Integrand size = 21, antiderivative size = 101 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^5+\frac {1}{6} \left (b^2 B+2 A b c+2 a B c\right ) x^6+\frac {1}{7} c (2 b B+A c) x^7+\frac {1}{8} B c^2 x^8 \] Output:
1/3*a^2*A*x^3+1/4*a*(2*A*b+B*a)*x^4+1/5*(2*a*b*B+A*(2*a*c+b^2))*x^5+1/6*(2 *A*b*c+2*B*a*c+B*b^2)*x^6+1/7*c*(A*c+2*B*b)*x^7+1/8*B*c^2*x^8
Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{4} a (2 A b+a B) x^4+\frac {1}{5} \left (A b^2+2 a b B+2 a A c\right ) x^5+\frac {1}{6} \left (b^2 B+2 A b c+2 a B c\right ) x^6+\frac {1}{7} c (2 b B+A c) x^7+\frac {1}{8} B c^2 x^8 \] Input:
Integrate[x^2*(A + B*x)*(a + b*x + c*x^2)^2,x]
Output:
(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + ((A*b^2 + 2*a*b*B + 2*a*A*c)*x^5 )/5 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c ^2*x^8)/8
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1195, 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.
\(\displaystyle \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (a^2 A x^2+x^5 \left (2 a B c+2 A b c+b^2 B\right )+x^4 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x^3 (a B+2 A b)+c x^6 (A c+2 b B)+B c^2 x^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^2 A x^3+\frac {1}{6} x^6 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{4} a x^4 (a B+2 A b)+\frac {1}{7} c x^7 (A c+2 b B)+\frac {1}{8} B c^2 x^8\) |
Input:
Int[x^2*(A + B*x)*(a + b*x + c*x^2)^2,x]
Output:
(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^4)/4 + ((2*a*b*B + A*(b^2 + 2*a*c))*x^5 )/5 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c ^2*x^8)/8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {B \,c^{2} x^{8}}{8}+\frac {\left (A \,c^{2}+2 B b c \right ) x^{7}}{7}+\frac {\left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(94\) |
norman | \(\frac {B \,c^{2} x^{8}}{8}+\left (\frac {1}{7} A \,c^{2}+\frac {2}{7} B b c \right ) x^{7}+\left (\frac {1}{3} A b c +\frac {1}{3} a B c +\frac {1}{6} B \,b^{2}\right ) x^{6}+\left (\frac {2}{5} A a c +\frac {1}{5} b^{2} A +\frac {2}{5} a b B \right ) x^{5}+\left (\frac {1}{2} a b A +\frac {1}{4} a^{2} B \right ) x^{4}+\frac {a^{2} A \,x^{3}}{3}\) | \(94\) |
orering | \(\frac {x^{3} \left (105 B \,c^{2} x^{5}+120 x^{4} A \,c^{2}+240 x^{4} B b c +280 x^{3} A b c +280 B a c \,x^{3}+140 x^{3} B \,b^{2}+336 A a c \,x^{2}+168 x^{2} b^{2} A +336 B a \,x^{2} b +420 a b A x +210 a^{2} B x +280 a^{2} A \right )}{840}\) | \(102\) |
gosper | \(\frac {1}{8} B \,c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2}+\frac {2}{7} x^{7} B b c +\frac {1}{3} A b c \,x^{6}+\frac {1}{3} x^{6} a B c +\frac {1}{6} B \,b^{2} x^{6}+\frac {2}{5} x^{5} A a c +\frac {1}{5} A \,b^{2} x^{5}+\frac {2}{5} B a b \,x^{5}+\frac {1}{2} A a b \,x^{4}+\frac {1}{4} a^{2} B \,x^{4}+\frac {1}{3} a^{2} A \,x^{3}\) | \(104\) |
risch | \(\frac {1}{8} B \,c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2}+\frac {2}{7} x^{7} B b c +\frac {1}{3} A b c \,x^{6}+\frac {1}{3} x^{6} a B c +\frac {1}{6} B \,b^{2} x^{6}+\frac {2}{5} x^{5} A a c +\frac {1}{5} A \,b^{2} x^{5}+\frac {2}{5} B a b \,x^{5}+\frac {1}{2} A a b \,x^{4}+\frac {1}{4} a^{2} B \,x^{4}+\frac {1}{3} a^{2} A \,x^{3}\) | \(104\) |
parallelrisch | \(\frac {1}{8} B \,c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2}+\frac {2}{7} x^{7} B b c +\frac {1}{3} A b c \,x^{6}+\frac {1}{3} x^{6} a B c +\frac {1}{6} B \,b^{2} x^{6}+\frac {2}{5} x^{5} A a c +\frac {1}{5} A \,b^{2} x^{5}+\frac {2}{5} B a b \,x^{5}+\frac {1}{2} A a b \,x^{4}+\frac {1}{4} a^{2} B \,x^{4}+\frac {1}{3} a^{2} A \,x^{3}\) | \(104\) |
Input:
int(x^2*(B*x+A)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/8*B*c^2*x^8+1/7*(A*c^2+2*B*b*c)*x^7+1/6*(2*A*b*c+B*(2*a*c+b^2))*x^6+1/5* (2*a*b*B+A*(2*a*c+b^2))*x^5+1/4*(2*A*a*b+B*a^2)*x^4+1/3*a^2*A*x^3
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/8*B*c^2*x^8 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/6*(B*b^2 + 2*(B*a + A*b)*c)* x^6 + 1/3*A*a^2*x^3 + 1/5*(2*B*a*b + A*b^2 + 2*A*a*c)*x^5 + 1/4*(B*a^2 + 2 *A*a*b)*x^4
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {A a^{2} x^{3}}{3} + \frac {B c^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2}}{7} + \frac {2 B b c}{7}\right ) + x^{6} \left (\frac {A b c}{3} + \frac {B a c}{3} + \frac {B b^{2}}{6}\right ) + x^{5} \cdot \left (\frac {2 A a c}{5} + \frac {A b^{2}}{5} + \frac {2 B a b}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \] Input:
integrate(x**2*(B*x+A)*(c*x**2+b*x+a)**2,x)
Output:
A*a**2*x**3/3 + B*c**2*x**8/8 + x**7*(A*c**2/7 + 2*B*b*c/7) + x**6*(A*b*c/ 3 + B*a*c/3 + B*b**2/6) + x**5*(2*A*a*c/5 + A*b**2/5 + 2*B*a*b/5) + x**4*( A*a*b/2 + B*a**2/4)
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/8*B*c^2*x^8 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/6*(B*b^2 + 2*(B*a + A*b)*c)* x^6 + 1/3*A*a^2*x^3 + 1/5*(2*B*a*b + A*b^2 + 2*A*a*c)*x^5 + 1/4*(B*a^2 + 2 *A*a*b)*x^4
Time = 0.19 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} x^{8} + \frac {2}{7} \, B b c x^{7} + \frac {1}{7} \, A c^{2} x^{7} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, B a c x^{6} + \frac {1}{3} \, A b c x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {2}{5} \, A a c x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, A a^{2} x^{3} \] Input:
integrate(x^2*(B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/8*B*c^2*x^8 + 2/7*B*b*c*x^7 + 1/7*A*c^2*x^7 + 1/6*B*b^2*x^6 + 1/3*B*a*c* x^6 + 1/3*A*b*c*x^6 + 2/5*B*a*b*x^5 + 1/5*A*b^2*x^5 + 2/5*A*a*c*x^5 + 1/4* B*a^2*x^4 + 1/2*A*a*b*x^4 + 1/3*A*a^2*x^3
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^7\,\left (\frac {A\,c^2}{7}+\frac {2\,B\,b\,c}{7}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^6\,\left (\frac {B\,b^2}{6}+\frac {A\,c\,b}{3}+\frac {B\,a\,c}{3}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,c^2\,x^8}{8} \] Input:
int(x^2*(A + B*x)*(a + b*x + c*x^2)^2,x)
Output:
x^4*((B*a^2)/4 + (A*a*b)/2) + x^7*((A*c^2)/7 + (2*B*b*c)/7) + x^5*((A*b^2) /5 + (2*A*a*c)/5 + (2*B*a*b)/5) + x^6*((B*b^2)/6 + (A*b*c)/3 + (B*a*c)/3) + (A*a^2*x^3)/3 + (B*c^2*x^8)/8
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {x^{3} \left (105 b \,c^{2} x^{5}+120 a \,c^{2} x^{4}+240 b^{2} c \,x^{4}+560 a b c \,x^{3}+140 b^{3} x^{3}+336 a^{2} c \,x^{2}+504 a \,b^{2} x^{2}+630 a^{2} b x +280 a^{3}\right )}{840} \] Input:
int(x^2*(B*x+A)*(c*x^2+b*x+a)^2,x)
Output:
(x**3*(280*a**3 + 630*a**2*b*x + 336*a**2*c*x**2 + 504*a*b**2*x**2 + 560*a *b*c*x**3 + 120*a*c**2*x**4 + 140*b**3*x**3 + 240*b**2*c*x**4 + 105*b*c**2 *x**5))/840