Integrand size = 27, antiderivative size = 87 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {a^2 (A b-a B) (a+b x)^7}{7 b^4}-\frac {a (2 A b-3 a B) (a+b x)^8}{8 b^4}+\frac {(A b-3 a B) (a+b x)^9}{9 b^4}+\frac {B (a+b x)^{10}}{10 b^4} \] Output:
1/7*a^2*(A*b-B*a)*(b*x+a)^7/b^4-1/8*a*(2*A*b-3*B*a)*(b*x+a)^8/b^4+1/9*(A*b -3*B*a)*(b*x+a)^9/b^4+1/10*B*(b*x+a)^10/b^4
Time = 0.02 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.64 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{3} a^6 A x^3+\frac {1}{4} a^5 (6 A b+a B) x^4+\frac {3}{5} a^4 b (5 A b+2 a B) x^5+\frac {5}{6} a^3 b^2 (4 A b+3 a B) x^6+\frac {5}{7} a^2 b^3 (3 A b+4 a B) x^7+\frac {3}{8} a b^4 (2 A b+5 a B) x^8+\frac {1}{9} b^5 (A b+6 a B) x^9+\frac {1}{10} b^6 B x^{10} \] Input:
Integrate[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(a^6*A*x^3)/3 + (a^5*(6*A*b + a*B)*x^4)/4 + (3*a^4*b*(5*A*b + 2*a*B)*x^5)/ 5 + (5*a^3*b^2*(4*A*b + 3*a*B)*x^6)/6 + (5*a^2*b^3*(3*A*b + 4*a*B)*x^7)/7 + (3*a*b^4*(2*A*b + 5*a*B)*x^8)/8 + (b^5*(A*b + 6*a*B)*x^9)/9 + (b^6*B*x^1 0)/10
Time = 0.45 (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.
\(\displaystyle \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int b^6 x^2 (a+b x)^6 (A+B x)dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^2 (a+b x)^6 (A+B x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (-\frac {a^2 (a+b x)^6 (a B-A b)}{b^3}+\frac {(a+b x)^8 (A b-3 a B)}{b^3}+\frac {a (a+b x)^7 (3 a B-2 A b)}{b^3}+\frac {B (a+b x)^9}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac {(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac {a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac {B (a+b x)^{10}}{10 b^4}\) |
Input:
Int[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(a^2*(A*b - a*B)*(a + b*x)^7)/(7*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^8)/(8 *b^4) + ((A*b - 3*a*B)*(a + b*x)^9)/(9*b^4) + (B*(a + b*x)^10)/(10*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 = 0.93 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.66
method | result | size |
norman | \(\frac {B \,b^{6} x^{10}}{10}+\left (\frac {1}{9} A \,b^{6}+\frac {2}{3} B a \,b^{5}\right ) x^{9}+\left (\frac {3}{4} A a \,b^{5}+\frac {15}{8} B \,a^{2} b^{4}\right ) x^{8}+\left (\frac {15}{7} A \,a^{2} b^{4}+\frac {20}{7} B \,a^{3} b^{3}\right ) x^{7}+\left (\frac {10}{3} A \,a^{3} b^{3}+\frac {5}{2} B \,a^{4} b^{2}\right ) x^{6}+\left (3 A \,a^{4} b^{2}+\frac {6}{5} B \,a^{5} b \right ) x^{5}+\left (\frac {3}{2} A \,a^{5} b +\frac {1}{4} B \,a^{6}\right ) x^{4}+\frac {A \,a^{6} x^{3}}{3}\) | \(144\) |
gosper | \(\frac {x^{3} \left (252 b^{6} B \,x^{7}+280 A \,b^{6} x^{6}+1680 B a \,b^{5} x^{6}+1890 A a \,b^{5} x^{5}+4725 B \,a^{2} b^{4} x^{5}+5400 A \,a^{2} b^{4} x^{4}+7200 B \,a^{3} b^{3} x^{4}+8400 A \,a^{3} b^{3} x^{3}+6300 B \,a^{4} b^{2} x^{3}+7560 A \,a^{4} b^{2} x^{2}+3024 B \,a^{5} b \,x^{2}+3780 A \,a^{5} b x +630 B \,a^{6} x +840 A \,a^{6}\right )}{2520}\) | \(148\) |
default | \(\frac {B \,b^{6} x^{10}}{10}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{9}}{9}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{8}}{8}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{7}}{7}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{6}}{6}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{5}}{5}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{4}}{4}+\frac {A \,a^{6} x^{3}}{3}\) | \(148\) |
risch | \(\frac {1}{10} B \,b^{6} x^{10}+\frac {1}{9} x^{9} A \,b^{6}+\frac {2}{3} x^{9} B a \,b^{5}+\frac {3}{4} x^{8} A a \,b^{5}+\frac {15}{8} x^{8} B \,a^{2} b^{4}+\frac {15}{7} x^{7} A \,a^{2} b^{4}+\frac {20}{7} x^{7} B \,a^{3} b^{3}+\frac {10}{3} x^{6} A \,a^{3} b^{3}+\frac {5}{2} x^{6} B \,a^{4} b^{2}+3 x^{5} A \,a^{4} b^{2}+\frac {6}{5} x^{5} B \,a^{5} b +\frac {3}{2} x^{4} A \,a^{5} b +\frac {1}{4} x^{4} B \,a^{6}+\frac {1}{3} A \,a^{6} x^{3}\) | \(150\) |
parallelrisch | \(\frac {1}{10} B \,b^{6} x^{10}+\frac {1}{9} x^{9} A \,b^{6}+\frac {2}{3} x^{9} B a \,b^{5}+\frac {3}{4} x^{8} A a \,b^{5}+\frac {15}{8} x^{8} B \,a^{2} b^{4}+\frac {15}{7} x^{7} A \,a^{2} b^{4}+\frac {20}{7} x^{7} B \,a^{3} b^{3}+\frac {10}{3} x^{6} A \,a^{3} b^{3}+\frac {5}{2} x^{6} B \,a^{4} b^{2}+3 x^{5} A \,a^{4} b^{2}+\frac {6}{5} x^{5} B \,a^{5} b +\frac {3}{2} x^{4} A \,a^{5} b +\frac {1}{4} x^{4} B \,a^{6}+\frac {1}{3} A \,a^{6} x^{3}\) | \(150\) |
orering | \(\frac {x^{3} \left (252 b^{6} B \,x^{7}+280 A \,b^{6} x^{6}+1680 B a \,b^{5} x^{6}+1890 A a \,b^{5} x^{5}+4725 B \,a^{2} b^{4} x^{5}+5400 A \,a^{2} b^{4} x^{4}+7200 B \,a^{3} b^{3} x^{4}+8400 A \,a^{3} b^{3} x^{3}+6300 B \,a^{4} b^{2} x^{3}+7560 A \,a^{4} b^{2} x^{2}+3024 B \,a^{5} b \,x^{2}+3780 A \,a^{5} b x +630 B \,a^{6} x +840 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{2520 \left (b x +a \right )^{6}}\) | \(173\) |
Input:
int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/10*B*b^6*x^10+(1/9*A*b^6+2/3*B*a*b^5)*x^9+(3/4*A*a*b^5+15/8*B*a^2*b^4)*x ^8+(15/7*A*a^2*b^4+20/7*B*a^3*b^3)*x^7+(10/3*A*a^3*b^3+5/2*B*a^4*b^2)*x^6+ (3*A*a^4*b^2+6/5*B*a^5*b)*x^5+(3/2*A*a^5*b+1/4*B*a^6)*x^4+1/3*A*a^6*x^3
Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.69 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, B b^{6} x^{10} + \frac {1}{3} \, A a^{6} x^{3} + \frac {1}{9} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + \frac {3}{8} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + \frac {5}{6} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/10*B*b^6*x^10 + 1/3*A*a^6*x^3 + 1/9*(6*B*a*b^5 + A*b^6)*x^9 + 3/8*(5*B*a ^2*b^4 + 2*A*a*b^5)*x^8 + 5/7*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^7 + 5/6*(3*B*a ^4*b^2 + 4*A*a^3*b^3)*x^6 + 3/5*(2*B*a^5*b + 5*A*a^4*b^2)*x^5 + 1/4*(B*a^6 + 6*A*a^5*b)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (78) = 156\).
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.87 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {A a^{6} x^{3}}{3} + \frac {B b^{6} x^{10}}{10} + x^{9} \left (\frac {A b^{6}}{9} + \frac {2 B a b^{5}}{3}\right ) + x^{8} \cdot \left (\frac {3 A a b^{5}}{4} + \frac {15 B a^{2} b^{4}}{8}\right ) + x^{7} \cdot \left (\frac {15 A a^{2} b^{4}}{7} + \frac {20 B a^{3} b^{3}}{7}\right ) + x^{6} \cdot \left (\frac {10 A a^{3} b^{3}}{3} + \frac {5 B a^{4} b^{2}}{2}\right ) + x^{5} \cdot \left (3 A a^{4} b^{2} + \frac {6 B a^{5} b}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{5} b}{2} + \frac {B a^{6}}{4}\right ) \] Input:
integrate(x**2*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
A*a**6*x**3/3 + B*b**6*x**10/10 + x**9*(A*b**6/9 + 2*B*a*b**5/3) + x**8*(3 *A*a*b**5/4 + 15*B*a**2*b**4/8) + x**7*(15*A*a**2*b**4/7 + 20*B*a**3*b**3/ 7) + x**6*(10*A*a**3*b**3/3 + 5*B*a**4*b**2/2) + x**5*(3*A*a**4*b**2 + 6*B *a**5*b/5) + x**4*(3*A*a**5*b/2 + B*a**6/4)
Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.69 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, B b^{6} x^{10} + \frac {1}{3} \, A a^{6} x^{3} + \frac {1}{9} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + \frac {3}{8} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + \frac {5}{6} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
1/10*B*b^6*x^10 + 1/3*A*a^6*x^3 + 1/9*(6*B*a*b^5 + A*b^6)*x^9 + 3/8*(5*B*a ^2*b^4 + 2*A*a*b^5)*x^8 + 5/7*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^7 + 5/6*(3*B*a ^4*b^2 + 4*A*a^3*b^3)*x^6 + 3/5*(2*B*a^5*b + 5*A*a^4*b^2)*x^5 + 1/4*(B*a^6 + 6*A*a^5*b)*x^4
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.71 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{10} \, B b^{6} x^{10} + \frac {2}{3} \, B a b^{5} x^{9} + \frac {1}{9} \, A b^{6} x^{9} + \frac {15}{8} \, B a^{2} b^{4} x^{8} + \frac {3}{4} \, A a b^{5} x^{8} + \frac {20}{7} \, B a^{3} b^{3} x^{7} + \frac {15}{7} \, A a^{2} b^{4} x^{7} + \frac {5}{2} \, B a^{4} b^{2} x^{6} + \frac {10}{3} \, A a^{3} b^{3} x^{6} + \frac {6}{5} \, B a^{5} b x^{5} + 3 \, A a^{4} b^{2} x^{5} + \frac {1}{4} \, B a^{6} x^{4} + \frac {3}{2} \, A a^{5} b x^{4} + \frac {1}{3} \, A a^{6} x^{3} \] Input:
integrate(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
1/10*B*b^6*x^10 + 2/3*B*a*b^5*x^9 + 1/9*A*b^6*x^9 + 15/8*B*a^2*b^4*x^8 + 3 /4*A*a*b^5*x^8 + 20/7*B*a^3*b^3*x^7 + 15/7*A*a^2*b^4*x^7 + 5/2*B*a^4*b^2*x ^6 + 10/3*A*a^3*b^3*x^6 + 6/5*B*a^5*b*x^5 + 3*A*a^4*b^2*x^5 + 1/4*B*a^6*x^ 4 + 3/2*A*a^5*b*x^4 + 1/3*A*a^6*x^3
Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^4\,\left (\frac {B\,a^6}{4}+\frac {3\,A\,b\,a^5}{2}\right )+x^9\,\left (\frac {A\,b^6}{9}+\frac {2\,B\,a\,b^5}{3}\right )+\frac {A\,a^6\,x^3}{3}+\frac {B\,b^6\,x^{10}}{10}+\frac {5\,a^3\,b^2\,x^6\,\left (4\,A\,b+3\,B\,a\right )}{6}+\frac {5\,a^2\,b^3\,x^7\,\left (3\,A\,b+4\,B\,a\right )}{7}+\frac {3\,a^4\,b\,x^5\,\left (5\,A\,b+2\,B\,a\right )}{5}+\frac {3\,a\,b^4\,x^8\,\left (2\,A\,b+5\,B\,a\right )}{8} \] Input:
int(x^2*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
x^4*((B*a^6)/4 + (3*A*a^5*b)/2) + x^9*((A*b^6)/9 + (2*B*a*b^5)/3) + (A*a^6 *x^3)/3 + (B*b^6*x^10)/10 + (5*a^3*b^2*x^6*(4*A*b + 3*B*a))/6 + (5*a^2*b^3 *x^7*(3*A*b + 4*B*a))/7 + (3*a^4*b*x^5*(5*A*b + 2*B*a))/5 + (3*a*b^4*x^8*( 2*A*b + 5*B*a))/8
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x^{3} \left (36 b^{7} x^{7}+280 a \,b^{6} x^{6}+945 a^{2} b^{5} x^{5}+1800 a^{3} b^{4} x^{4}+2100 a^{4} b^{3} x^{3}+1512 a^{5} b^{2} x^{2}+630 a^{6} b x +120 a^{7}\right )}{360} \] Input:
int(x^2*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
(x**3*(120*a**7 + 630*a**6*b*x + 1512*a**5*b**2*x**2 + 2100*a**4*b**3*x**3 + 1800*a**3*b**4*x**4 + 945*a**2*b**5*x**5 + 280*a*b**6*x**6 + 36*b**7*x* *7))/360