Integrand size = 23, antiderivative size = 138 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{4} a^4 d x^4+\frac {1}{5} a^4 e x^5+\frac {4}{7} a^3 b d x^7+\frac {1}{2} a^3 b e x^8+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 d x^{13}+\frac {2}{7} a b^3 e x^{14}+\frac {1}{16} b^4 d x^{16}+\frac {1}{17} b^4 e x^{17}+\frac {c \left (a+b x^3\right )^5}{15 b} \]
1/4*a^4*d*x^4+1/5*a^4*e*x^5+4/7*a^3*b*d*x^7+1/2*a^3*b*e*x^8+3/5*a^2*b^2*d* x^10+6/11*a^2*b^2*e*x^11+4/13*a*b^3*d*x^13+2/7*a*b^3*e*x^14+1/16*b^4*d*x^1 6+1/17*b^4*e*x^17+1/15*c*(b*x^3+a)^5/b
Time = 0.01 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.31 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{3} a^4 c x^3+\frac {1}{4} a^4 d x^4+\frac {1}{5} a^4 e x^5+\frac {2}{3} a^3 b c x^6+\frac {4}{7} a^3 b d x^7+\frac {1}{2} a^3 b e x^8+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {1}{3} a b^3 c x^{12}+\frac {4}{13} a b^3 d x^{13}+\frac {2}{7} a b^3 e x^{14}+\frac {1}{15} b^4 c x^{15}+\frac {1}{16} b^4 d x^{16}+\frac {1}{17} b^4 e x^{17} \]
(a^4*c*x^3)/3 + (a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (2*a^3*b*c*x^6)/3 + (4*a^3 *b*d*x^7)/7 + (a^3*b*e*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*b^3*c*x^12)/3 + (4*a*b^3*d*x^13)/13 + (2*a*b ^3*e*x^14)/7 + (b^4*c*x^15)/15 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
| \(\displaystyle \int x^2 \left (a+b x^3\right )^4 \left (c+d x+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (b x^3+a\right )^4 \left (x^2 \left (e x^2+d x+c\right )-c x^2\right )dx+\frac {c \left (a+b x^3\right )^5}{15 b}\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (b^4 e x^{16}+b^4 d x^{15}+4 a b^3 e x^{13}+4 a b^3 d x^{12}+6 a^2 b^2 e x^{10}+6 a^2 b^2 d x^9+4 a^3 b e x^7+4 a^3 b d x^6+a^4 e x^4+a^4 d x^3\right )dx+\frac {c \left (a+b x^3\right )^5}{15 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^4 d x^4+\frac {1}{5} a^4 e x^5+\frac {4}{7} a^3 b d x^7+\frac {1}{2} a^3 b e x^8+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 d x^{13}+\frac {2}{7} a b^3 e x^{14}+\frac {c \left (a+b x^3\right )^5}{15 b}+\frac {1}{16} b^4 d x^{16}+\frac {1}{17} b^4 e x^{17}\) |
(a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (4*a^3*b*d*x^7)/7 + (a^3*b*e*x^8)/2 + (3*a ^2*b^2*d*x^10)/5 + (6*a^2*b^2*e*x^11)/11 + (4*a*b^3*d*x^13)/13 + (2*a*b^3* e*x^14)/7 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17 + (c*(a + b*x^3)^5)/(15*b)
3.4.31.3.1 Defintions of rubi rules used
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 = 1.62 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10
| method | result | size |
| gosper | \(\frac {1}{3} a^{4} c \,x^{3}+\frac {1}{4} a^{4} d \,x^{4}+\frac {1}{5} a^{4} e \,x^{5}+\frac {2}{3} a^{3} b c \,x^{6}+\frac {4}{7} a^{3} b d \,x^{7}+\frac {1}{2} a^{3} b e \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{3} a \,b^{3} c \,x^{12}+\frac {4}{13} a \,b^{3} d \,x^{13}+\frac {2}{7} a \,b^{3} e \,x^{14}+\frac {1}{15} b^{4} c \,x^{15}+\frac {1}{16} b^{4} d \,x^{16}+\frac {1}{17} b^{4} e \,x^{17}\) | \(152\) |
| default | \(\frac {1}{3} a^{4} c \,x^{3}+\frac {1}{4} a^{4} d \,x^{4}+\frac {1}{5} a^{4} e \,x^{5}+\frac {2}{3} a^{3} b c \,x^{6}+\frac {4}{7} a^{3} b d \,x^{7}+\frac {1}{2} a^{3} b e \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{3} a \,b^{3} c \,x^{12}+\frac {4}{13} a \,b^{3} d \,x^{13}+\frac {2}{7} a \,b^{3} e \,x^{14}+\frac {1}{15} b^{4} c \,x^{15}+\frac {1}{16} b^{4} d \,x^{16}+\frac {1}{17} b^{4} e \,x^{17}\) | \(152\) |
| norman | \(\frac {1}{3} a^{4} c \,x^{3}+\frac {1}{4} a^{4} d \,x^{4}+\frac {1}{5} a^{4} e \,x^{5}+\frac {2}{3} a^{3} b c \,x^{6}+\frac {4}{7} a^{3} b d \,x^{7}+\frac {1}{2} a^{3} b e \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{3} a \,b^{3} c \,x^{12}+\frac {4}{13} a \,b^{3} d \,x^{13}+\frac {2}{7} a \,b^{3} e \,x^{14}+\frac {1}{15} b^{4} c \,x^{15}+\frac {1}{16} b^{4} d \,x^{16}+\frac {1}{17} b^{4} e \,x^{17}\) | \(152\) |
| risch | \(\frac {1}{3} a^{4} c \,x^{3}+\frac {1}{4} a^{4} d \,x^{4}+\frac {1}{5} a^{4} e \,x^{5}+\frac {2}{3} a^{3} b c \,x^{6}+\frac {4}{7} a^{3} b d \,x^{7}+\frac {1}{2} a^{3} b e \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{3} a \,b^{3} c \,x^{12}+\frac {4}{13} a \,b^{3} d \,x^{13}+\frac {2}{7} a \,b^{3} e \,x^{14}+\frac {1}{15} b^{4} c \,x^{15}+\frac {1}{16} b^{4} d \,x^{16}+\frac {1}{17} b^{4} e \,x^{17}\) | \(152\) |
| parallelrisch | \(\frac {1}{3} a^{4} c \,x^{3}+\frac {1}{4} a^{4} d \,x^{4}+\frac {1}{5} a^{4} e \,x^{5}+\frac {2}{3} a^{3} b c \,x^{6}+\frac {4}{7} a^{3} b d \,x^{7}+\frac {1}{2} a^{3} b e \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{3} a \,b^{3} c \,x^{12}+\frac {4}{13} a \,b^{3} d \,x^{13}+\frac {2}{7} a \,b^{3} e \,x^{14}+\frac {1}{15} b^{4} c \,x^{15}+\frac {1}{16} b^{4} d \,x^{16}+\frac {1}{17} b^{4} e \,x^{17}\) | \(152\) |
1/3*a^4*c*x^3+1/4*a^4*d*x^4+1/5*a^4*e*x^5+2/3*a^3*b*c*x^6+4/7*a^3*b*d*x^7+ 1/2*a^3*b*e*x^8+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+1 /3*a*b^3*c*x^12+4/13*a*b^3*d*x^13+2/7*a*b^3*e*x^14+1/15*b^4*c*x^15+1/16*b^ 4*d*x^16+1/17*b^4*e*x^17
Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{17} \, b^{4} e x^{17} + \frac {1}{16} \, b^{4} d x^{16} + \frac {1}{15} \, b^{4} c x^{15} + \frac {2}{7} \, a b^{3} e x^{14} + \frac {4}{13} \, a b^{3} d x^{13} + \frac {1}{3} \, a b^{3} c x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b e x^{8} + \frac {4}{7} \, a^{3} b d x^{7} + \frac {2}{3} \, a^{3} b c x^{6} + \frac {1}{5} \, a^{4} e x^{5} + \frac {1}{4} \, a^{4} d x^{4} + \frac {1}{3} \, a^{4} c x^{3} \]
1/17*b^4*e*x^17 + 1/16*b^4*d*x^16 + 1/15*b^4*c*x^15 + 2/7*a*b^3*e*x^14 + 4 /13*a*b^3*d*x^13 + 1/3*a*b^3*c*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*e*x^8 + 4/7*a^3*b*d*x^7 + 2/3*a^3*b*c *x^6 + 1/5*a^4*e*x^5 + 1/4*a^4*d*x^4 + 1/3*a^4*c*x^3
Time = 0.02 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.33 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {a^{4} c x^{3}}{3} + \frac {a^{4} d x^{4}}{4} + \frac {a^{4} e x^{5}}{5} + \frac {2 a^{3} b c x^{6}}{3} + \frac {4 a^{3} b d x^{7}}{7} + \frac {a^{3} b e x^{8}}{2} + \frac {2 a^{2} b^{2} c x^{9}}{3} + \frac {3 a^{2} b^{2} d x^{10}}{5} + \frac {6 a^{2} b^{2} e x^{11}}{11} + \frac {a b^{3} c x^{12}}{3} + \frac {4 a b^{3} d x^{13}}{13} + \frac {2 a b^{3} e x^{14}}{7} + \frac {b^{4} c x^{15}}{15} + \frac {b^{4} d x^{16}}{16} + \frac {b^{4} e x^{17}}{17} \]
a**4*c*x**3/3 + a**4*d*x**4/4 + a**4*e*x**5/5 + 2*a**3*b*c*x**6/3 + 4*a**3 *b*d*x**7/7 + a**3*b*e*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*b**3*c*x**12/3 + 4*a*b**3*d*x**13/13 + 2*a *b**3*e*x**14/7 + b**4*c*x**15/15 + b**4*d*x**16/16 + b**4*e*x**17/17
Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{17} \, b^{4} e x^{17} + \frac {1}{16} \, b^{4} d x^{16} + \frac {1}{15} \, b^{4} c x^{15} + \frac {2}{7} \, a b^{3} e x^{14} + \frac {4}{13} \, a b^{3} d x^{13} + \frac {1}{3} \, a b^{3} c x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b e x^{8} + \frac {4}{7} \, a^{3} b d x^{7} + \frac {2}{3} \, a^{3} b c x^{6} + \frac {1}{5} \, a^{4} e x^{5} + \frac {1}{4} \, a^{4} d x^{4} + \frac {1}{3} \, a^{4} c x^{3} \]
1/17*b^4*e*x^17 + 1/16*b^4*d*x^16 + 1/15*b^4*c*x^15 + 2/7*a*b^3*e*x^14 + 4 /13*a*b^3*d*x^13 + 1/3*a*b^3*c*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*e*x^8 + 4/7*a^3*b*d*x^7 + 2/3*a^3*b*c *x^6 + 1/5*a^4*e*x^5 + 1/4*a^4*d*x^4 + 1/3*a^4*c*x^3
Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{17} \, b^{4} e x^{17} + \frac {1}{16} \, b^{4} d x^{16} + \frac {1}{15} \, b^{4} c x^{15} + \frac {2}{7} \, a b^{3} e x^{14} + \frac {4}{13} \, a b^{3} d x^{13} + \frac {1}{3} \, a b^{3} c x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b e x^{8} + \frac {4}{7} \, a^{3} b d x^{7} + \frac {2}{3} \, a^{3} b c x^{6} + \frac {1}{5} \, a^{4} e x^{5} + \frac {1}{4} \, a^{4} d x^{4} + \frac {1}{3} \, a^{4} c x^{3} \]
1/17*b^4*e*x^17 + 1/16*b^4*d*x^16 + 1/15*b^4*c*x^15 + 2/7*a*b^3*e*x^14 + 4 /13*a*b^3*d*x^13 + 1/3*a*b^3*c*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*e*x^8 + 4/7*a^3*b*d*x^7 + 2/3*a^3*b*c *x^6 + 1/5*a^4*e*x^5 + 1/4*a^4*d*x^4 + 1/3*a^4*c*x^3
Time = 9.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {e\,a^4\,x^5}{5}+\frac {d\,a^4\,x^4}{4}+\frac {c\,a^4\,x^3}{3}+\frac {e\,a^3\,b\,x^8}{2}+\frac {4\,d\,a^3\,b\,x^7}{7}+\frac {2\,c\,a^3\,b\,x^6}{3}+\frac {6\,e\,a^2\,b^2\,x^{11}}{11}+\frac {3\,d\,a^2\,b^2\,x^{10}}{5}+\frac {2\,c\,a^2\,b^2\,x^9}{3}+\frac {2\,e\,a\,b^3\,x^{14}}{7}+\frac {4\,d\,a\,b^3\,x^{13}}{13}+\frac {c\,a\,b^3\,x^{12}}{3}+\frac {e\,b^4\,x^{17}}{17}+\frac {d\,b^4\,x^{16}}{16}+\frac {c\,b^4\,x^{15}}{15} \]