Integrand size = 36, antiderivative size = 97 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a c x^5+\frac {1}{6} a d x^6+\frac {1}{7} a e x^7+\frac {1}{8} (b c+a f) x^8+\frac {1}{9} (b d+a g) x^9+\frac {1}{10} (b e+a h) x^{10}+\frac {1}{11} b f x^{11}+\frac {1}{12} b g x^{12}+\frac {1}{13} b h x^{13} \] Output:
1/5*a*c*x^5+1/6*a*d*x^6+1/7*a*e*x^7+1/8*(a*f+b*c)*x^8+1/9*(a*g+b*d)*x^9+1/ 10*(a*h+b*e)*x^10+1/11*b*f*x^11+1/12*b*g*x^12+1/13*b*h*x^13
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a c x^5+\frac {1}{6} a d x^6+\frac {1}{7} a e x^7+\frac {1}{8} (b c+a f) x^8+\frac {1}{9} (b d+a g) x^9+\frac {1}{10} (b e+a h) x^{10}+\frac {1}{11} b f x^{11}+\frac {1}{12} b g x^{12}+\frac {1}{13} b h x^{13} \] Input:
Integrate[x^4*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]
Output:
(a*c*x^5)/5 + (a*d*x^6)/6 + (a*e*x^7)/7 + ((b*c + a*f)*x^8)/8 + ((b*d + a* g)*x^9)/9 + ((b*e + a*h)*x^10)/10 + (b*f*x^11)/11 + (b*g*x^12)/12 + (b*h*x ^13)/13
Time = 0.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, 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.
\(\displaystyle \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\) |
\(\Big \downarrow \) 2360 |
\(\displaystyle \int \left (x^7 (a f+b c)+x^8 (a g+b d)+x^9 (a h+b e)+a c x^4+a d x^5+a e x^6+b f x^{10}+b g x^{11}+b h x^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} x^8 (a f+b c)+\frac {1}{9} x^9 (a g+b d)+\frac {1}{10} x^{10} (a h+b e)+\frac {1}{5} a c x^5+\frac {1}{6} a d x^6+\frac {1}{7} a e x^7+\frac {1}{11} b f x^{11}+\frac {1}{12} b g x^{12}+\frac {1}{13} b h x^{13}\) |
Input:
Int[x^4*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]
Output:
(a*c*x^5)/5 + (a*d*x^6)/6 + (a*e*x^7)/7 + ((b*c + a*f)*x^8)/8 + ((b*d + a* g)*x^9)/9 + ((b*e + a*h)*x^10)/10 + (b*f*x^11)/11 + (b*g*x^12)/12 + (b*h*x ^13)/13
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 = 1.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {a c \,x^{5}}{5}+\frac {a d \,x^{6}}{6}+\frac {a e \,x^{7}}{7}+\frac {\left (a f +c b \right ) x^{8}}{8}+\frac {\left (a g +b d \right ) x^{9}}{9}+\frac {\left (a h +b e \right ) x^{10}}{10}+\frac {b f \,x^{11}}{11}+\frac {b g \,x^{12}}{12}+\frac {b h \,x^{13}}{13}\) | \(80\) |
norman | \(\frac {b h \,x^{13}}{13}+\frac {b g \,x^{12}}{12}+\frac {b f \,x^{11}}{11}+\left (\frac {a h}{10}+\frac {b e}{10}\right ) x^{10}+\left (\frac {a g}{9}+\frac {b d}{9}\right ) x^{9}+\left (\frac {a f}{8}+\frac {c b}{8}\right ) x^{8}+\frac {a e \,x^{7}}{7}+\frac {a d \,x^{6}}{6}+\frac {a c \,x^{5}}{5}\) | \(83\) |
gosper | \(\frac {1}{13} b h \,x^{13}+\frac {1}{12} b g \,x^{12}+\frac {1}{11} b f \,x^{11}+\frac {1}{10} x^{10} a h +\frac {1}{10} x^{10} b e +\frac {1}{9} x^{9} a g +\frac {1}{9} x^{9} b d +\frac {1}{8} x^{8} a f +\frac {1}{8} b c \,x^{8}+\frac {1}{7} a e \,x^{7}+\frac {1}{6} a d \,x^{6}+\frac {1}{5} a c \,x^{5}\) | \(86\) |
risch | \(\frac {1}{13} b h \,x^{13}+\frac {1}{12} b g \,x^{12}+\frac {1}{11} b f \,x^{11}+\frac {1}{10} x^{10} a h +\frac {1}{10} x^{10} b e +\frac {1}{9} x^{9} a g +\frac {1}{9} x^{9} b d +\frac {1}{8} x^{8} a f +\frac {1}{8} b c \,x^{8}+\frac {1}{7} a e \,x^{7}+\frac {1}{6} a d \,x^{6}+\frac {1}{5} a c \,x^{5}\) | \(86\) |
parallelrisch | \(\frac {1}{13} b h \,x^{13}+\frac {1}{12} b g \,x^{12}+\frac {1}{11} b f \,x^{11}+\frac {1}{10} x^{10} a h +\frac {1}{10} x^{10} b e +\frac {1}{9} x^{9} a g +\frac {1}{9} x^{9} b d +\frac {1}{8} x^{8} a f +\frac {1}{8} b c \,x^{8}+\frac {1}{7} a e \,x^{7}+\frac {1}{6} a d \,x^{6}+\frac {1}{5} a c \,x^{5}\) | \(86\) |
orering | \(\frac {x^{5} \left (27720 b h \,x^{8}+30030 b g \,x^{7}+32760 b f \,x^{6}+36036 a h \,x^{5}+36036 b e \,x^{5}+40040 a g \,x^{4}+40040 d b \,x^{4}+45045 a f \,x^{3}+45045 b c \,x^{3}+51480 a e \,x^{2}+60060 a d x +72072 a c \right )}{360360}\) | \(86\) |
Input:
int(x^4*(b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x,method=_RETURNVERBOSE)
Output:
1/5*a*c*x^5+1/6*a*d*x^6+1/7*a*e*x^7+1/8*(a*f+b*c)*x^8+1/9*(a*g+b*d)*x^9+1/ 10*(a*h+b*e)*x^10+1/11*b*f*x^11+1/12*b*g*x^12+1/13*b*h*x^13
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{13} \, b h x^{13} + \frac {1}{12} \, b g x^{12} + \frac {1}{11} \, b f x^{11} + \frac {1}{10} \, {\left (b e + a h\right )} x^{10} + \frac {1}{9} \, {\left (b d + a g\right )} x^{9} + \frac {1}{7} \, a e x^{7} + \frac {1}{8} \, {\left (b c + a f\right )} x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{5} \, a c x^{5} \] Input:
integrate(x^4*(b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fric as")
Output:
1/13*b*h*x^13 + 1/12*b*g*x^12 + 1/11*b*f*x^11 + 1/10*(b*e + a*h)*x^10 + 1/ 9*(b*d + a*g)*x^9 + 1/7*a*e*x^7 + 1/8*(b*c + a*f)*x^8 + 1/6*a*d*x^6 + 1/5* a*c*x^5
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a c x^{5}}{5} + \frac {a d x^{6}}{6} + \frac {a e x^{7}}{7} + \frac {b f x^{11}}{11} + \frac {b g x^{12}}{12} + \frac {b h x^{13}}{13} + x^{10} \left (\frac {a h}{10} + \frac {b e}{10}\right ) + x^{9} \left (\frac {a g}{9} + \frac {b d}{9}\right ) + x^{8} \left (\frac {a f}{8} + \frac {b c}{8}\right ) \] Input:
integrate(x**4*(b*x**3+a)*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)
Output:
a*c*x**5/5 + a*d*x**6/6 + a*e*x**7/7 + b*f*x**11/11 + b*g*x**12/12 + b*h*x **13/13 + x**10*(a*h/10 + b*e/10) + x**9*(a*g/9 + b*d/9) + x**8*(a*f/8 + b *c/8)
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{13} \, b h x^{13} + \frac {1}{12} \, b g x^{12} + \frac {1}{11} \, b f x^{11} + \frac {1}{10} \, {\left (b e + a h\right )} x^{10} + \frac {1}{9} \, {\left (b d + a g\right )} x^{9} + \frac {1}{7} \, a e x^{7} + \frac {1}{8} \, {\left (b c + a f\right )} x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{5} \, a c x^{5} \] Input:
integrate(x^4*(b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="maxi ma")
Output:
1/13*b*h*x^13 + 1/12*b*g*x^12 + 1/11*b*f*x^11 + 1/10*(b*e + a*h)*x^10 + 1/ 9*(b*d + a*g)*x^9 + 1/7*a*e*x^7 + 1/8*(b*c + a*f)*x^8 + 1/6*a*d*x^6 + 1/5* a*c*x^5
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{13} \, b h x^{13} + \frac {1}{12} \, b g x^{12} + \frac {1}{11} \, b f x^{11} + \frac {1}{10} \, b e x^{10} + \frac {1}{10} \, a h x^{10} + \frac {1}{9} \, b d x^{9} + \frac {1}{9} \, a g x^{9} + \frac {1}{8} \, b c x^{8} + \frac {1}{8} \, a f x^{8} + \frac {1}{7} \, a e x^{7} + \frac {1}{6} \, a d x^{6} + \frac {1}{5} \, a c x^{5} \] Input:
integrate(x^4*(b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="giac ")
Output:
1/13*b*h*x^13 + 1/12*b*g*x^12 + 1/11*b*f*x^11 + 1/10*b*e*x^10 + 1/10*a*h*x ^10 + 1/9*b*d*x^9 + 1/9*a*g*x^9 + 1/8*b*c*x^8 + 1/8*a*f*x^8 + 1/7*a*e*x^7 + 1/6*a*d*x^6 + 1/5*a*c*x^5
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {b\,h\,x^{13}}{13}+\frac {b\,g\,x^{12}}{12}+\frac {b\,f\,x^{11}}{11}+\left (\frac {b\,e}{10}+\frac {a\,h}{10}\right )\,x^{10}+\left (\frac {b\,d}{9}+\frac {a\,g}{9}\right )\,x^9+\left (\frac {b\,c}{8}+\frac {a\,f}{8}\right )\,x^8+\frac {a\,e\,x^7}{7}+\frac {a\,d\,x^6}{6}+\frac {a\,c\,x^5}{5} \] Input:
int(x^4*(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x)
Output:
x^8*((b*c)/8 + (a*f)/8) + x^9*((b*d)/9 + (a*g)/9) + x^10*((b*e)/10 + (a*h) /10) + (b*h*x^13)/13 + (a*c*x^5)/5 + (a*d*x^6)/6 + (a*e*x^7)/7 + (b*f*x^11 )/11 + (b*g*x^12)/12
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {x^{5} \left (27720 b h \,x^{8}+30030 b g \,x^{7}+32760 b f \,x^{6}+36036 a h \,x^{5}+36036 b e \,x^{5}+40040 a g \,x^{4}+40040 b d \,x^{4}+45045 a f \,x^{3}+45045 b c \,x^{3}+51480 a e \,x^{2}+60060 a d x +72072 a c \right )}{360360} \] Input:
int(x^4*(b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x)
Output:
(x**5*(72072*a*c + 60060*a*d*x + 51480*a*e*x**2 + 45045*a*f*x**3 + 40040*a *g*x**4 + 36036*a*h*x**5 + 45045*b*c*x**3 + 40040*b*d*x**4 + 36036*b*e*x** 5 + 32760*b*f*x**6 + 30030*b*g*x**7 + 27720*b*h*x**8))/360360