Integrand size = 38, antiderivative size = 163 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a (2 b c+a f) x^7+\frac {1}{8} a (2 b d+a g) x^8+\frac {1}{9} a (2 b e+a h) x^9+\frac {1}{10} b (b c+2 a f) x^{10}+\frac {1}{11} b (b d+2 a g) x^{11}+\frac {1}{12} b (b e+2 a h) x^{12}+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15} \]
1/4*a^2*c*x^4+1/5*a^2*d*x^5+1/6*a^2*e*x^6+1/7*a*(a*f+2*b*c)*x^7+1/8*a*(a*g +2*b*d)*x^8+1/9*a*(a*h+2*b*e)*x^9+1/10*b*(2*a*f+b*c)*x^10+1/11*b*(2*a*g+b* d)*x^11+1/12*b*(2*a*h+b*e)*x^12+1/13*b^2*f*x^13+1/14*b^2*g*x^14+1/15*b^2*h *x^15
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a (2 b c+a f) x^7+\frac {1}{8} a (2 b d+a g) x^8+\frac {1}{9} a (2 b e+a h) x^9+\frac {1}{10} b (b c+2 a f) x^{10}+\frac {1}{11} b (b d+2 a g) x^{11}+\frac {1}{12} b (b e+2 a h) x^{12}+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15} \]
(a^2*c*x^4)/4 + (a^2*d*x^5)/5 + (a^2*e*x^6)/6 + (a*(2*b*c + a*f)*x^7)/7 + (a*(2*b*d + a*g)*x^8)/8 + (a*(2*b*e + a*h)*x^9)/9 + (b*(b*c + 2*a*f)*x^10) /10 + (b*(b*d + 2*a*g)*x^11)/11 + (b*(b*e + 2*a*h)*x^12)/12 + (b^2*f*x^13) /13 + (b^2*g*x^14)/14 + (b^2*h*x^15)/15
Time = 0.39 (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.
| \(\displaystyle \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\) |
| \(\Big \downarrow \) 2360 |
| \(\displaystyle \int \left (a^2 c x^3+a^2 d x^4+a^2 e x^5+b x^9 (2 a f+b c)+a x^6 (a f+2 b c)+b x^{10} (2 a g+b d)+a x^7 (a g+2 b d)+b x^{11} (2 a h+b e)+a x^8 (a h+2 b e)+b^2 f x^{12}+b^2 g x^{13}+b^2 h x^{14}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{10} b x^{10} (2 a f+b c)+\frac {1}{7} a x^7 (a f+2 b c)+\frac {1}{11} b x^{11} (2 a g+b d)+\frac {1}{8} a x^8 (a g+2 b d)+\frac {1}{12} b x^{12} (2 a h+b e)+\frac {1}{9} a x^9 (a h+2 b e)+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15}\) |
(a^2*c*x^4)/4 + (a^2*d*x^5)/5 + (a^2*e*x^6)/6 + (a*(2*b*c + a*f)*x^7)/7 + (a*(2*b*d + a*g)*x^8)/8 + (a*(2*b*e + a*h)*x^9)/9 + (b*(b*c + 2*a*f)*x^10) /10 + (b*(b*d + 2*a*g)*x^11)/11 + (b*(b*e + 2*a*h)*x^12)/12 + (b^2*f*x^13) /13 + (b^2*g*x^14)/14 + (b^2*h*x^15)/15
3.4.84.3.1 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 = 2.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {b^{2} h \,x^{15}}{15}+\frac {b^{2} g \,x^{14}}{14}+\frac {b^{2} f \,x^{13}}{13}+\frac {\left (2 a b h +b^{2} e \right ) x^{12}}{12}+\frac {\left (2 a b g +b^{2} d \right ) x^{11}}{11}+\frac {\left (2 a f b +b^{2} c \right ) x^{10}}{10}+\frac {\left (a^{2} h +2 a e b \right ) x^{9}}{9}+\frac {\left (a^{2} g +2 a b d \right ) x^{8}}{8}+\frac {\left (a^{2} f +2 a b c \right ) x^{7}}{7}+\frac {a^{2} e \,x^{6}}{6}+\frac {a^{2} d \,x^{5}}{5}+\frac {a^{2} c \,x^{4}}{4}\) | \(152\) |
| norman | \(\frac {a^{2} c \,x^{4}}{4}+\frac {a^{2} d \,x^{5}}{5}+\frac {a^{2} e \,x^{6}}{6}+\left (\frac {1}{7} a^{2} f +\frac {2}{7} a b c \right ) x^{7}+\left (\frac {1}{8} a^{2} g +\frac {1}{4} a b d \right ) x^{8}+\left (\frac {1}{9} a^{2} h +\frac {2}{9} a e b \right ) x^{9}+\left (\frac {1}{5} a f b +\frac {1}{10} b^{2} c \right ) x^{10}+\left (\frac {2}{11} a b g +\frac {1}{11} b^{2} d \right ) x^{11}+\left (\frac {1}{6} a b h +\frac {1}{12} b^{2} e \right ) x^{12}+\frac {b^{2} f \,x^{13}}{13}+\frac {b^{2} g \,x^{14}}{14}+\frac {b^{2} h \,x^{15}}{15}\) | \(152\) |
| gosper | \(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} a^{2} d \,x^{5}+\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {2}{7} x^{7} a b c +\frac {1}{8} x^{8} a^{2} g +\frac {1}{4} x^{8} a b d +\frac {1}{9} x^{9} a^{2} h +\frac {2}{9} x^{9} a e b +\frac {1}{5} x^{10} a f b +\frac {1}{10} x^{10} b^{2} c +\frac {2}{11} x^{11} a b g +\frac {1}{11} b^{2} d \,x^{11}+\frac {1}{6} x^{12} a b h +\frac {1}{12} x^{12} b^{2} e +\frac {1}{13} b^{2} f \,x^{13}+\frac {1}{14} b^{2} g \,x^{14}+\frac {1}{15} b^{2} h \,x^{15}\) | \(158\) |
| risch | \(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} a^{2} d \,x^{5}+\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {2}{7} x^{7} a b c +\frac {1}{8} x^{8} a^{2} g +\frac {1}{4} x^{8} a b d +\frac {1}{9} x^{9} a^{2} h +\frac {2}{9} x^{9} a e b +\frac {1}{5} x^{10} a f b +\frac {1}{10} x^{10} b^{2} c +\frac {2}{11} x^{11} a b g +\frac {1}{11} b^{2} d \,x^{11}+\frac {1}{6} x^{12} a b h +\frac {1}{12} x^{12} b^{2} e +\frac {1}{13} b^{2} f \,x^{13}+\frac {1}{14} b^{2} g \,x^{14}+\frac {1}{15} b^{2} h \,x^{15}\) | \(158\) |
| parallelrisch | \(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} a^{2} d \,x^{5}+\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {2}{7} x^{7} a b c +\frac {1}{8} x^{8} a^{2} g +\frac {1}{4} x^{8} a b d +\frac {1}{9} x^{9} a^{2} h +\frac {2}{9} x^{9} a e b +\frac {1}{5} x^{10} a f b +\frac {1}{10} x^{10} b^{2} c +\frac {2}{11} x^{11} a b g +\frac {1}{11} b^{2} d \,x^{11}+\frac {1}{6} x^{12} a b h +\frac {1}{12} x^{12} b^{2} e +\frac {1}{13} b^{2} f \,x^{13}+\frac {1}{14} b^{2} g \,x^{14}+\frac {1}{15} b^{2} h \,x^{15}\) | \(158\) |
1/15*b^2*h*x^15+1/14*b^2*g*x^14+1/13*b^2*f*x^13+1/12*(2*a*b*h+b^2*e)*x^12+ 1/11*(2*a*b*g+b^2*d)*x^11+1/10*(2*a*b*f+b^2*c)*x^10+1/9*(a^2*h+2*a*b*e)*x^ 9+1/8*(a^2*g+2*a*b*d)*x^8+1/7*(a^2*f+2*a*b*c)*x^7+1/6*a^2*e*x^6+1/5*a^2*d* x^5+1/4*a^2*c*x^4
Time = 0.33 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{15} \, b^{2} h x^{15} + \frac {1}{14} \, b^{2} g x^{14} + \frac {1}{13} \, b^{2} f x^{13} + \frac {1}{12} \, {\left (b^{2} e + 2 \, a b h\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} d + 2 \, a b g\right )} x^{11} + \frac {1}{10} \, {\left (b^{2} c + 2 \, a b f\right )} x^{10} + \frac {1}{9} \, {\left (2 \, a b e + a^{2} h\right )} x^{9} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{8} \, {\left (2 \, a b d + a^{2} g\right )} x^{8} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} f\right )} x^{7} + \frac {1}{4} \, a^{2} c x^{4} \]
1/15*b^2*h*x^15 + 1/14*b^2*g*x^14 + 1/13*b^2*f*x^13 + 1/12*(b^2*e + 2*a*b* h)*x^12 + 1/11*(b^2*d + 2*a*b*g)*x^11 + 1/10*(b^2*c + 2*a*b*f)*x^10 + 1/9* (2*a*b*e + a^2*h)*x^9 + 1/6*a^2*e*x^6 + 1/8*(2*a*b*d + a^2*g)*x^8 + 1/5*a^ 2*d*x^5 + 1/7*(2*a*b*c + a^2*f)*x^7 + 1/4*a^2*c*x^4
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a^{2} c x^{4}}{4} + \frac {a^{2} d x^{5}}{5} + \frac {a^{2} e x^{6}}{6} + \frac {b^{2} f x^{13}}{13} + \frac {b^{2} g x^{14}}{14} + \frac {b^{2} h x^{15}}{15} + x^{12} \left (\frac {a b h}{6} + \frac {b^{2} e}{12}\right ) + x^{11} \cdot \left (\frac {2 a b g}{11} + \frac {b^{2} d}{11}\right ) + x^{10} \left (\frac {a b f}{5} + \frac {b^{2} c}{10}\right ) + x^{9} \left (\frac {a^{2} h}{9} + \frac {2 a b e}{9}\right ) + x^{8} \left (\frac {a^{2} g}{8} + \frac {a b d}{4}\right ) + x^{7} \left (\frac {a^{2} f}{7} + \frac {2 a b c}{7}\right ) \]
a**2*c*x**4/4 + a**2*d*x**5/5 + a**2*e*x**6/6 + b**2*f*x**13/13 + b**2*g*x **14/14 + b**2*h*x**15/15 + x**12*(a*b*h/6 + b**2*e/12) + x**11*(2*a*b*g/1 1 + b**2*d/11) + x**10*(a*b*f/5 + b**2*c/10) + x**9*(a**2*h/9 + 2*a*b*e/9) + x**8*(a**2*g/8 + a*b*d/4) + x**7*(a**2*f/7 + 2*a*b*c/7)
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{15} \, b^{2} h x^{15} + \frac {1}{14} \, b^{2} g x^{14} + \frac {1}{13} \, b^{2} f x^{13} + \frac {1}{12} \, {\left (b^{2} e + 2 \, a b h\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} d + 2 \, a b g\right )} x^{11} + \frac {1}{10} \, {\left (b^{2} c + 2 \, a b f\right )} x^{10} + \frac {1}{9} \, {\left (2 \, a b e + a^{2} h\right )} x^{9} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{8} \, {\left (2 \, a b d + a^{2} g\right )} x^{8} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} f\right )} x^{7} + \frac {1}{4} \, a^{2} c x^{4} \]
1/15*b^2*h*x^15 + 1/14*b^2*g*x^14 + 1/13*b^2*f*x^13 + 1/12*(b^2*e + 2*a*b* h)*x^12 + 1/11*(b^2*d + 2*a*b*g)*x^11 + 1/10*(b^2*c + 2*a*b*f)*x^10 + 1/9* (2*a*b*e + a^2*h)*x^9 + 1/6*a^2*e*x^6 + 1/8*(2*a*b*d + a^2*g)*x^8 + 1/5*a^ 2*d*x^5 + 1/7*(2*a*b*c + a^2*f)*x^7 + 1/4*a^2*c*x^4
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{15} \, b^{2} h x^{15} + \frac {1}{14} \, b^{2} g x^{14} + \frac {1}{13} \, b^{2} f x^{13} + \frac {1}{12} \, b^{2} e x^{12} + \frac {1}{6} \, a b h x^{12} + \frac {1}{11} \, b^{2} d x^{11} + \frac {2}{11} \, a b g x^{11} + \frac {1}{10} \, b^{2} c x^{10} + \frac {1}{5} \, a b f x^{10} + \frac {2}{9} \, a b e x^{9} + \frac {1}{9} \, a^{2} h x^{9} + \frac {1}{4} \, a b d x^{8} + \frac {1}{8} \, a^{2} g x^{8} + \frac {2}{7} \, a b c x^{7} + \frac {1}{7} \, a^{2} f x^{7} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{4} \, a^{2} c x^{4} \]
1/15*b^2*h*x^15 + 1/14*b^2*g*x^14 + 1/13*b^2*f*x^13 + 1/12*b^2*e*x^12 + 1/ 6*a*b*h*x^12 + 1/11*b^2*d*x^11 + 2/11*a*b*g*x^11 + 1/10*b^2*c*x^10 + 1/5*a *b*f*x^10 + 2/9*a*b*e*x^9 + 1/9*a^2*h*x^9 + 1/4*a*b*d*x^8 + 1/8*a^2*g*x^8 + 2/7*a*b*c*x^7 + 1/7*a^2*f*x^7 + 1/6*a^2*e*x^6 + 1/5*a^2*d*x^5 + 1/4*a^2* c*x^4
Time = 10.53 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^3 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^7\,\left (\frac {f\,a^2}{7}+\frac {2\,b\,c\,a}{7}\right )+x^{10}\,\left (\frac {c\,b^2}{10}+\frac {a\,f\,b}{5}\right )+x^8\,\left (\frac {g\,a^2}{8}+\frac {b\,d\,a}{4}\right )+x^{11}\,\left (\frac {d\,b^2}{11}+\frac {2\,a\,g\,b}{11}\right )+x^9\,\left (\frac {h\,a^2}{9}+\frac {2\,b\,e\,a}{9}\right )+x^{12}\,\left (\frac {e\,b^2}{12}+\frac {a\,h\,b}{6}\right )+\frac {a^2\,c\,x^4}{4}+\frac {a^2\,d\,x^5}{5}+\frac {a^2\,e\,x^6}{6}+\frac {b^2\,f\,x^{13}}{13}+\frac {b^2\,g\,x^{14}}{14}+\frac {b^2\,h\,x^{15}}{15} \]
x^7*((a^2*f)/7 + (2*a*b*c)/7) + x^10*((b^2*c)/10 + (a*b*f)/5) + x^8*((a^2* g)/8 + (a*b*d)/4) + x^11*((b^2*d)/11 + (2*a*b*g)/11) + x^9*((a^2*h)/9 + (2 *a*b*e)/9) + x^12*((b^2*e)/12 + (a*b*h)/6) + (a^2*c*x^4)/4 + (a^2*d*x^5)/5 + (a^2*e*x^6)/6 + (b^2*f*x^13)/13 + (b^2*g*x^14)/14 + (b^2*h*x^15)/15