Integrand size = 27, antiderivative size = 87 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {a^2 (A b-a B) (a+b x)^5}{5 b^4}-\frac {a (2 A b-3 a B) (a+b x)^6}{6 b^4}+\frac {(A b-3 a B) (a+b x)^7}{7 b^4}+\frac {B (a+b x)^8}{8 b^4} \] Output:
1/5*a^2*(A*b-B*a)*(b*x+a)^5/b^4-1/6*a*(2*A*b-3*B*a)*(b*x+a)^6/b^4+1/7*(A*b -3*B*a)*(b*x+a)^7/b^4+1/8*B*(b*x+a)^8/b^4
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{840} x^3 \left (70 a^4 (4 A+3 B x)+168 a^3 b x (5 A+4 B x)+168 a^2 b^2 x^2 (6 A+5 B x)+80 a b^3 x^3 (7 A+6 B x)+15 b^4 x^4 (8 A+7 B x)\right ) \] Input:
Integrate[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(x^3*(70*a^4*(4*A + 3*B*x) + 168*a^3*b*x*(5*A + 4*B*x) + 168*a^2*b^2*x^2*( 6*A + 5*B*x) + 80*a*b^3*x^3*(7*A + 6*B*x) + 15*b^4*x^4*(8*A + 7*B*x)))/840
Time = 0.44 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 85, 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.
\(\displaystyle \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int b^4 x^2 (a+b x)^4 (A+B x)dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^2 (a+b x)^4 (A+B x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (-\frac {a^2 (a+b x)^4 (a B-A b)}{b^3}+\frac {(a+b x)^6 (A b-3 a B)}{b^3}+\frac {a (a+b x)^5 (3 a B-2 A b)}{b^3}+\frac {B (a+b x)^7}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac {(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac {a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac {B (a+b x)^8}{8 b^4}\) |
Input:
Int[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(a^2*(A*b - a*B)*(a + b*x)^5)/(5*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^6)/(6 *b^4) + ((A*b - 3*a*B)*(a + b*x)^7)/(7*b^4) + (B*(a + b*x)^8)/(8*b^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.10
method | result | size |
norman | \(\frac {B \,b^{4} x^{8}}{8}+\left (\frac {1}{7} A \,b^{4}+\frac {4}{7} B a \,b^{3}\right ) x^{7}+\left (\frac {2}{3} A a \,b^{3}+B \,a^{2} b^{2}\right ) x^{6}+\left (\frac {6}{5} a^{2} A \,b^{2}+\frac {4}{5} B \,a^{3} b \right ) x^{5}+\left (A \,a^{3} b +\frac {1}{4} a^{4} B \right ) x^{4}+\frac {a^{4} A \,x^{3}}{3}\) | \(96\) |
gosper | \(\frac {x^{3} \left (105 b^{4} B \,x^{5}+120 A \,b^{4} x^{4}+480 B a \,b^{3} x^{4}+560 A a \,b^{3} x^{3}+840 B \,a^{2} b^{2} x^{3}+1008 A \,a^{2} b^{2} x^{2}+672 B \,a^{3} b \,x^{2}+840 A \,a^{3} b x +210 a^{4} B x +280 a^{4} A \right )}{840}\) | \(100\) |
default | \(\frac {B \,b^{4} x^{8}}{8}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{7}}{7}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{6}}{6}+\frac {\left (6 a^{2} A \,b^{2}+4 B \,a^{3} b \right ) x^{5}}{5}+\frac {\left (4 A \,a^{3} b +a^{4} B \right ) x^{4}}{4}+\frac {a^{4} A \,x^{3}}{3}\) | \(100\) |
risch | \(\frac {1}{8} B \,b^{4} x^{8}+\frac {1}{7} A \,b^{4} x^{7}+\frac {4}{7} a \,b^{3} B \,x^{7}+\frac {2}{3} a \,b^{3} A \,x^{6}+x^{6} B \,a^{2} b^{2}+\frac {6}{5} x^{5} a^{2} A \,b^{2}+\frac {4}{5} x^{5} B \,a^{3} b +x^{4} A \,a^{3} b +\frac {1}{4} x^{4} a^{4} B +\frac {1}{3} a^{4} A \,x^{3}\) | \(100\) |
parallelrisch | \(\frac {1}{8} B \,b^{4} x^{8}+\frac {1}{7} A \,b^{4} x^{7}+\frac {4}{7} a \,b^{3} B \,x^{7}+\frac {2}{3} a \,b^{3} A \,x^{6}+x^{6} B \,a^{2} b^{2}+\frac {6}{5} x^{5} a^{2} A \,b^{2}+\frac {4}{5} x^{5} B \,a^{3} b +x^{4} A \,a^{3} b +\frac {1}{4} x^{4} a^{4} B +\frac {1}{3} a^{4} A \,x^{3}\) | \(100\) |
orering | \(\frac {x^{3} \left (105 b^{4} B \,x^{5}+120 A \,b^{4} x^{4}+480 B a \,b^{3} x^{4}+560 A a \,b^{3} x^{3}+840 B \,a^{2} b^{2} x^{3}+1008 A \,a^{2} b^{2} x^{2}+672 B \,a^{3} b \,x^{2}+840 A \,a^{3} b x +210 a^{4} B x +280 a^{4} A \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{840 \left (b x +a \right )^{4}}\) | \(125\) |
Input:
int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/8*B*b^4*x^8+(1/7*A*b^4+4/7*B*a*b^3)*x^7+(2/3*A*a*b^3+B*a^2*b^2)*x^6+(6/5 *a^2*A*b^2+4/5*B*a^3*b)*x^5+(A*a^3*b+1/4*a^4*B)*x^4+1/3*a^4*A*x^3
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} x^{8} + \frac {1}{3} \, A a^{4} x^{3} + \frac {1}{7} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
1/8*B*b^4*x^8 + 1/3*A*a^4*x^3 + 1/7*(4*B*a*b^3 + A*b^4)*x^7 + 1/3*(3*B*a^2 *b^2 + 2*A*a*b^3)*x^6 + 2/5*(2*B*a^3*b + 3*A*a^2*b^2)*x^5 + 1/4*(B*a^4 + 4 *A*a^3*b)*x^4
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {A a^{4} x^{3}}{3} + \frac {B b^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{4}}{7} + \frac {4 B a b^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 A a b^{3}}{3} + B a^{2} b^{2}\right ) + x^{5} \cdot \left (\frac {6 A a^{2} b^{2}}{5} + \frac {4 B a^{3} b}{5}\right ) + x^{4} \left (A a^{3} b + \frac {B a^{4}}{4}\right ) \] Input:
integrate(x**2*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
A*a**4*x**3/3 + B*b**4*x**8/8 + x**7*(A*b**4/7 + 4*B*a*b**3/7) + x**6*(2*A *a*b**3/3 + B*a**2*b**2) + x**5*(6*A*a**2*b**2/5 + 4*B*a**3*b/5) + x**4*(A *a**3*b + B*a**4/4)
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} x^{8} + \frac {1}{3} \, A a^{4} x^{3} + \frac {1}{7} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
1/8*B*b^4*x^8 + 1/3*A*a^4*x^3 + 1/7*(4*B*a*b^3 + A*b^4)*x^7 + 1/3*(3*B*a^2 *b^2 + 2*A*a*b^3)*x^6 + 2/5*(2*B*a^3*b + 3*A*a^2*b^2)*x^5 + 1/4*(B*a^4 + 4 *A*a^3*b)*x^4
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} x^{8} + \frac {4}{7} \, B a b^{3} x^{7} + \frac {1}{7} \, A b^{4} x^{7} + B a^{2} b^{2} x^{6} + \frac {2}{3} \, A a b^{3} x^{6} + \frac {4}{5} \, B a^{3} b x^{5} + \frac {6}{5} \, A a^{2} b^{2} x^{5} + \frac {1}{4} \, B a^{4} x^{4} + A a^{3} b x^{4} + \frac {1}{3} \, A a^{4} x^{3} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
1/8*B*b^4*x^8 + 4/7*B*a*b^3*x^7 + 1/7*A*b^4*x^7 + B*a^2*b^2*x^6 + 2/3*A*a* b^3*x^6 + 4/5*B*a^3*b*x^5 + 6/5*A*a^2*b^2*x^5 + 1/4*B*a^4*x^4 + A*a^3*b*x^ 4 + 1/3*A*a^4*x^3
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^4\,\left (\frac {B\,a^4}{4}+A\,b\,a^3\right )+x^7\,\left (\frac {A\,b^4}{7}+\frac {4\,B\,a\,b^3}{7}\right )+\frac {A\,a^4\,x^3}{3}+\frac {B\,b^4\,x^8}{8}+\frac {2\,a^2\,b\,x^5\,\left (3\,A\,b+2\,B\,a\right )}{5}+\frac {a\,b^2\,x^6\,\left (2\,A\,b+3\,B\,a\right )}{3} \] Input:
int(x^2*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
x^4*((B*a^4)/4 + A*a^3*b) + x^7*((A*b^4)/7 + (4*B*a*b^3)/7) + (A*a^4*x^3)/ 3 + (B*b^4*x^8)/8 + (2*a^2*b*x^5*(3*A*b + 2*B*a))/5 + (a*b^2*x^6*(2*A*b + 3*B*a))/3
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {x^{3} \left (21 b^{5} x^{5}+120 a \,b^{4} x^{4}+280 a^{2} b^{3} x^{3}+336 a^{3} b^{2} x^{2}+210 a^{4} b x +56 a^{5}\right )}{168} \] Input:
int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(x**3*(56*a**5 + 210*a**4*b*x + 336*a**3*b**2*x**2 + 280*a**2*b**3*x**3 + 120*a*b**4*x**4 + 21*b**5*x**5))/168