^{3} f \,x^{15}+15519504 a \,b^{3} e \,x^{14}+16628040 a \,b^{3} d \,x^{13}+17907120 a \,b^{3} c \,x^{12}+29099070 a^{2} b^{2} f \,x^{11}+31744440 a^{2} b^{2} e \,x^{10}+34918884 a^{2} b^{2} d \,x^{9}+38798760 a^{2} b^{2} c \,x^{8}+29099070 a^{3} b f \,x^{7}+33256080 a^{3} b e \,x^{6}+38798760 a^{3} b d \,x^{5}+46558512 a^{3} b c \,x^{4}+14549535 a^{4} f \,x^{3}+19399380 a^{4} e \,x^{2}+29099070 a^{4} d x +58198140 a^{4} c \right )}{58198140} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x)
Output:
(x*(58198140*a**4*c + 29099070*a**4*d*x + 19399380*a**4*e*x**2 + 14549535* a**4*f*x**3 + 46558512*a**3*b*c*x**4 + 38798760*a**3*b*d*x**5 + 33256080*a **3*b*e*x**6 + 29099070*a**3*b*f*x**7 + 38798760*a**2*b**2*c*x**8 + 349188 84*a**2*b**2*d*x**9 + 31744440*a**2*b**2*e*x**10 + 29099070*a**2*b**2*f*x* *11 + 17907120*a*b**3*c*x**12 + 16628040*a*b**3*d*x**13 + 15519504*a*b**3* e*x**14 + 14549535*a*b**3*f*x**15 + 3423420*b**4*c*x**16 + 3233230*b**4*d* x**17 + 3063060*b**4*e*x**18 + 2909907*b**4*f*x**19))/58198140