Integrand size = 20, antiderivative size = 96 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {A b^3 x^{4+m}}{4+m}+\frac {b^2 (b B+3 A c) x^{5+m}}{5+m}+\frac {3 b c (b B+A c) x^{6+m}}{6+m}+\frac {c^2 (3 b B+A c) x^{7+m}}{7+m}+\frac {B c^3 x^{8+m}}{8+m} \]
A*b^3*x^(4+m)/(4+m)+b^2*(3*A*c+B*b)*x^(5+m)/(5+m)+3*b*c*(A*c+B*b)*x^(6+m)/ (6+m)+c^2*(A*c+3*B*b)*x^(7+m)/(7+m)+B*c^3*x^(8+m)/(8+m)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {x^{4+m} \left (B (b+c x)^4+(-b B (4+m)+A c (8+m)) \left (\frac {b^3}{4+m}+\frac {3 b^2 c x}{5+m}+\frac {3 b c^2 x^2}{6+m}+\frac {c^3 x^3}{7+m}\right )\right )}{c (8+m)} \]
(x^(4 + m)*(B*(b + c*x)^4 + (-(b*B*(4 + m)) + A*c*(8 + m))*(b^3/(4 + m) + (3*b^2*c*x)/(5 + m) + (3*b*c^2*x^2)/(6 + m) + (c^3*x^3)/(7 + m))))/(c*(8 + m))
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {9, 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.
| \(\displaystyle \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int x^{m+3} (A+B x) (b+c x)^3dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (A b^3 x^{m+3}+b^2 x^{m+4} (3 A c+b B)+c^2 x^{m+6} (A c+3 b B)+3 b c x^{m+5} (A c+b B)+B c^3 x^{m+7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A b^3 x^{m+4}}{m+4}+\frac {b^2 x^{m+5} (3 A c+b B)}{m+5}+\frac {c^2 x^{m+7} (A c+3 b B)}{m+7}+\frac {3 b c x^{m+6} (A c+b B)}{m+6}+\frac {B c^3 x^{m+8}}{m+8}\) |
(A*b^3*x^(4 + m))/(4 + m) + (b^2*(b*B + 3*A*c)*x^(5 + m))/(5 + m) + (3*b*c *(b*B + A*c)*x^(6 + m))/(6 + m) + (c^2*(3*b*B + A*c)*x^(7 + m))/(7 + m) + (B*c^3*x^(8 + m))/(8 + m)
3.1.27.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, 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])
Time = 0.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17
| method | result | size |
| norman | \(\frac {A \,b^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {B \,c^{3} x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}+\frac {b^{2} \left (3 A c +B b \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {c^{2} \left (A c +3 B b \right ) x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {3 b c \left (A c +B b \right ) x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}\) | \(112\) |
| gosper | \(\frac {x^{4+m} \left (B \,c^{3} m^{4} x^{4}+A \,c^{3} m^{4} x^{3}+3 B b \,c^{2} m^{4} x^{3}+22 B \,c^{3} m^{3} x^{4}+3 A b \,c^{2} m^{4} x^{2}+23 A \,c^{3} m^{3} x^{3}+3 B \,b^{2} c \,m^{4} x^{2}+69 B b \,c^{2} m^{3} x^{3}+179 B \,c^{3} m^{2} x^{4}+3 A \,b^{2} c \,m^{4} x +72 A b \,c^{2} m^{3} x^{2}+194 A \,c^{3} m^{2} x^{3}+B \,b^{3} m^{4} x +72 B \,b^{2} c \,m^{3} x^{2}+582 B b \,c^{2} m^{2} x^{3}+638 m \,x^{4} B \,c^{3}+A \,b^{3} m^{4}+75 A \,b^{2} c \,m^{3} x +633 A b \,c^{2} m^{2} x^{2}+712 A \,c^{3} m \,x^{3}+25 B \,b^{3} m^{3} x +633 B \,b^{2} c \,m^{2} x^{2}+2136 B b \,c^{2} m \,x^{3}+840 B \,c^{3} x^{4}+26 A \,b^{3} m^{3}+690 A \,b^{2} c \,m^{2} x +2412 A b \,c^{2} m \,x^{2}+960 A \,c^{3} x^{3}+230 B \,b^{3} m^{2} x +2412 B \,b^{2} c m \,x^{2}+2880 B b \,c^{2} x^{3}+251 A \,b^{3} m^{2}+2760 A \,b^{2} c m x +3360 A b \,c^{2} x^{2}+920 B \,b^{3} m x +3360 B \,b^{2} c \,x^{2}+1066 A \,b^{3} m +4032 A \,b^{2} c x +1344 B \,b^{3} x +1680 A \,b^{3}\right )}{\left (4+m \right ) \left (5+m \right ) \left (6+m \right ) \left (7+m \right ) \left (8+m \right )}\) | \(454\) |
| risch | \(\frac {x^{m} \left (B \,c^{3} m^{4} x^{4}+A \,c^{3} m^{4} x^{3}+3 B b \,c^{2} m^{4} x^{3}+22 B \,c^{3} m^{3} x^{4}+3 A b \,c^{2} m^{4} x^{2}+23 A \,c^{3} m^{3} x^{3}+3 B \,b^{2} c \,m^{4} x^{2}+69 B b \,c^{2} m^{3} x^{3}+179 B \,c^{3} m^{2} x^{4}+3 A \,b^{2} c \,m^{4} x +72 A b \,c^{2} m^{3} x^{2}+194 A \,c^{3} m^{2} x^{3}+B \,b^{3} m^{4} x +72 B \,b^{2} c \,m^{3} x^{2}+582 B b \,c^{2} m^{2} x^{3}+638 m \,x^{4} B \,c^{3}+A \,b^{3} m^{4}+75 A \,b^{2} c \,m^{3} x +633 A b \,c^{2} m^{2} x^{2}+712 A \,c^{3} m \,x^{3}+25 B \,b^{3} m^{3} x +633 B \,b^{2} c \,m^{2} x^{2}+2136 B b \,c^{2} m \,x^{3}+840 B \,c^{3} x^{4}+26 A \,b^{3} m^{3}+690 A \,b^{2} c \,m^{2} x +2412 A b \,c^{2} m \,x^{2}+960 A \,c^{3} x^{3}+230 B \,b^{3} m^{2} x +2412 B \,b^{2} c m \,x^{2}+2880 B b \,c^{2} x^{3}+251 A \,b^{3} m^{2}+2760 A \,b^{2} c m x +3360 A b \,c^{2} x^{2}+920 B \,b^{3} m x +3360 B \,b^{2} c \,x^{2}+1066 A \,b^{3} m +4032 A \,b^{2} c x +1344 B \,b^{3} x +1680 A \,b^{3}\right ) x^{4}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(455\) |
| parallelrisch | \(\frac {840 B \,x^{8} x^{m} c^{3}+960 A \,x^{7} x^{m} c^{3}+1344 B \,x^{5} x^{m} b^{3}+1680 A \,x^{4} x^{m} b^{3}+69 B \,x^{7} x^{m} b \,c^{2} m^{3}+75 A \,x^{5} x^{m} b^{2} c \,m^{3}+2136 B \,x^{7} x^{m} b \,c^{2} m +633 B \,x^{6} x^{m} b^{2} c \,m^{2}+2412 A \,x^{6} x^{m} b \,c^{2} m +690 A \,x^{5} x^{m} b^{2} c \,m^{2}+2412 B \,x^{6} x^{m} b^{2} c m +2760 A \,x^{5} x^{m} b^{2} c m +3 B \,x^{7} x^{m} b \,c^{2} m^{4}+3 A \,x^{6} x^{m} b \,c^{2} m^{4}+3 B \,x^{6} x^{m} b^{2} c \,m^{4}+72 A \,x^{6} x^{m} b \,c^{2} m^{3}+3 A \,x^{5} x^{m} b^{2} c \,m^{4}+582 B \,x^{7} x^{m} b \,c^{2} m^{2}+72 B \,x^{6} x^{m} b^{2} c \,m^{3}+633 A \,x^{6} x^{m} b \,c^{2} m^{2}+920 B \,x^{5} x^{m} b^{3} m +4032 A \,x^{5} x^{m} b^{2} c +1066 A \,x^{4} x^{m} b^{3} m +B \,x^{8} x^{m} c^{3} m^{4}+A \,x^{7} x^{m} c^{3} m^{4}+22 B \,x^{8} x^{m} c^{3} m^{3}+23 A \,x^{7} x^{m} c^{3} m^{3}+179 B \,x^{8} x^{m} c^{3} m^{2}+194 A \,x^{7} x^{m} c^{3} m^{2}+638 B \,x^{8} x^{m} c^{3} m +B \,x^{5} x^{m} b^{3} m^{4}+712 A \,x^{7} x^{m} c^{3} m +A \,x^{4} x^{m} b^{3} m^{4}+25 B \,x^{5} x^{m} b^{3} m^{3}+26 A \,x^{4} x^{m} b^{3} m^{3}+2880 B \,x^{7} x^{m} b \,c^{2}+230 B \,x^{5} x^{m} b^{3} m^{2}+3360 A \,x^{6} x^{m} b \,c^{2}+251 A \,x^{4} x^{m} b^{3} m^{2}+3360 B \,x^{6} x^{m} b^{2} c}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(604\) |
A*b^3/(4+m)*x^4*exp(m*ln(x))+B*c^3/(8+m)*x^8*exp(m*ln(x))+b^2*(3*A*c+B*b)/ (5+m)*x^5*exp(m*ln(x))+c^2*(A*c+3*B*b)/(7+m)*x^7*exp(m*ln(x))+3*b*c*(A*c+B *b)/(6+m)*x^6*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (96) = 192\).
Time = 0.27 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.97 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {{\left ({\left (B c^{3} m^{4} + 22 \, B c^{3} m^{3} + 179 \, B c^{3} m^{2} + 638 \, B c^{3} m + 840 \, B c^{3}\right )} x^{8} + {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} m^{4} + 2880 \, B b c^{2} + 960 \, A c^{3} + 23 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m^{3} + 194 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m^{2} + 712 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B b^{2} c + A b c^{2}\right )} m^{4} + 1120 \, B b^{2} c + 1120 \, A b c^{2} + 24 \, {\left (B b^{2} c + A b c^{2}\right )} m^{3} + 211 \, {\left (B b^{2} c + A b c^{2}\right )} m^{2} + 804 \, {\left (B b^{2} c + A b c^{2}\right )} m\right )} x^{6} + {\left ({\left (B b^{3} + 3 \, A b^{2} c\right )} m^{4} + 1344 \, B b^{3} + 4032 \, A b^{2} c + 25 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m^{3} + 230 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m^{2} + 920 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m\right )} x^{5} + {\left (A b^{3} m^{4} + 26 \, A b^{3} m^{3} + 251 \, A b^{3} m^{2} + 1066 \, A b^{3} m + 1680 \, A b^{3}\right )} x^{4}\right )} x^{m}}{m^{5} + 30 \, m^{4} + 355 \, m^{3} + 2070 \, m^{2} + 5944 \, m + 6720} \]
((B*c^3*m^4 + 22*B*c^3*m^3 + 179*B*c^3*m^2 + 638*B*c^3*m + 840*B*c^3)*x^8 + ((3*B*b*c^2 + A*c^3)*m^4 + 2880*B*b*c^2 + 960*A*c^3 + 23*(3*B*b*c^2 + A* c^3)*m^3 + 194*(3*B*b*c^2 + A*c^3)*m^2 + 712*(3*B*b*c^2 + A*c^3)*m)*x^7 + 3*((B*b^2*c + A*b*c^2)*m^4 + 1120*B*b^2*c + 1120*A*b*c^2 + 24*(B*b^2*c + A *b*c^2)*m^3 + 211*(B*b^2*c + A*b*c^2)*m^2 + 804*(B*b^2*c + A*b*c^2)*m)*x^6 + ((B*b^3 + 3*A*b^2*c)*m^4 + 1344*B*b^3 + 4032*A*b^2*c + 25*(B*b^3 + 3*A* b^2*c)*m^3 + 230*(B*b^3 + 3*A*b^2*c)*m^2 + 920*(B*b^3 + 3*A*b^2*c)*m)*x^5 + (A*b^3*m^4 + 26*A*b^3*m^3 + 251*A*b^3*m^2 + 1066*A*b^3*m + 1680*A*b^3)*x ^4)*x^m/(m^5 + 30*m^4 + 355*m^3 + 2070*m^2 + 5944*m + 6720)
Leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (87) = 174\).
Time = 0.53 (sec) , antiderivative size = 2026, normalized size of antiderivative = 21.10 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((-A*b**3/(4*x**4) - A*b**2*c/x**3 - 3*A*b*c**2/(2*x**2) - A*c**3 /x - B*b**3/(3*x**3) - 3*B*b**2*c/(2*x**2) - 3*B*b*c**2/x + B*c**3*log(x), Eq(m, -8)), (-A*b**3/(3*x**3) - 3*A*b**2*c/(2*x**2) - 3*A*b*c**2/x + A*c* *3*log(x) - B*b**3/(2*x**2) - 3*B*b**2*c/x + 3*B*b*c**2*log(x) + B*c**3*x, Eq(m, -7)), (-A*b**3/(2*x**2) - 3*A*b**2*c/x + 3*A*b*c**2*log(x) + A*c**3 *x - B*b**3/x + 3*B*b**2*c*log(x) + 3*B*b*c**2*x + B*c**3*x**2/2, Eq(m, -6 )), (-A*b**3/x + 3*A*b**2*c*log(x) + 3*A*b*c**2*x + A*c**3*x**2/2 + B*b**3 *log(x) + 3*B*b**2*c*x + 3*B*b*c**2*x**2/2 + B*c**3*x**3/3, Eq(m, -5)), (A *b**3*log(x) + 3*A*b**2*c*x + 3*A*b*c**2*x**2/2 + A*c**3*x**3/3 + B*b**3*x + 3*B*b**2*c*x**2/2 + B*b*c**2*x**3 + B*c**3*x**4/4, Eq(m, -4)), (A*b**3* m**4*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 2 6*A*b**3*m**3*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 251*A*b**3*m**2*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 1066*A*b**3*m*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 207 0*m**2 + 5944*m + 6720) + 1680*A*b**3*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 3*A*b**2*c*m**4*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 75*A*b**2*c*m**3*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 690*A*b**2*c*m**2*x** 5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 2760*A*b* *2*c*m*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720...
Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {B c^{3} x^{m + 8}}{m + 8} + \frac {3 \, B b c^{2} x^{m + 7}}{m + 7} + \frac {A c^{3} x^{m + 7}}{m + 7} + \frac {3 \, B b^{2} c x^{m + 6}}{m + 6} + \frac {3 \, A b c^{2} x^{m + 6}}{m + 6} + \frac {B b^{3} x^{m + 5}}{m + 5} + \frac {3 \, A b^{2} c x^{m + 5}}{m + 5} + \frac {A b^{3} x^{m + 4}}{m + 4} \]
B*c^3*x^(m + 8)/(m + 8) + 3*B*b*c^2*x^(m + 7)/(m + 7) + A*c^3*x^(m + 7)/(m + 7) + 3*B*b^2*c*x^(m + 6)/(m + 6) + 3*A*b*c^2*x^(m + 6)/(m + 6) + B*b^3* x^(m + 5)/(m + 5) + 3*A*b^2*c*x^(m + 5)/(m + 5) + A*b^3*x^(m + 4)/(m + 4)
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (96) = 192\).
Time = 0.29 (sec) , antiderivative size = 603, normalized size of antiderivative = 6.28 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {B c^{3} m^{4} x^{8} x^{m} + 3 \, B b c^{2} m^{4} x^{7} x^{m} + A c^{3} m^{4} x^{7} x^{m} + 22 \, B c^{3} m^{3} x^{8} x^{m} + 3 \, B b^{2} c m^{4} x^{6} x^{m} + 3 \, A b c^{2} m^{4} x^{6} x^{m} + 69 \, B b c^{2} m^{3} x^{7} x^{m} + 23 \, A c^{3} m^{3} x^{7} x^{m} + 179 \, B c^{3} m^{2} x^{8} x^{m} + B b^{3} m^{4} x^{5} x^{m} + 3 \, A b^{2} c m^{4} x^{5} x^{m} + 72 \, B b^{2} c m^{3} x^{6} x^{m} + 72 \, A b c^{2} m^{3} x^{6} x^{m} + 582 \, B b c^{2} m^{2} x^{7} x^{m} + 194 \, A c^{3} m^{2} x^{7} x^{m} + 638 \, B c^{3} m x^{8} x^{m} + A b^{3} m^{4} x^{4} x^{m} + 25 \, B b^{3} m^{3} x^{5} x^{m} + 75 \, A b^{2} c m^{3} x^{5} x^{m} + 633 \, B b^{2} c m^{2} x^{6} x^{m} + 633 \, A b c^{2} m^{2} x^{6} x^{m} + 2136 \, B b c^{2} m x^{7} x^{m} + 712 \, A c^{3} m x^{7} x^{m} + 840 \, B c^{3} x^{8} x^{m} + 26 \, A b^{3} m^{3} x^{4} x^{m} + 230 \, B b^{3} m^{2} x^{5} x^{m} + 690 \, A b^{2} c m^{2} x^{5} x^{m} + 2412 \, B b^{2} c m x^{6} x^{m} + 2412 \, A b c^{2} m x^{6} x^{m} + 2880 \, B b c^{2} x^{7} x^{m} + 960 \, A c^{3} x^{7} x^{m} + 251 \, A b^{3} m^{2} x^{4} x^{m} + 920 \, B b^{3} m x^{5} x^{m} + 2760 \, A b^{2} c m x^{5} x^{m} + 3360 \, B b^{2} c x^{6} x^{m} + 3360 \, A b c^{2} x^{6} x^{m} + 1066 \, A b^{3} m x^{4} x^{m} + 1344 \, B b^{3} x^{5} x^{m} + 4032 \, A b^{2} c x^{5} x^{m} + 1680 \, A b^{3} x^{4} x^{m}}{m^{5} + 30 \, m^{4} + 355 \, m^{3} + 2070 \, m^{2} + 5944 \, m + 6720} \]
(B*c^3*m^4*x^8*x^m + 3*B*b*c^2*m^4*x^7*x^m + A*c^3*m^4*x^7*x^m + 22*B*c^3* m^3*x^8*x^m + 3*B*b^2*c*m^4*x^6*x^m + 3*A*b*c^2*m^4*x^6*x^m + 69*B*b*c^2*m ^3*x^7*x^m + 23*A*c^3*m^3*x^7*x^m + 179*B*c^3*m^2*x^8*x^m + B*b^3*m^4*x^5* x^m + 3*A*b^2*c*m^4*x^5*x^m + 72*B*b^2*c*m^3*x^6*x^m + 72*A*b*c^2*m^3*x^6* x^m + 582*B*b*c^2*m^2*x^7*x^m + 194*A*c^3*m^2*x^7*x^m + 638*B*c^3*m*x^8*x^ m + A*b^3*m^4*x^4*x^m + 25*B*b^3*m^3*x^5*x^m + 75*A*b^2*c*m^3*x^5*x^m + 63 3*B*b^2*c*m^2*x^6*x^m + 633*A*b*c^2*m^2*x^6*x^m + 2136*B*b*c^2*m*x^7*x^m + 712*A*c^3*m*x^7*x^m + 840*B*c^3*x^8*x^m + 26*A*b^3*m^3*x^4*x^m + 230*B*b^ 3*m^2*x^5*x^m + 690*A*b^2*c*m^2*x^5*x^m + 2412*B*b^2*c*m*x^6*x^m + 2412*A* b*c^2*m*x^6*x^m + 2880*B*b*c^2*x^7*x^m + 960*A*c^3*x^7*x^m + 251*A*b^3*m^2 *x^4*x^m + 920*B*b^3*m*x^5*x^m + 2760*A*b^2*c*m*x^5*x^m + 3360*B*b^2*c*x^6 *x^m + 3360*A*b*c^2*x^6*x^m + 1066*A*b^3*m*x^4*x^m + 1344*B*b^3*x^5*x^m + 4032*A*b^2*c*x^5*x^m + 1680*A*b^3*x^4*x^m)/(m^5 + 30*m^4 + 355*m^3 + 2070* m^2 + 5944*m + 6720)
Time = 10.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.03 \[ \int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {A\,b^3\,x^m\,x^4\,\left (m^4+26\,m^3+251\,m^2+1066\,m+1680\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {B\,c^3\,x^m\,x^8\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {b^2\,x^m\,x^5\,\left (3\,A\,c+B\,b\right )\,\left (m^4+25\,m^3+230\,m^2+920\,m+1344\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {c^2\,x^m\,x^7\,\left (A\,c+3\,B\,b\right )\,\left (m^4+23\,m^3+194\,m^2+712\,m+960\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {3\,b\,c\,x^m\,x^6\,\left (A\,c+B\,b\right )\,\left (m^4+24\,m^3+211\,m^2+804\,m+1120\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720} \]
(A*b^3*x^m*x^4*(1066*m + 251*m^2 + 26*m^3 + m^4 + 1680))/(5944*m + 2070*m^ 2 + 355*m^3 + 30*m^4 + m^5 + 6720) + (B*c^3*x^m*x^8*(638*m + 179*m^2 + 22* m^3 + m^4 + 840))/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720) + (b ^2*x^m*x^5*(3*A*c + B*b)*(920*m + 230*m^2 + 25*m^3 + m^4 + 1344))/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720) + (c^2*x^m*x^7*(A*c + 3*B*b)*( 712*m + 194*m^2 + 23*m^3 + m^4 + 960))/(5944*m + 2070*m^2 + 355*m^3 + 30*m ^4 + m^5 + 6720) + (3*b*c*x^m*x^6*(A*c + B*b)*(804*m + 211*m^2 + 24*m^3 + m^4 + 1120))/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720)