Integrand size = 28, antiderivative size = 198 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23}+\frac {c \left (a+b x^4\right )^5}{20 b} \] Output:
1/5*a^4*d*x^5+1/6*a^4*e*x^6+1/7*a^4*f*x^7+4/9*a^3*b*d*x^9+2/5*a^3*b*e*x^10 +4/11*a^3*b*f*x^11+6/13*a^2*b^2*d*x^13+3/7*a^2*b^2*e*x^14+2/5*a^2*b^2*f*x^ 15+4/17*a*b^3*d*x^17+2/9*a*b^3*e*x^18+4/19*a*b^3*f*x^19+1/21*b^4*d*x^21+1/ 22*b^4*e*x^22+1/23*b^4*f*x^23+1/20*c*(b*x^4+a)^5/b
Time = 0.00 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.22 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{4} a^4 c x^4+\frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {1}{2} a^3 b c x^8+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {1}{2} a^2 b^2 c x^{12}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {1}{4} a b^3 c x^{16}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {1}{20} b^4 c x^{20}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23} \] Input:
Integrate[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
Output:
(a^4*c*x^4)/4 + (a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (a^3*b*c*x ^8)/2 + (4*a^3*b*d*x^9)/9 + (2*a^3*b*e*x^10)/5 + (4*a^3*b*f*x^11)/11 + (a^ 2*b^2*c*x^12)/2 + (6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*b^ 2*f*x^15)/5 + (a*b^3*c*x^16)/4 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/9 + (4*a*b^3*f*x^19)/19 + (b^4*c*x^20)/20 + (b^4*d*x^21)/21 + (b^4*e*x^22)/2 2 + (b^4*f*x^23)/23
Time = 0.68 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2017, 2389, 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.
| \(\displaystyle \int x^3 \left (a+b x^4\right )^4 \left (c+d x+e x^2+f x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (b x^4+a\right )^4 \left (x^3 \left (f x^3+e x^2+d x+c\right )-c x^3\right )dx+\frac {c \left (a+b x^4\right )^5}{20 b}\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (b^4 f x^{22}+b^4 e x^{21}+b^4 d x^{20}+4 a b^3 f x^{18}+4 a b^3 e x^{17}+4 a b^3 d x^{16}+6 a^2 b^2 f x^{14}+6 a^2 b^2 e x^{13}+6 a^2 b^2 d x^{12}+4 a^3 b f x^{10}+4 a^3 b e x^9+4 a^3 b d x^8+a^4 f x^6+a^4 e x^5+a^4 d x^4\right )dx+\frac {c \left (a+b x^4\right )^5}{20 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {c \left (a+b x^4\right )^5}{20 b}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23}\) |
Input:
Int[x^3*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^4,x]
Output:
(a^4*d*x^5)/5 + (a^4*e*x^6)/6 + (a^4*f*x^7)/7 + (4*a^3*b*d*x^9)/9 + (2*a^3 *b*e*x^10)/5 + (4*a^3*b*f*x^11)/11 + (6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e* x^14)/7 + (2*a^2*b^2*f*x^15)/5 + (4*a*b^3*d*x^17)/17 + (2*a*b^3*e*x^18)/9 + (4*a*b^3*f*x^19)/19 + (b^4*d*x^21)/21 + (b^4*e*x^22)/22 + (b^4*f*x^23)/2 3 + (c*(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_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02
| method | result | size |
| gosper | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} b^{4} c \,x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
| default | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} b^{4} c \,x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
| norman | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} b^{4} c \,x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
| risch | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} b^{4} c \,x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
| parallelrisch | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} b^{4} c \,x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
| orering | \(\frac {x^{4} \left (58198140 f \,b^{4} x^{19}+60843510 b^{4} e \,x^{18}+63740820 d \,b^{4} x^{17}+66927861 b^{4} c \,x^{16}+281801520 f a \,b^{3} x^{15}+297457160 a \,b^{3} e \,x^{14}+314954640 a \,b^{3} d \,x^{13}+334639305 a \,b^{3} c \,x^{12}+535422888 f \,a^{2} b^{2} x^{11}+573667380 a^{2} b^{2} e \,x^{10}+617795640 a^{2} b^{2} d \,x^{9}+669278610 a^{2} b^{2} c \,x^{8}+486748080 a^{3} b f \,x^{7}+535422888 a^{3} e b \,x^{6}+594914320 a^{3} b d \,x^{5}+669278610 a^{3} b c \,x^{4}+191222460 a^{4} f \,x^{3}+223092870 a^{4} e \,x^{2}+267711444 a^{4} d x +334639305 c \,a^{4}\right )}{1338557220}\) | \(202\) |
Input:
int(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x,method=_RETURNVERBOSE)
Output:
1/4*c*a^4*x^4+1/5*a^4*d*x^5+1/6*a^4*e*x^6+1/7*a^4*f*x^7+1/2*c*a^3*b*x^8+4/ 9*a^3*b*d*x^9+2/5*a^3*b*e*x^10+4/11*a^3*b*f*x^11+1/2*c*a^2*b^2*x^12+6/13*a ^2*b^2*d*x^13+3/7*a^2*b^2*e*x^14+2/5*a^2*b^2*f*x^15+1/4*a*b^3*c*x^16+4/17* a*b^3*d*x^17+2/9*a*b^3*e*x^18+4/19*a*b^3*f*x^19+1/20*b^4*c*x^20+1/21*b^4*d *x^21+1/22*b^4*e*x^22+1/23*b^4*f*x^23
Time = 0.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} e x^{22} + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} e x^{18} + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} e x^{14} + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b e x^{10} + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} e x^{6} + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="fricas")
Output:
1/23*b^4*f*x^23 + 1/22*b^4*e*x^22 + 1/21*b^4*d*x^21 + 1/20*b^4*c*x^20 + 4/ 19*a*b^3*f*x^19 + 2/9*a*b^3*e*x^18 + 4/17*a*b^3*d*x^17 + 1/4*a*b^3*c*x^16 + 2/5*a^2*b^2*f*x^15 + 3/7*a^2*b^2*e*x^14 + 6/13*a^2*b^2*d*x^13 + 1/2*a^2* b^2*c*x^12 + 4/11*a^3*b*f*x^11 + 2/5*a^3*b*e*x^10 + 4/9*a^3*b*d*x^9 + 1/2* a^3*b*c*x^8 + 1/7*a^4*f*x^7 + 1/6*a^4*e*x^6 + 1/5*a^4*d*x^5 + 1/4*a^4*c*x^ 4
Time = 0.03 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.24 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {a^{4} c x^{4}}{4} + \frac {a^{4} d x^{5}}{5} + \frac {a^{4} e x^{6}}{6} + \frac {a^{4} f x^{7}}{7} + \frac {a^{3} b c x^{8}}{2} + \frac {4 a^{3} b d x^{9}}{9} + \frac {2 a^{3} b e x^{10}}{5} + \frac {4 a^{3} b f x^{11}}{11} + \frac {a^{2} b^{2} c x^{12}}{2} + \frac {6 a^{2} b^{2} d x^{13}}{13} + \frac {3 a^{2} b^{2} e x^{14}}{7} + \frac {2 a^{2} b^{2} f x^{15}}{5} + \frac {a b^{3} c x^{16}}{4} + \frac {4 a b^{3} d x^{17}}{17} + \frac {2 a b^{3} e x^{18}}{9} + \frac {4 a b^{3} f x^{19}}{19} + \frac {b^{4} c x^{20}}{20} + \frac {b^{4} d x^{21}}{21} + \frac {b^{4} e x^{22}}{22} + \frac {b^{4} f x^{23}}{23} \] Input:
integrate(x**3*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**4,x)
Output:
a**4*c*x**4/4 + a**4*d*x**5/5 + a**4*e*x**6/6 + a**4*f*x**7/7 + a**3*b*c*x **8/2 + 4*a**3*b*d*x**9/9 + 2*a**3*b*e*x**10/5 + 4*a**3*b*f*x**11/11 + a** 2*b**2*c*x**12/2 + 6*a**2*b**2*d*x**13/13 + 3*a**2*b**2*e*x**14/7 + 2*a**2 *b**2*f*x**15/5 + a*b**3*c*x**16/4 + 4*a*b**3*d*x**17/17 + 2*a*b**3*e*x**1 8/9 + 4*a*b**3*f*x**19/19 + b**4*c*x**20/20 + b**4*d*x**21/21 + b**4*e*x** 22/22 + b**4*f*x**23/23
Time = 0.03 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} e x^{22} + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} e x^{18} + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} e x^{14} + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b e x^{10} + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} e x^{6} + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="maxima")
Output:
1/23*b^4*f*x^23 + 1/22*b^4*e*x^22 + 1/21*b^4*d*x^21 + 1/20*b^4*c*x^20 + 4/ 19*a*b^3*f*x^19 + 2/9*a*b^3*e*x^18 + 4/17*a*b^3*d*x^17 + 1/4*a*b^3*c*x^16 + 2/5*a^2*b^2*f*x^15 + 3/7*a^2*b^2*e*x^14 + 6/13*a^2*b^2*d*x^13 + 1/2*a^2* b^2*c*x^12 + 4/11*a^3*b*f*x^11 + 2/5*a^3*b*e*x^10 + 4/9*a^3*b*d*x^9 + 1/2* a^3*b*c*x^8 + 1/7*a^4*f*x^7 + 1/6*a^4*e*x^6 + 1/5*a^4*d*x^5 + 1/4*a^4*c*x^ 4
Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} e x^{22} + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} e x^{18} + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} e x^{14} + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b e x^{10} + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} e x^{6} + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x, algorithm="giac")
Output:
1/23*b^4*f*x^23 + 1/22*b^4*e*x^22 + 1/21*b^4*d*x^21 + 1/20*b^4*c*x^20 + 4/ 19*a*b^3*f*x^19 + 2/9*a*b^3*e*x^18 + 4/17*a*b^3*d*x^17 + 1/4*a*b^3*c*x^16 + 2/5*a^2*b^2*f*x^15 + 3/7*a^2*b^2*e*x^14 + 6/13*a^2*b^2*d*x^13 + 1/2*a^2* b^2*c*x^12 + 4/11*a^3*b*f*x^11 + 2/5*a^3*b*e*x^10 + 4/9*a^3*b*d*x^9 + 1/2* a^3*b*c*x^8 + 1/7*a^4*f*x^7 + 1/6*a^4*e*x^6 + 1/5*a^4*d*x^5 + 1/4*a^4*c*x^ 4
Time = 0.39 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {f\,a^4\,x^7}{7}+\frac {e\,a^4\,x^6}{6}+\frac {d\,a^4\,x^5}{5}+\frac {c\,a^4\,x^4}{4}+\frac {4\,f\,a^3\,b\,x^{11}}{11}+\frac {2\,e\,a^3\,b\,x^{10}}{5}+\frac {4\,d\,a^3\,b\,x^9}{9}+\frac {c\,a^3\,b\,x^8}{2}+\frac {2\,f\,a^2\,b^2\,x^{15}}{5}+\frac {3\,e\,a^2\,b^2\,x^{14}}{7}+\frac {6\,d\,a^2\,b^2\,x^{13}}{13}+\frac {c\,a^2\,b^2\,x^{12}}{2}+\frac {4\,f\,a\,b^3\,x^{19}}{19}+\frac {2\,e\,a\,b^3\,x^{18}}{9}+\frac {4\,d\,a\,b^3\,x^{17}}{17}+\frac {c\,a\,b^3\,x^{16}}{4}+\frac {f\,b^4\,x^{23}}{23}+\frac {e\,b^4\,x^{22}}{22}+\frac {d\,b^4\,x^{21}}{21}+\frac {c\,b^4\,x^{20}}{20} \] Input:
int(x^3*(a + b*x^4)^4*(c + d*x + e*x^2 + f*x^3),x)
Output:
(a^4*c*x^4)/4 + (a^4*d*x^5)/5 + (b^4*c*x^20)/20 + (a^4*e*x^6)/6 + (b^4*d*x ^21)/21 + (a^4*f*x^7)/7 + (b^4*e*x^22)/22 + (b^4*f*x^23)/23 + (a^2*b^2*c*x ^12)/2 + (6*a^2*b^2*d*x^13)/13 + (3*a^2*b^2*e*x^14)/7 + (2*a^2*b^2*f*x^15) /5 + (a^3*b*c*x^8)/2 + (a*b^3*c*x^16)/4 + (4*a^3*b*d*x^9)/9 + (4*a*b^3*d*x ^17)/17 + (2*a^3*b*e*x^10)/5 + (2*a*b^3*e*x^18)/9 + (4*a^3*b*f*x^11)/11 + (4*a*b^3*f*x^19)/19
Time = 0.18 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx=\frac {x^{4} \left (58198140 b^{4} f \,x^{19}+60843510 b^{4} e \,x^{18}+63740820 b^{4} d \,x^{17}+66927861 b^{4} c \,x^{16}+281801520 a \,b^{3} f \,x^{15}+297457160 a \,b^{3} e \,x^{14}+314954640 a \,b^{3} d \,x^{13}+334639305 a \,b^{3} c \,x^{12}+535422888 a^{2} b^{2} f \,x^{11}+573667380 a^{2} b^{2} e \,x^{10}+617795640 a^{2} b^{2} d \,x^{9}+669278610 a^{2} b^{2} c \,x^{8}+486748080 a^{3} b f \,x^{7}+535422888 a^{3} b e \,x^{6}+594914320 a^{3} b d \,x^{5}+669278610 a^{3} b c \,x^{4}+191222460 a^{4} f \,x^{3}+223092870 a^{4} e \,x^{2}+267711444 a^{4} d x +334639305 a^{4} c \right )}{1338557220} \] Input:
int(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^4,x)
Output:
(x**4*(334639305*a**4*c + 267711444*a**4*d*x + 223092870*a**4*e*x**2 + 191 222460*a**4*f*x**3 + 669278610*a**3*b*c*x**4 + 594914320*a**3*b*d*x**5 + 5 35422888*a**3*b*e*x**6 + 486748080*a**3*b*f*x**7 + 669278610*a**2*b**2*c*x **8 + 617795640*a**2*b**2*d*x**9 + 573667380*a**2*b**2*e*x**10 + 535422888 *a**2*b**2*f*x**11 + 334639305*a*b**3*c*x**12 + 314954640*a*b**3*d*x**13 + 297457160*a*b**3*e*x**14 + 281801520*a*b**3*f*x**15 + 66927861*b**4*c*x** 16 + 63740820*b**4*d*x**17 + 60843510*b**4*e*x**18 + 58198140*b**4*f*x**19 ))/1338557220