Integrand size = 16, antiderivative size = 96 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{2+m}}{2+m}+\frac {3 a b (A b+a B) x^{3+m}}{3+m}+\frac {b^2 (A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 B x^{5+m}}{5+m} \]
a^3*A*x^(1+m)/(1+m)+a^2*(3*A*b+B*a)*x^(2+m)/(2+m)+3*a*b*(A*b+B*a)*x^(3+m)/ (3+m)+b^2*(A*b+3*B*a)*x^(4+m)/(4+m)+b^3*B*x^(5+m)/(5+m)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {x^{1+m} \left (B (a+b x)^4+(-a B (1+m)+A b (5+m)) \left (\frac {a^3}{1+m}+\frac {3 a^2 b x}{2+m}+\frac {3 a b^2 x^2}{3+m}+\frac {b^3 x^3}{4+m}\right )\right )}{b (5+m)} \]
(x^(1 + m)*(B*(a + b*x)^4 + (-(a*B*(1 + m)) + A*b*(5 + m))*(a^3/(1 + m) + (3*a^2*b*x)/(2 + m) + (3*a*b^2*x^2)/(3 + m) + (b^3*x^3)/(4 + m))))/(b*(5 + m))
Time = 0.24 (sec) , antiderivative size = 96, 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 = {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.
| \(\displaystyle \int x^m (a+b x)^3 (A+B x) \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (a^3 A x^m+a^2 x^{m+1} (a B+3 A b)+b^2 x^{m+3} (3 a B+A b)+3 a b x^{m+2} (a B+A b)+b^3 B x^{m+4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A x^{m+1}}{m+1}+\frac {a^2 x^{m+2} (a B+3 A b)}{m+2}+\frac {b^2 x^{m+4} (3 a B+A b)}{m+4}+\frac {3 a b x^{m+3} (a B+A b)}{m+3}+\frac {b^3 B x^{m+5}}{m+5}\) |
(a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(2 + m))/(2 + m) + (3*a*b *(A*b + a*B)*x^(3 + m))/(3 + m) + (b^2*(A*b + 3*a*B)*x^(4 + m))/(4 + m) + (b^3*B*x^(5 + m))/(5 + m)
3.4.72.3.1 Defintions of rubi rules used
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 = 0.46 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15
| method | result | size |
| norman | \(\frac {a^{2} \left (3 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {a^{3} A x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{2} \left (A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {b^{3} B \,x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {3 a b \left (A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\) | \(110\) |
| risch | \(\frac {x \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right ) x^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(453\) |
| gosper | \(\frac {x^{1+m} \left (B \,b^{3} m^{4} x^{4}+A \,b^{3} m^{4} x^{3}+3 B a \,b^{2} m^{4} x^{3}+10 B \,b^{3} m^{3} x^{4}+3 A a \,b^{2} m^{4} x^{2}+11 A \,b^{3} m^{3} x^{3}+3 B \,a^{2} b \,m^{4} x^{2}+33 B a \,b^{2} m^{3} x^{3}+35 B \,b^{3} m^{2} x^{4}+3 A \,a^{2} b \,m^{4} x +36 A a \,b^{2} m^{3} x^{2}+41 A \,b^{3} m^{2} x^{3}+B \,a^{3} m^{4} x +36 B \,a^{2} b \,m^{3} x^{2}+123 B a \,b^{2} m^{2} x^{3}+50 m \,x^{4} b^{3} B +A \,a^{3} m^{4}+39 A \,a^{2} b \,m^{3} x +147 A a \,b^{2} m^{2} x^{2}+61 A \,b^{3} x^{3} m +13 B \,a^{3} m^{3} x +147 B \,a^{2} b \,m^{2} x^{2}+183 B a \,b^{2} x^{3} m +24 b^{3} B \,x^{4}+14 A \,a^{3} m^{3}+177 A \,a^{2} b \,m^{2} x +234 a A \,b^{2} x^{2} m +30 A \,b^{3} x^{3}+59 B \,a^{3} m^{2} x +234 B \,a^{2} b \,x^{2} m +90 B a \,b^{2} x^{3}+71 A \,a^{3} m^{2}+321 a^{2} A b x m +120 a A \,b^{2} x^{2}+107 a^{3} B x m +120 B \,a^{2} b \,x^{2}+154 a^{3} A m +180 a^{2} A b x +60 a^{3} B x +120 a^{3} A \right )}{\left (1+m \right ) \left (2+m \right ) \left (3+m \right ) \left (4+m \right ) \left (5+m \right )}\) | \(454\) |
| parallelrisch | \(\frac {60 B \,x^{2} x^{m} a^{3}+120 A x \,x^{m} a^{3}+24 B \,x^{5} x^{m} b^{3}+30 A \,x^{4} x^{m} b^{3}+123 B \,x^{4} x^{m} a \,b^{2} m^{2}+36 B \,x^{3} x^{m} a^{2} b \,m^{3}+147 A \,x^{3} x^{m} a \,b^{2} m^{2}+39 A \,x^{2} x^{m} a^{2} b \,m^{3}+183 B \,x^{4} x^{m} a \,b^{2} m +147 B \,x^{3} x^{m} a^{2} b \,m^{2}+234 A \,x^{3} x^{m} a \,b^{2} m +177 A \,x^{2} x^{m} a^{2} b \,m^{2}+234 B \,x^{3} x^{m} a^{2} b m +321 A \,x^{2} x^{m} a^{2} b m +3 B \,x^{4} x^{m} a \,b^{2} m^{4}+3 A \,x^{3} x^{m} a \,b^{2} m^{4}+33 B \,x^{4} x^{m} a \,b^{2} m^{3}+3 B \,x^{3} x^{m} a^{2} b \,m^{4}+36 A \,x^{3} x^{m} a \,b^{2} m^{3}+3 A \,x^{2} x^{m} a^{2} b \,m^{4}+B \,x^{5} x^{m} b^{3} m^{4}+A \,x^{4} x^{m} b^{3} m^{4}+10 B \,x^{5} x^{m} b^{3} m^{3}+11 A \,x^{4} x^{m} b^{3} m^{3}+35 B \,x^{5} x^{m} b^{3} m^{2}+41 A \,x^{4} x^{m} b^{3} m^{2}+50 B \,x^{5} x^{m} b^{3} m +B \,x^{2} x^{m} a^{3} m^{4}+61 A \,x^{4} x^{m} b^{3} m +A x \,x^{m} a^{3} m^{4}+13 B \,x^{2} x^{m} a^{3} m^{3}+14 A x \,x^{m} a^{3} m^{3}+90 B \,x^{4} x^{m} a \,b^{2}+59 B \,x^{2} x^{m} a^{3} m^{2}+120 A \,x^{3} x^{m} a \,b^{2}+71 A x \,x^{m} a^{3} m^{2}+120 B \,x^{3} x^{m} a^{2} b +107 B \,x^{2} x^{m} a^{3} m +180 A \,x^{2} x^{m} a^{2} b +154 A x \,x^{m} a^{3} m}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(594\) |
a^2*(3*A*b+B*a)/(2+m)*x^2*exp(m*ln(x))+a^3*A/(1+m)*x*exp(m*ln(x))+b^2*(A*b +3*B*a)/(4+m)*x^4*exp(m*ln(x))+b^3*B/(5+m)*x^5*exp(m*ln(x))+3*a*b*(A*b+B*a )/(3+m)*x^3*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (96) = 192\).
Time = 0.23 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.95 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {{\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} + {\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]
((B*b^3*m^4 + 10*B*b^3*m^3 + 35*B*b^3*m^2 + 50*B*b^3*m + 24*B*b^3)*x^5 + ( (3*B*a*b^2 + A*b^3)*m^4 + 90*B*a*b^2 + 30*A*b^3 + 11*(3*B*a*b^2 + A*b^3)*m ^3 + 41*(3*B*a*b^2 + A*b^3)*m^2 + 61*(3*B*a*b^2 + A*b^3)*m)*x^4 + 3*((B*a^ 2*b + A*a*b^2)*m^4 + 40*B*a^2*b + 40*A*a*b^2 + 12*(B*a^2*b + A*a*b^2)*m^3 + 49*(B*a^2*b + A*a*b^2)*m^2 + 78*(B*a^2*b + A*a*b^2)*m)*x^3 + ((B*a^3 + 3 *A*a^2*b)*m^4 + 60*B*a^3 + 180*A*a^2*b + 13*(B*a^3 + 3*A*a^2*b)*m^3 + 59*( B*a^3 + 3*A*a^2*b)*m^2 + 107*(B*a^3 + 3*A*a^2*b)*m)*x^2 + (A*a^3*m^4 + 14* A*a^3*m^3 + 71*A*a^3*m^2 + 154*A*a^3*m + 120*A*a^3)*x)*x^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
Leaf count of result is larger than twice the leaf count of optimal. 2018 vs. \(2 (87) = 174\).
Time = 0.42 (sec) , antiderivative size = 2018, normalized size of antiderivative = 21.02 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\text {Too large to display} \]
Piecewise((-A*a**3/(4*x**4) - A*a**2*b/x**3 - 3*A*a*b**2/(2*x**2) - A*b**3 /x - B*a**3/(3*x**3) - 3*B*a**2*b/(2*x**2) - 3*B*a*b**2/x + B*b**3*log(x), Eq(m, -5)), (-A*a**3/(3*x**3) - 3*A*a**2*b/(2*x**2) - 3*A*a*b**2/x + A*b* *3*log(x) - B*a**3/(2*x**2) - 3*B*a**2*b/x + 3*B*a*b**2*log(x) + B*b**3*x, Eq(m, -4)), (-A*a**3/(2*x**2) - 3*A*a**2*b/x + 3*A*a*b**2*log(x) + A*b**3 *x - B*a**3/x + 3*B*a**2*b*log(x) + 3*B*a*b**2*x + B*b**3*x**2/2, Eq(m, -3 )), (-A*a**3/x + 3*A*a**2*b*log(x) + 3*A*a*b**2*x + A*b**3*x**2/2 + B*a**3 *log(x) + 3*B*a**2*b*x + 3*B*a*b**2*x**2/2 + B*b**3*x**3/3, Eq(m, -2)), (A *a**3*log(x) + 3*A*a**2*b*x + 3*A*a*b**2*x**2/2 + A*b**3*x**3/3 + B*a**3*x + 3*B*a**2*b*x**2/2 + B*a*b**2*x**3 + B*b**3*x**4/4, Eq(m, -1)), (A*a**3* m**4*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 14*A*a** 3*m**3*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 71*A*a **3*m**2*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 154* A*a**3*m*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 120* A*a**3*x*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 3*A*a* *2*b*m**4*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 39*A*a**2*b*m**3*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 177*A*a**2*b*m**2*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 321*A*a**2*b*m*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + 225*m* *2 + 274*m + 120) + 180*A*a**2*b*x**2*x**m/(m**5 + 15*m**4 + 85*m**3 + ...
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {B b^{3} x^{m + 5}}{m + 5} + \frac {3 \, B a b^{2} x^{m + 4}}{m + 4} + \frac {A b^{3} x^{m + 4}}{m + 4} + \frac {3 \, B a^{2} b x^{m + 3}}{m + 3} + \frac {3 \, A a b^{2} x^{m + 3}}{m + 3} + \frac {B a^{3} x^{m + 2}}{m + 2} + \frac {3 \, A a^{2} b x^{m + 2}}{m + 2} + \frac {A a^{3} x^{m + 1}}{m + 1} \]
B*b^3*x^(m + 5)/(m + 5) + 3*B*a*b^2*x^(m + 4)/(m + 4) + A*b^3*x^(m + 4)/(m + 4) + 3*B*a^2*b*x^(m + 3)/(m + 3) + 3*A*a*b^2*x^(m + 3)/(m + 3) + B*a^3* x^(m + 2)/(m + 2) + 3*A*a^2*b*x^(m + 2)/(m + 2) + A*a^3*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (96) = 192\).
Time = 0.29 (sec) , antiderivative size = 593, normalized size of antiderivative = 6.18 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {B b^{3} m^{4} x^{5} x^{m} + 3 \, B a b^{2} m^{4} x^{4} x^{m} + A b^{3} m^{4} x^{4} x^{m} + 10 \, B b^{3} m^{3} x^{5} x^{m} + 3 \, B a^{2} b m^{4} x^{3} x^{m} + 3 \, A a b^{2} m^{4} x^{3} x^{m} + 33 \, B a b^{2} m^{3} x^{4} x^{m} + 11 \, A b^{3} m^{3} x^{4} x^{m} + 35 \, B b^{3} m^{2} x^{5} x^{m} + B a^{3} m^{4} x^{2} x^{m} + 3 \, A a^{2} b m^{4} x^{2} x^{m} + 36 \, B a^{2} b m^{3} x^{3} x^{m} + 36 \, A a b^{2} m^{3} x^{3} x^{m} + 123 \, B a b^{2} m^{2} x^{4} x^{m} + 41 \, A b^{3} m^{2} x^{4} x^{m} + 50 \, B b^{3} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 13 \, B a^{3} m^{3} x^{2} x^{m} + 39 \, A a^{2} b m^{3} x^{2} x^{m} + 147 \, B a^{2} b m^{2} x^{3} x^{m} + 147 \, A a b^{2} m^{2} x^{3} x^{m} + 183 \, B a b^{2} m x^{4} x^{m} + 61 \, A b^{3} m x^{4} x^{m} + 24 \, B b^{3} x^{5} x^{m} + 14 \, A a^{3} m^{3} x x^{m} + 59 \, B a^{3} m^{2} x^{2} x^{m} + 177 \, A a^{2} b m^{2} x^{2} x^{m} + 234 \, B a^{2} b m x^{3} x^{m} + 234 \, A a b^{2} m x^{3} x^{m} + 90 \, B a b^{2} x^{4} x^{m} + 30 \, A b^{3} x^{4} x^{m} + 71 \, A a^{3} m^{2} x x^{m} + 107 \, B a^{3} m x^{2} x^{m} + 321 \, A a^{2} b m x^{2} x^{m} + 120 \, B a^{2} b x^{3} x^{m} + 120 \, A a b^{2} x^{3} x^{m} + 154 \, A a^{3} m x x^{m} + 60 \, B a^{3} x^{2} x^{m} + 180 \, A a^{2} b x^{2} x^{m} + 120 \, A a^{3} x x^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]
(B*b^3*m^4*x^5*x^m + 3*B*a*b^2*m^4*x^4*x^m + A*b^3*m^4*x^4*x^m + 10*B*b^3* m^3*x^5*x^m + 3*B*a^2*b*m^4*x^3*x^m + 3*A*a*b^2*m^4*x^3*x^m + 33*B*a*b^2*m ^3*x^4*x^m + 11*A*b^3*m^3*x^4*x^m + 35*B*b^3*m^2*x^5*x^m + B*a^3*m^4*x^2*x ^m + 3*A*a^2*b*m^4*x^2*x^m + 36*B*a^2*b*m^3*x^3*x^m + 36*A*a*b^2*m^3*x^3*x ^m + 123*B*a*b^2*m^2*x^4*x^m + 41*A*b^3*m^2*x^4*x^m + 50*B*b^3*m*x^5*x^m + A*a^3*m^4*x*x^m + 13*B*a^3*m^3*x^2*x^m + 39*A*a^2*b*m^3*x^2*x^m + 147*B*a ^2*b*m^2*x^3*x^m + 147*A*a*b^2*m^2*x^3*x^m + 183*B*a*b^2*m*x^4*x^m + 61*A* b^3*m*x^4*x^m + 24*B*b^3*x^5*x^m + 14*A*a^3*m^3*x*x^m + 59*B*a^3*m^2*x^2*x ^m + 177*A*a^2*b*m^2*x^2*x^m + 234*B*a^2*b*m*x^3*x^m + 234*A*a*b^2*m*x^3*x ^m + 90*B*a*b^2*x^4*x^m + 30*A*b^3*x^4*x^m + 71*A*a^3*m^2*x*x^m + 107*B*a^ 3*m*x^2*x^m + 321*A*a^2*b*m*x^2*x^m + 120*B*a^2*b*x^3*x^m + 120*A*a*b^2*x^ 3*x^m + 154*A*a^3*m*x*x^m + 60*B*a^3*x^2*x^m + 180*A*a^2*b*x^2*x^m + 120*A *a^3*x*x^m)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
Time = 0.59 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.01 \[ \int x^m (a+b x)^3 (A+B x) \, dx=\frac {A\,a^3\,x\,x^m\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {B\,b^3\,x^m\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {a^2\,x^m\,x^2\,\left (3\,A\,b+B\,a\right )\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b^2\,x^m\,x^4\,\left (A\,b+3\,B\,a\right )\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {3\,a\,b\,x^m\,x^3\,\left (A\,b+B\,a\right )\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120} \]
(A*a^3*x*x^m*(154*m + 71*m^2 + 14*m^3 + m^4 + 120))/(274*m + 225*m^2 + 85* m^3 + 15*m^4 + m^5 + 120) + (B*b^3*x^m*x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (a^2*x^m*x^2*(3*A* b + B*a)*(107*m + 59*m^2 + 13*m^3 + m^4 + 60))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (b^2*x^m*x^4*(A*b + 3*B*a)*(61*m + 41*m^2 + 11*m^3 + m^4 + 30))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (3*a*b*x^m* x^3*(A*b + B*a)*(78*m + 49*m^2 + 12*m^3 + m^4 + 40))/(274*m + 225*m^2 + 85 *m^3 + 15*m^4 + m^5 + 120)