Integrand size = 23, antiderivative size = 110 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {3}{7} a^2 b d x^7+\frac {3}{8} a^2 b e x^8+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {1}{13} b^3 d x^{13}+\frac {1}{14} b^3 e x^{14}+\frac {c \left (a+b x^3\right )^4}{12 b} \]
1/4*a^3*d*x^4+1/5*a^3*e*x^5+3/7*a^2*b*d*x^7+3/8*a^2*b*e*x^8+3/10*a*b^2*d*x ^10+3/11*a*b^2*e*x^11+1/13*b^3*d*x^13+1/14*b^3*e*x^14+1/12*c*(b*x^3+a)^4/b
Time = 0.01 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.26 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {1}{3} a^3 c x^3+\frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {1}{2} a^2 b c x^6+\frac {3}{7} a^2 b d x^7+\frac {3}{8} a^2 b e x^8+\frac {1}{3} a b^2 c x^9+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {1}{12} b^3 c x^{12}+\frac {1}{13} b^3 d x^{13}+\frac {1}{14} b^3 e x^{14} \]
(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b *d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3 *a*b^2*e*x^11)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14
Time = 0.30 (sec) , antiderivative size = 110, 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.
\(\displaystyle \int x^2 \left (a+b x^3\right )^3 \left (c+d x+e x^2\right ) \, dx\) |
\(\Big \downarrow \) 2017 |
\(\displaystyle \int \left (b x^3+a\right )^3 \left (x^2 \left (e x^2+d x+c\right )-c x^2\right )dx+\frac {c \left (a+b x^3\right )^4}{12 b}\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (b^3 e x^{13}+b^3 d x^{12}+3 a b^2 e x^{10}+3 a b^2 d x^9+3 a^2 b e x^7+3 a^2 b d x^6+a^3 e x^4+a^3 d x^3\right )dx+\frac {c \left (a+b x^3\right )^4}{12 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {3}{7} a^2 b d x^7+\frac {3}{8} a^2 b e x^8+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {c \left (a+b x^3\right )^4}{12 b}+\frac {1}{13} b^3 d x^{13}+\frac {1}{14} b^3 e x^{14}\) |
(a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (3*a^2*b*d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (3 *a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)/11 + (b^3*d*x^13)/13 + (b^3*e*x^14)/1 4 + (c*(a + b*x^3)^4)/(12*b)
3.4.25.3.1 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 = 1.53 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(\frac {1}{3} c \,a^{3} x^{3}+\frac {1}{4} a^{3} d \,x^{4}+\frac {1}{5} a^{3} e \,x^{5}+\frac {1}{2} a^{2} b c \,x^{6}+\frac {3}{7} a^{2} b d \,x^{7}+\frac {3}{8} a^{2} b e \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{12} b^{3} c \,x^{12}+\frac {1}{13} b^{3} d \,x^{13}+\frac {1}{14} b^{3} e \,x^{14}\) | \(116\) |
default | \(\frac {1}{3} c \,a^{3} x^{3}+\frac {1}{4} a^{3} d \,x^{4}+\frac {1}{5} a^{3} e \,x^{5}+\frac {1}{2} a^{2} b c \,x^{6}+\frac {3}{7} a^{2} b d \,x^{7}+\frac {3}{8} a^{2} b e \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{12} b^{3} c \,x^{12}+\frac {1}{13} b^{3} d \,x^{13}+\frac {1}{14} b^{3} e \,x^{14}\) | \(116\) |
norman | \(\frac {1}{3} c \,a^{3} x^{3}+\frac {1}{4} a^{3} d \,x^{4}+\frac {1}{5} a^{3} e \,x^{5}+\frac {1}{2} a^{2} b c \,x^{6}+\frac {3}{7} a^{2} b d \,x^{7}+\frac {3}{8} a^{2} b e \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{12} b^{3} c \,x^{12}+\frac {1}{13} b^{3} d \,x^{13}+\frac {1}{14} b^{3} e \,x^{14}\) | \(116\) |
risch | \(\frac {1}{3} c \,a^{3} x^{3}+\frac {1}{4} a^{3} d \,x^{4}+\frac {1}{5} a^{3} e \,x^{5}+\frac {1}{2} a^{2} b c \,x^{6}+\frac {3}{7} a^{2} b d \,x^{7}+\frac {3}{8} a^{2} b e \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{12} b^{3} c \,x^{12}+\frac {1}{13} b^{3} d \,x^{13}+\frac {1}{14} b^{3} e \,x^{14}\) | \(116\) |
parallelrisch | \(\frac {1}{3} c \,a^{3} x^{3}+\frac {1}{4} a^{3} d \,x^{4}+\frac {1}{5} a^{3} e \,x^{5}+\frac {1}{2} a^{2} b c \,x^{6}+\frac {3}{7} a^{2} b d \,x^{7}+\frac {3}{8} a^{2} b e \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{12} b^{3} c \,x^{12}+\frac {1}{13} b^{3} d \,x^{13}+\frac {1}{14} b^{3} e \,x^{14}\) | \(116\) |
1/3*c*a^3*x^3+1/4*a^3*d*x^4+1/5*a^3*e*x^5+1/2*a^2*b*c*x^6+3/7*a^2*b*d*x^7+ 3/8*a^2*b*e*x^8+1/3*a*b^2*c*x^9+3/10*a*b^2*d*x^10+3/11*a*b^2*e*x^11+1/12*b ^3*c*x^12+1/13*b^3*d*x^13+1/14*b^3*e*x^14
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {1}{14} \, b^{3} e x^{14} + \frac {1}{13} \, b^{3} d x^{13} + \frac {1}{12} \, b^{3} c x^{12} + \frac {3}{11} \, a b^{2} e x^{11} + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b e x^{8} + \frac {3}{7} \, a^{2} b d x^{7} + \frac {1}{2} \, a^{2} b c x^{6} + \frac {1}{5} \, a^{3} e x^{5} + \frac {1}{4} \, a^{3} d x^{4} + \frac {1}{3} \, a^{3} c x^{3} \]
1/14*b^3*e*x^14 + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*e*x^11 + 3/10*a*b^2*d*x^10 + 1/3*a*b^2*c*x^9 + 3/8*a^2*b*e*x^8 + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*x^6 + 1/5*a^3*e*x^5 + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3
Time = 0.02 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {a^{3} c x^{3}}{3} + \frac {a^{3} d x^{4}}{4} + \frac {a^{3} e x^{5}}{5} + \frac {a^{2} b c x^{6}}{2} + \frac {3 a^{2} b d x^{7}}{7} + \frac {3 a^{2} b e x^{8}}{8} + \frac {a b^{2} c x^{9}}{3} + \frac {3 a b^{2} d x^{10}}{10} + \frac {3 a b^{2} e x^{11}}{11} + \frac {b^{3} c x^{12}}{12} + \frac {b^{3} d x^{13}}{13} + \frac {b^{3} e x^{14}}{14} \]
a**3*c*x**3/3 + a**3*d*x**4/4 + a**3*e*x**5/5 + a**2*b*c*x**6/2 + 3*a**2*b *d*x**7/7 + 3*a**2*b*e*x**8/8 + a*b**2*c*x**9/3 + 3*a*b**2*d*x**10/10 + 3* a*b**2*e*x**11/11 + b**3*c*x**12/12 + b**3*d*x**13/13 + b**3*e*x**14/14
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {1}{14} \, b^{3} e x^{14} + \frac {1}{13} \, b^{3} d x^{13} + \frac {1}{12} \, b^{3} c x^{12} + \frac {3}{11} \, a b^{2} e x^{11} + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b e x^{8} + \frac {3}{7} \, a^{2} b d x^{7} + \frac {1}{2} \, a^{2} b c x^{6} + \frac {1}{5} \, a^{3} e x^{5} + \frac {1}{4} \, a^{3} d x^{4} + \frac {1}{3} \, a^{3} c x^{3} \]
1/14*b^3*e*x^14 + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*e*x^11 + 3/10*a*b^2*d*x^10 + 1/3*a*b^2*c*x^9 + 3/8*a^2*b*e*x^8 + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*x^6 + 1/5*a^3*e*x^5 + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {1}{14} \, b^{3} e x^{14} + \frac {1}{13} \, b^{3} d x^{13} + \frac {1}{12} \, b^{3} c x^{12} + \frac {3}{11} \, a b^{2} e x^{11} + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b e x^{8} + \frac {3}{7} \, a^{2} b d x^{7} + \frac {1}{2} \, a^{2} b c x^{6} + \frac {1}{5} \, a^{3} e x^{5} + \frac {1}{4} \, a^{3} d x^{4} + \frac {1}{3} \, a^{3} c x^{3} \]
1/14*b^3*e*x^14 + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*e*x^11 + 3/10*a*b^2*d*x^10 + 1/3*a*b^2*c*x^9 + 3/8*a^2*b*e*x^8 + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*x^6 + 1/5*a^3*e*x^5 + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^3 \, dx=\frac {e\,a^3\,x^5}{5}+\frac {d\,a^3\,x^4}{4}+\frac {c\,a^3\,x^3}{3}+\frac {3\,e\,a^2\,b\,x^8}{8}+\frac {3\,d\,a^2\,b\,x^7}{7}+\frac {c\,a^2\,b\,x^6}{2}+\frac {3\,e\,a\,b^2\,x^{11}}{11}+\frac {3\,d\,a\,b^2\,x^{10}}{10}+\frac {c\,a\,b^2\,x^9}{3}+\frac {e\,b^3\,x^{14}}{14}+\frac {d\,b^3\,x^{13}}{13}+\frac {c\,b^3\,x^{12}}{12} \]