Integrand size = 16, antiderivative size = 89 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {a^3 x^4}{4}+\frac {3}{5} a^2 b x^5+\frac {1}{2} a \left (b^2+a c\right ) x^6+\frac {1}{7} b \left (b^2+6 a c\right ) x^7+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{10}}{10} \] Output:
1/4*a^3*x^4+3/5*a^2*b*x^5+1/2*a*(a*c+b^2)*x^6+1/7*b*(6*a*c+b^2)*x^7+3/8*c* (a*c+b^2)*x^8+1/3*b*c^2*x^9+1/10*c^3*x^10
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {a^3 x^4}{4}+\frac {3}{5} a^2 b x^5+\frac {1}{2} a \left (b^2+a c\right ) x^6+\frac {1}{7} b \left (b^2+6 a c\right ) x^7+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{10}}{10} \] Input:
Integrate[x^3*(a + b*x + c*x^2)^3,x]
Output:
(a^3*x^4)/4 + (3*a^2*b*x^5)/5 + (a*(b^2 + a*c)*x^6)/2 + (b*(b^2 + 6*a*c)*x ^7)/7 + (3*c*(b^2 + a*c)*x^8)/8 + (b*c^2*x^9)/3 + (c^3*x^10)/10
Time = 0.40 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1140, 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.
\(\displaystyle \int x^3 \left (a+b x+c x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (a^3 x^3+3 a^2 b x^4+3 c x^7 \left (a c+b^2\right )+b x^6 \left (6 a c+b^2\right )+3 a x^5 \left (a c+b^2\right )+3 b c^2 x^8+c^3 x^9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 x^4}{4}+\frac {3}{5} a^2 b x^5+\frac {3}{8} c x^8 \left (a c+b^2\right )+\frac {1}{7} b x^7 \left (6 a c+b^2\right )+\frac {1}{2} a x^6 \left (a c+b^2\right )+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{10}}{10}\) |
Input:
Int[x^3*(a + b*x + c*x^2)^3,x]
Output:
(a^3*x^4)/4 + (3*a^2*b*x^5)/5 + (a*(b^2 + a*c)*x^6)/2 + (b*(b^2 + 6*a*c)*x ^7)/7 + (3*c*(b^2 + a*c)*x^8)/8 + (b*c^2*x^9)/3 + (c^3*x^10)/10
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.61 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {c^{3} x^{10}}{10}+\frac {b \,c^{2} x^{9}}{3}+\left (\frac {3}{8} a \,c^{2}+\frac {3}{8} b^{2} c \right ) x^{8}+\left (\frac {6}{7} a b c +\frac {1}{7} b^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} c +\frac {1}{2} a \,b^{2}\right ) x^{6}+\frac {3 a^{2} b \,x^{5}}{5}+\frac {a^{3} x^{4}}{4}\) | \(85\) |
gosper | \(\frac {1}{10} c^{3} x^{10}+\frac {1}{3} b \,c^{2} x^{9}+\frac {3}{8} x^{8} a \,c^{2}+\frac {3}{8} b^{2} c \,x^{8}+\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {1}{2} x^{6} a^{2} c +\frac {1}{2} a \,b^{2} x^{6}+\frac {3}{5} a^{2} b \,x^{5}+\frac {1}{4} a^{3} x^{4}\) | \(88\) |
risch | \(\frac {1}{10} c^{3} x^{10}+\frac {1}{3} b \,c^{2} x^{9}+\frac {3}{8} x^{8} a \,c^{2}+\frac {3}{8} b^{2} c \,x^{8}+\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {1}{2} x^{6} a^{2} c +\frac {1}{2} a \,b^{2} x^{6}+\frac {3}{5} a^{2} b \,x^{5}+\frac {1}{4} a^{3} x^{4}\) | \(88\) |
parallelrisch | \(\frac {1}{10} c^{3} x^{10}+\frac {1}{3} b \,c^{2} x^{9}+\frac {3}{8} x^{8} a \,c^{2}+\frac {3}{8} b^{2} c \,x^{8}+\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {1}{2} x^{6} a^{2} c +\frac {1}{2} a \,b^{2} x^{6}+\frac {3}{5} a^{2} b \,x^{5}+\frac {1}{4} a^{3} x^{4}\) | \(88\) |
orering | \(\frac {x^{4} \left (84 c^{3} x^{6}+280 b \,c^{2} x^{5}+315 a \,c^{2} x^{4}+315 b^{2} c \,x^{4}+720 a b c \,x^{3}+120 b^{3} x^{3}+420 a^{2} c \,x^{2}+420 a \,b^{2} x^{2}+504 a^{2} b x +210 a^{3}\right )}{840}\) | \(88\) |
default | \(\frac {c^{3} x^{10}}{10}+\frac {b \,c^{2} x^{9}}{3}+\frac {\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) x^{6}}{6}+\frac {3 a^{2} b \,x^{5}}{5}+\frac {a^{3} x^{4}}{4}\) | \(111\) |
Input:
int(x^3*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/10*c^3*x^10+1/3*b*c^2*x^9+(3/8*a*c^2+3/8*b^2*c)*x^8+(6/7*a*b*c+1/7*b^3)* x^7+(1/2*a^2*c+1/2*a*b^2)*x^6+3/5*a^2*b*x^5+1/4*a^3*x^4
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} x^{10} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{7} + \frac {1}{4} \, a^{3} x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{6} \] Input:
integrate(x^3*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/10*c^3*x^10 + 1/3*b*c^2*x^9 + 3/8*(b^2*c + a*c^2)*x^8 + 3/5*a^2*b*x^5 + 1/7*(b^3 + 6*a*b*c)*x^7 + 1/4*a^3*x^4 + 1/2*(a*b^2 + a^2*c)*x^6
Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{5}}{5} + \frac {b c^{2} x^{9}}{3} + \frac {c^{3} x^{10}}{10} + x^{8} \cdot \left (\frac {3 a c^{2}}{8} + \frac {3 b^{2} c}{8}\right ) + x^{7} \cdot \left (\frac {6 a b c}{7} + \frac {b^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c}{2} + \frac {a b^{2}}{2}\right ) \] Input:
integrate(x**3*(c*x**2+b*x+a)**3,x)
Output:
a**3*x**4/4 + 3*a**2*b*x**5/5 + b*c**2*x**9/3 + c**3*x**10/10 + x**8*(3*a* c**2/8 + 3*b**2*c/8) + x**7*(6*a*b*c/7 + b**3/7) + x**6*(a**2*c/2 + a*b**2 /2)
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} x^{10} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{7} + \frac {1}{4} \, a^{3} x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{6} \] Input:
integrate(x^3*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
1/10*c^3*x^10 + 1/3*b*c^2*x^9 + 3/8*(b^2*c + a*c^2)*x^8 + 3/5*a^2*b*x^5 + 1/7*(b^3 + 6*a*b*c)*x^7 + 1/4*a^3*x^4 + 1/2*(a*b^2 + a^2*c)*x^6
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} x^{10} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{8} \, b^{2} c x^{8} + \frac {3}{8} \, a c^{2} x^{8} + \frac {1}{7} \, b^{3} x^{7} + \frac {6}{7} \, a b c x^{7} + \frac {1}{2} \, a b^{2} x^{6} + \frac {1}{2} \, a^{2} c x^{6} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{4} \, a^{3} x^{4} \] Input:
integrate(x^3*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
1/10*c^3*x^10 + 1/3*b*c^2*x^9 + 3/8*b^2*c*x^8 + 3/8*a*c^2*x^8 + 1/7*b^3*x^ 7 + 6/7*a*b*c*x^7 + 1/2*a*b^2*x^6 + 1/2*a^2*c*x^6 + 3/5*a^2*b*x^5 + 1/4*a^ 3*x^4
Time = 8.85 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {b^3}{7}+\frac {6\,a\,c\,b}{7}\right )+\frac {a^3\,x^4}{4}+\frac {c^3\,x^{10}}{10}+\frac {3\,a^2\,b\,x^5}{5}+\frac {b\,c^2\,x^9}{3}+\frac {a\,x^6\,\left (b^2+a\,c\right )}{2}+\frac {3\,c\,x^8\,\left (b^2+a\,c\right )}{8} \] Input:
int(x^3*(a + b*x + c*x^2)^3,x)
Output:
x^7*(b^3/7 + (6*a*b*c)/7) + (a^3*x^4)/4 + (c^3*x^10)/10 + (3*a^2*b*x^5)/5 + (b*c^2*x^9)/3 + (a*x^6*(a*c + b^2))/2 + (3*c*x^8*(a*c + b^2))/8
Time = 0.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {x^{4} \left (84 c^{3} x^{6}+280 b \,c^{2} x^{5}+315 a \,c^{2} x^{4}+315 b^{2} c \,x^{4}+720 a b c \,x^{3}+120 b^{3} x^{3}+420 a^{2} c \,x^{2}+420 a \,b^{2} x^{2}+504 a^{2} b x +210 a^{3}\right )}{840} \] Input:
int(x^3*(c*x^2+b*x+a)^3,x)
Output:
(x**4*(210*a**3 + 504*a**2*b*x + 420*a**2*c*x**2 + 420*a*b**2*x**2 + 720*a *b*c*x**3 + 315*a*c**2*x**4 + 120*b**3*x**3 + 315*b**2*c*x**4 + 280*b*c**2 *x**5 + 84*c**3*x**6))/840