Integrand size = 38, antiderivative size = 163 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{8} a (2 b c+a f) x^8+\frac {1}{9} a (2 b d+a g) x^9+\frac {1}{10} a (2 b e+a h) x^{10}+\frac {1}{11} b (b c+2 a f) x^{11}+\frac {1}{12} b (b d+2 a g) x^{12}+\frac {1}{13} b (b e+2 a h) x^{13}+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \]
1/5*a^2*c*x^5+1/6*a^2*d*x^6+1/7*a^2*e*x^7+1/8*a*(a*f+2*b*c)*x^8+1/9*a*(a*g +2*b*d)*x^9+1/10*a*(a*h+2*b*e)*x^10+1/11*b*(2*a*f+b*c)*x^11+1/12*b*(2*a*g+ b*d)*x^12+1/13*b*(2*a*h+b*e)*x^13+1/14*b^2*f*x^14+1/15*b^2*g*x^15+1/16*b^2 *h*x^16
Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{8} a (2 b c+a f) x^8+\frac {1}{9} a (2 b d+a g) x^9+\frac {1}{10} a (2 b e+a h) x^{10}+\frac {1}{11} b (b c+2 a f) x^{11}+\frac {1}{12} b (b d+2 a g) x^{12}+\frac {1}{13} b (b e+2 a h) x^{13}+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \]
(a^2*c*x^5)/5 + (a^2*d*x^6)/6 + (a^2*e*x^7)/7 + (a*(2*b*c + a*f)*x^8)/8 + (a*(2*b*d + a*g)*x^9)/9 + (a*(2*b*e + a*h)*x^10)/10 + (b*(b*c + 2*a*f)*x^1 1)/11 + (b*(b*d + 2*a*g)*x^12)/12 + (b*(b*e + 2*a*h)*x^13)/13 + (b^2*f*x^1 4)/14 + (b^2*g*x^15)/15 + (b^2*h*x^16)/16
Time = 0.42 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2360, 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.
| \(\displaystyle \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\) |
| \(\Big \downarrow \) 2360 |
| \(\displaystyle \int \left (a^2 c x^4+a^2 d x^5+a^2 e x^6+b x^{10} (2 a f+b c)+a x^7 (a f+2 b c)+b x^{11} (2 a g+b d)+a x^8 (a g+2 b d)+b x^{12} (2 a h+b e)+a x^9 (a h+2 b e)+b^2 f x^{13}+b^2 g x^{14}+b^2 h x^{15}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{11} b x^{11} (2 a f+b c)+\frac {1}{8} a x^8 (a f+2 b c)+\frac {1}{12} b x^{12} (2 a g+b d)+\frac {1}{9} a x^9 (a g+2 b d)+\frac {1}{13} b x^{13} (2 a h+b e)+\frac {1}{10} a x^{10} (a h+2 b e)+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16}\) |
(a^2*c*x^5)/5 + (a^2*d*x^6)/6 + (a^2*e*x^7)/7 + (a*(2*b*c + a*f)*x^8)/8 + (a*(2*b*d + a*g)*x^9)/9 + (a*(2*b*e + a*h)*x^10)/10 + (b*(b*c + 2*a*f)*x^1 1)/11 + (b*(b*d + 2*a*g)*x^12)/12 + (b*(b*e + 2*a*h)*x^13)/13 + (b^2*f*x^1 4)/14 + (b^2*g*x^15)/15 + (b^2*h*x^16)/16
3.4.83.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
Time = 2.06 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {b^{2} h \,x^{16}}{16}+\frac {b^{2} g \,x^{15}}{15}+\frac {b^{2} f \,x^{14}}{14}+\frac {\left (2 a b h +b^{2} e \right ) x^{13}}{13}+\frac {\left (2 a b g +b^{2} d \right ) x^{12}}{12}+\frac {\left (2 a f b +b^{2} c \right ) x^{11}}{11}+\frac {\left (a^{2} h +2 a e b \right ) x^{10}}{10}+\frac {\left (a^{2} g +2 a b d \right ) x^{9}}{9}+\frac {\left (a^{2} f +2 a b c \right ) x^{8}}{8}+\frac {a^{2} e \,x^{7}}{7}+\frac {a^{2} d \,x^{6}}{6}+\frac {a^{2} c \,x^{5}}{5}\) | \(152\) |
| norman | \(\frac {a^{2} c \,x^{5}}{5}+\frac {a^{2} d \,x^{6}}{6}+\frac {a^{2} e \,x^{7}}{7}+\left (\frac {1}{8} a^{2} f +\frac {1}{4} a b c \right ) x^{8}+\left (\frac {1}{9} a^{2} g +\frac {2}{9} a b d \right ) x^{9}+\left (\frac {1}{10} a^{2} h +\frac {1}{5} a e b \right ) x^{10}+\left (\frac {2}{11} a f b +\frac {1}{11} b^{2} c \right ) x^{11}+\left (\frac {1}{6} a b g +\frac {1}{12} b^{2} d \right ) x^{12}+\left (\frac {2}{13} a b h +\frac {1}{13} b^{2} e \right ) x^{13}+\frac {b^{2} f \,x^{14}}{14}+\frac {b^{2} g \,x^{15}}{15}+\frac {b^{2} h \,x^{16}}{16}\) | \(152\) |
| gosper | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
| risch | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
| parallelrisch | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
1/16*b^2*h*x^16+1/15*b^2*g*x^15+1/14*b^2*f*x^14+1/13*(2*a*b*h+b^2*e)*x^13+ 1/12*(2*a*b*g+b^2*d)*x^12+1/11*(2*a*b*f+b^2*c)*x^11+1/10*(a^2*h+2*a*b*e)*x ^10+1/9*(a^2*g+2*a*b*d)*x^9+1/8*(a^2*f+2*a*b*c)*x^8+1/7*a^2*e*x^7+1/6*a^2* d*x^6+1/5*a^2*c*x^5
Time = 0.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, {\left (b^{2} e + 2 \, a b h\right )} x^{13} + \frac {1}{12} \, {\left (b^{2} d + 2 \, a b g\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} c + 2 \, a b f\right )} x^{11} + \frac {1}{10} \, {\left (2 \, a b e + a^{2} h\right )} x^{10} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{9} \, {\left (2 \, a b d + a^{2} g\right )} x^{9} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{8} \, {\left (2 \, a b c + a^{2} f\right )} x^{8} + \frac {1}{5} \, a^{2} c x^{5} \]
1/16*b^2*h*x^16 + 1/15*b^2*g*x^15 + 1/14*b^2*f*x^14 + 1/13*(b^2*e + 2*a*b* h)*x^13 + 1/12*(b^2*d + 2*a*b*g)*x^12 + 1/11*(b^2*c + 2*a*b*f)*x^11 + 1/10 *(2*a*b*e + a^2*h)*x^10 + 1/7*a^2*e*x^7 + 1/9*(2*a*b*d + a^2*g)*x^9 + 1/6* a^2*d*x^6 + 1/8*(2*a*b*c + a^2*f)*x^8 + 1/5*a^2*c*x^5
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a^{2} c x^{5}}{5} + \frac {a^{2} d x^{6}}{6} + \frac {a^{2} e x^{7}}{7} + \frac {b^{2} f x^{14}}{14} + \frac {b^{2} g x^{15}}{15} + \frac {b^{2} h x^{16}}{16} + x^{13} \cdot \left (\frac {2 a b h}{13} + \frac {b^{2} e}{13}\right ) + x^{12} \left (\frac {a b g}{6} + \frac {b^{2} d}{12}\right ) + x^{11} \cdot \left (\frac {2 a b f}{11} + \frac {b^{2} c}{11}\right ) + x^{10} \left (\frac {a^{2} h}{10} + \frac {a b e}{5}\right ) + x^{9} \left (\frac {a^{2} g}{9} + \frac {2 a b d}{9}\right ) + x^{8} \left (\frac {a^{2} f}{8} + \frac {a b c}{4}\right ) \]
a**2*c*x**5/5 + a**2*d*x**6/6 + a**2*e*x**7/7 + b**2*f*x**14/14 + b**2*g*x **15/15 + b**2*h*x**16/16 + x**13*(2*a*b*h/13 + b**2*e/13) + x**12*(a*b*g/ 6 + b**2*d/12) + x**11*(2*a*b*f/11 + b**2*c/11) + x**10*(a**2*h/10 + a*b*e /5) + x**9*(a**2*g/9 + 2*a*b*d/9) + x**8*(a**2*f/8 + a*b*c/4)
Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, {\left (b^{2} e + 2 \, a b h\right )} x^{13} + \frac {1}{12} \, {\left (b^{2} d + 2 \, a b g\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} c + 2 \, a b f\right )} x^{11} + \frac {1}{10} \, {\left (2 \, a b e + a^{2} h\right )} x^{10} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{9} \, {\left (2 \, a b d + a^{2} g\right )} x^{9} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{8} \, {\left (2 \, a b c + a^{2} f\right )} x^{8} + \frac {1}{5} \, a^{2} c x^{5} \]
1/16*b^2*h*x^16 + 1/15*b^2*g*x^15 + 1/14*b^2*f*x^14 + 1/13*(b^2*e + 2*a*b* h)*x^13 + 1/12*(b^2*d + 2*a*b*g)*x^12 + 1/11*(b^2*c + 2*a*b*f)*x^11 + 1/10 *(2*a*b*e + a^2*h)*x^10 + 1/7*a^2*e*x^7 + 1/9*(2*a*b*d + a^2*g)*x^9 + 1/6* a^2*d*x^6 + 1/8*(2*a*b*c + a^2*f)*x^8 + 1/5*a^2*c*x^5
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, b^{2} e x^{13} + \frac {2}{13} \, a b h x^{13} + \frac {1}{12} \, b^{2} d x^{12} + \frac {1}{6} \, a b g x^{12} + \frac {1}{11} \, b^{2} c x^{11} + \frac {2}{11} \, a b f x^{11} + \frac {1}{5} \, a b e x^{10} + \frac {1}{10} \, a^{2} h x^{10} + \frac {2}{9} \, a b d x^{9} + \frac {1}{9} \, a^{2} g x^{9} + \frac {1}{4} \, a b c x^{8} + \frac {1}{8} \, a^{2} f x^{8} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{5} \, a^{2} c x^{5} \]
1/16*b^2*h*x^16 + 1/15*b^2*g*x^15 + 1/14*b^2*f*x^14 + 1/13*b^2*e*x^13 + 2/ 13*a*b*h*x^13 + 1/12*b^2*d*x^12 + 1/6*a*b*g*x^12 + 1/11*b^2*c*x^11 + 2/11* a*b*f*x^11 + 1/5*a*b*e*x^10 + 1/10*a^2*h*x^10 + 2/9*a*b*d*x^9 + 1/9*a^2*g* x^9 + 1/4*a*b*c*x^8 + 1/8*a^2*f*x^8 + 1/7*a^2*e*x^7 + 1/6*a^2*d*x^6 + 1/5* a^2*c*x^5
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^8\,\left (\frac {f\,a^2}{8}+\frac {b\,c\,a}{4}\right )+x^{11}\,\left (\frac {c\,b^2}{11}+\frac {2\,a\,f\,b}{11}\right )+x^9\,\left (\frac {g\,a^2}{9}+\frac {2\,b\,d\,a}{9}\right )+x^{12}\,\left (\frac {d\,b^2}{12}+\frac {a\,g\,b}{6}\right )+x^{10}\,\left (\frac {h\,a^2}{10}+\frac {b\,e\,a}{5}\right )+x^{13}\,\left (\frac {e\,b^2}{13}+\frac {2\,a\,h\,b}{13}\right )+\frac {a^2\,c\,x^5}{5}+\frac {a^2\,d\,x^6}{6}+\frac {a^2\,e\,x^7}{7}+\frac {b^2\,f\,x^{14}}{14}+\frac {b^2\,g\,x^{15}}{15}+\frac {b^2\,h\,x^{16}}{16} \]
x^8*((a^2*f)/8 + (a*b*c)/4) + x^11*((b^2*c)/11 + (2*a*b*f)/11) + x^9*((a^2 *g)/9 + (2*a*b*d)/9) + x^12*((b^2*d)/12 + (a*b*g)/6) + x^10*((a^2*h)/10 + (a*b*e)/5) + x^13*((b^2*e)/13 + (2*a*b*h)/13) + (a^2*c*x^5)/5 + (a^2*d*x^6 )/6 + (a^2*e*x^7)/7 + (b^2*f*x^14)/14 + (b^2*g*x^15)/15 + (b^2*h*x^16)/16