Integrand size = 13, antiderivative size = 97 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\frac {a^5 x^{1+m}}{1+m}+\frac {5 a^4 b x^{4+m}}{4+m}+\frac {10 a^3 b^2 x^{7+m}}{7+m}+\frac {10 a^2 b^3 x^{10+m}}{10+m}+\frac {5 a b^4 x^{13+m}}{13+m}+\frac {b^5 x^{16+m}}{16+m} \]
a^5*x^(1+m)/(1+m)+5*a^4*b*x^(4+m)/(4+m)+10*a^3*b^2*x^(7+m)/(7+m)+10*a^2*b^ 3*x^(10+m)/(10+m)+5*a*b^4*x^(13+m)/(13+m)+b^5*x^(16+m)/(16+m)
Time = 0.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int x^m \left (a+b x^3\right )^5 \, dx=x^{1+m} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^3}{4+m}+\frac {10 a^3 b^2 x^6}{7+m}+\frac {10 a^2 b^3 x^9}{10+m}+\frac {5 a b^4 x^{12}}{13+m}+\frac {b^5 x^{15}}{16+m}\right ) \]
x^(1 + m)*(a^5/(1 + m) + (5*a^4*b*x^3)/(4 + m) + (10*a^3*b^2*x^6)/(7 + m) + (10*a^2*b^3*x^9)/(10 + m) + (5*a*b^4*x^12)/(13 + m) + (b^5*x^15)/(16 + m ))
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {802, 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.
| \(\displaystyle \int x^m \left (a+b x^3\right )^5 \, dx\) |
| \(\Big \downarrow \) 802 |
| \(\displaystyle \int \left (a^5 x^m+5 a^4 b x^{m+3}+10 a^3 b^2 x^{m+6}+10 a^2 b^3 x^{m+9}+5 a b^4 x^{m+12}+b^5 x^{m+15}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5 x^{m+1}}{m+1}+\frac {5 a^4 b x^{m+4}}{m+4}+\frac {10 a^3 b^2 x^{m+7}}{m+7}+\frac {10 a^2 b^3 x^{m+10}}{m+10}+\frac {5 a b^4 x^{m+13}}{m+13}+\frac {b^5 x^{m+16}}{m+16}\) |
(a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(4 + m))/(4 + m) + (10*a^3*b^2*x^(7 + m))/(7 + m) + (10*a^2*b^3*x^(10 + m))/(10 + m) + (5*a*b^4*x^(13 + m))/(13 + m) + (b^5*x^(16 + m))/(16 + m)
3.6.84.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(97)=194\).
Time = 4.26 (sec) , antiderivative size = 431, normalized size of antiderivative = 4.44
| method | result | size |
| risch | \(\frac {x \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} a \,b^{4}+5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} a^{4} b +955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right ) x^{m}}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(431\) |
| gosper | \(\frac {x^{1+m} \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} a \,b^{4}+5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} a^{4} b +955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right )}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(432\) |
| parallelrisch | \(\frac {5 x^{13} x^{m} a \,b^{4} m^{5}+190 x^{13} x^{m} a \,b^{4} m^{4}+2555 x^{13} x^{m} a \,b^{4} m^{3}+14810 x^{13} x^{m} a \,b^{4} m^{2}+10 x^{10} x^{m} a^{2} b^{3} m^{5}+34840 x^{13} x^{m} a \,b^{4} m +410 x^{10} x^{m} a^{2} b^{3} m^{4}+5950 x^{10} x^{m} a^{2} b^{3} m^{3}+36550 x^{10} x^{m} a^{2} b^{3} m^{2}+10 x^{7} x^{m} a^{3} b^{2} m^{5}+89240 x^{10} x^{m} a^{2} b^{3} m +440 x^{7} x^{m} a^{3} b^{2} m^{4}+6970 x^{7} x^{m} a^{3} b^{2} m^{3}+47260 x^{7} x^{m} a^{3} b^{2} m^{2}+5 x^{4} x^{m} a^{4} b \,m^{5}+123920 x^{7} x^{m} a^{3} b^{2} m +235 x^{4} x^{m} a^{4} b \,m^{4}+4085 x^{4} x^{m} a^{4} b \,m^{3}+31685 x^{4} x^{m} a^{4} b \,m^{2}+100630 x^{4} x^{m} a^{4} b m +x^{16} x^{m} b^{5} m^{5}+35 x^{16} x^{m} b^{5} m^{4}+445 x^{16} x^{m} b^{5} m^{3}+2485 x^{16} x^{m} b^{5} m^{2}+5714 x^{16} x^{m} b^{5} m +22400 x^{13} x^{m} a \,b^{4}+58240 x^{10} x^{m} a^{2} b^{3}+83200 x^{7} x^{m} a^{3} b^{2}+x \,x^{m} a^{5} m^{5}+50 x \,x^{m} a^{5} m^{4}+72800 x^{4} x^{m} a^{4} b +955 x \,x^{m} a^{5} m^{3}+8650 x \,x^{m} a^{5} m^{2}+36824 x \,x^{m} a^{5} m +3640 x^{16} x^{m} b^{5}+58240 x \,x^{m} a^{5}}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(541\) |
x*(b^5*m^5*x^15+35*b^5*m^4*x^15+445*b^5*m^3*x^15+5*a*b^4*m^5*x^12+2485*b^5 *m^2*x^15+190*a*b^4*m^4*x^12+5714*b^5*m*x^15+2555*a*b^4*m^3*x^12+3640*b^5* x^15+10*a^2*b^3*m^5*x^9+14810*a*b^4*m^2*x^12+410*a^2*b^3*m^4*x^9+34840*a*b ^4*m*x^12+5950*a^2*b^3*m^3*x^9+22400*a*b^4*x^12+10*a^3*b^2*m^5*x^6+36550*a ^2*b^3*m^2*x^9+440*a^3*b^2*m^4*x^6+89240*a^2*b^3*m*x^9+6970*a^3*b^2*m^3*x^ 6+58240*a^2*b^3*x^9+5*a^4*b*m^5*x^3+47260*a^3*b^2*m^2*x^6+235*a^4*b*m^4*x^ 3+123920*a^3*b^2*m*x^6+4085*a^4*b*m^3*x^3+83200*a^3*b^2*x^6+a^5*m^5+31685* a^4*b*m^2*x^3+50*a^5*m^4+100630*a^4*b*m*x^3+955*a^5*m^3+72800*a^4*b*x^3+86 50*a^5*m^2+36824*a^5*m+58240*a^5)*x^m/(1+m)/(4+m)/(7+m)/(10+m)/(13+m)/(16+ m)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (97) = 194\).
Time = 0.42 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.78 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\frac {{\left ({\left (b^{5} m^{5} + 35 \, b^{5} m^{4} + 445 \, b^{5} m^{3} + 2485 \, b^{5} m^{2} + 5714 \, b^{5} m + 3640 \, b^{5}\right )} x^{16} + 5 \, {\left (a b^{4} m^{5} + 38 \, a b^{4} m^{4} + 511 \, a b^{4} m^{3} + 2962 \, a b^{4} m^{2} + 6968 \, a b^{4} m + 4480 \, a b^{4}\right )} x^{13} + 10 \, {\left (a^{2} b^{3} m^{5} + 41 \, a^{2} b^{3} m^{4} + 595 \, a^{2} b^{3} m^{3} + 3655 \, a^{2} b^{3} m^{2} + 8924 \, a^{2} b^{3} m + 5824 \, a^{2} b^{3}\right )} x^{10} + 10 \, {\left (a^{3} b^{2} m^{5} + 44 \, a^{3} b^{2} m^{4} + 697 \, a^{3} b^{2} m^{3} + 4726 \, a^{3} b^{2} m^{2} + 12392 \, a^{3} b^{2} m + 8320 \, a^{3} b^{2}\right )} x^{7} + 5 \, {\left (a^{4} b m^{5} + 47 \, a^{4} b m^{4} + 817 \, a^{4} b m^{3} + 6337 \, a^{4} b m^{2} + 20126 \, a^{4} b m + 14560 \, a^{4} b\right )} x^{4} + {\left (a^{5} m^{5} + 50 \, a^{5} m^{4} + 955 \, a^{5} m^{3} + 8650 \, a^{5} m^{2} + 36824 \, a^{5} m + 58240 \, a^{5}\right )} x\right )} x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \]
((b^5*m^5 + 35*b^5*m^4 + 445*b^5*m^3 + 2485*b^5*m^2 + 5714*b^5*m + 3640*b^ 5)*x^16 + 5*(a*b^4*m^5 + 38*a*b^4*m^4 + 511*a*b^4*m^3 + 2962*a*b^4*m^2 + 6 968*a*b^4*m + 4480*a*b^4)*x^13 + 10*(a^2*b^3*m^5 + 41*a^2*b^3*m^4 + 595*a^ 2*b^3*m^3 + 3655*a^2*b^3*m^2 + 8924*a^2*b^3*m + 5824*a^2*b^3)*x^10 + 10*(a ^3*b^2*m^5 + 44*a^3*b^2*m^4 + 697*a^3*b^2*m^3 + 4726*a^3*b^2*m^2 + 12392*a ^3*b^2*m + 8320*a^3*b^2)*x^7 + 5*(a^4*b*m^5 + 47*a^4*b*m^4 + 817*a^4*b*m^3 + 6337*a^4*b*m^2 + 20126*a^4*b*m + 14560*a^4*b)*x^4 + (a^5*m^5 + 50*a^5*m ^4 + 955*a^5*m^3 + 8650*a^5*m^2 + 36824*a^5*m + 58240*a^5)*x)*x^m/(m^6 + 5 1*m^5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*m + 58240)
Leaf count of result is larger than twice the leaf count of optimal. 2006 vs. \(2 (87) = 174\).
Time = 1.04 (sec) , antiderivative size = 2006, normalized size of antiderivative = 20.68 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\text {Too large to display} \]
Piecewise((-a**5/(15*x**15) - 5*a**4*b/(12*x**12) - 10*a**3*b**2/(9*x**9) - 5*a**2*b**3/(3*x**6) - 5*a*b**4/(3*x**3) + b**5*log(x), Eq(m, -16)), (-a **5/(12*x**12) - 5*a**4*b/(9*x**9) - 5*a**3*b**2/(3*x**6) - 10*a**2*b**3/( 3*x**3) + 5*a*b**4*log(x) + b**5*x**3/3, Eq(m, -13)), (-a**5/(9*x**9) - 5* a**4*b/(6*x**6) - 10*a**3*b**2/(3*x**3) + 10*a**2*b**3*log(x) + 5*a*b**4*x **3/3 + b**5*x**6/6, Eq(m, -10)), (-a**5/(6*x**6) - 5*a**4*b/(3*x**3) + 10 *a**3*b**2*log(x) + 10*a**2*b**3*x**3/3 + 5*a*b**4*x**6/6 + b**5*x**9/9, E q(m, -7)), (-a**5/(3*x**3) + 5*a**4*b*log(x) + 10*a**3*b**2*x**3/3 + 5*a** 2*b**3*x**6/3 + 5*a*b**4*x**9/9 + b**5*x**12/12, Eq(m, -4)), (a**5*log(x) + 5*a**4*b*x**3/3 + 5*a**3*b**2*x**6/3 + 10*a**2*b**3*x**9/9 + 5*a*b**4*x* *12/12 + b**5*x**15/15, Eq(m, -1)), (a**5*m**5*x*x**m/(m**6 + 51*m**5 + 10 05*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 50*a**5*m**4*x*x**m/ (m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 955*a**5*m**3*x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 8650*a**5*m**2*x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9 605*m**3 + 45474*m**2 + 95064*m + 58240) + 36824*a**5*m*x*x**m/(m**6 + 51* m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 58240*a**5* x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58 240) + 5*a**4*b*m**5*x**4*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 4 5474*m**2 + 95064*m + 58240) + 235*a**4*b*m**4*x**4*x**m/(m**6 + 51*m**...
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\frac {b^{5} x^{m + 16}}{m + 16} + \frac {5 \, a b^{4} x^{m + 13}}{m + 13} + \frac {10 \, a^{2} b^{3} x^{m + 10}}{m + 10} + \frac {10 \, a^{3} b^{2} x^{m + 7}}{m + 7} + \frac {5 \, a^{4} b x^{m + 4}}{m + 4} + \frac {a^{5} x^{m + 1}}{m + 1} \]
b^5*x^(m + 16)/(m + 16) + 5*a*b^4*x^(m + 13)/(m + 13) + 10*a^2*b^3*x^(m + 10)/(m + 10) + 10*a^3*b^2*x^(m + 7)/(m + 7) + 5*a^4*b*x^(m + 4)/(m + 4) + a^5*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 540, normalized size of antiderivative = 5.57 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\frac {b^{5} m^{5} x^{16} x^{m} + 35 \, b^{5} m^{4} x^{16} x^{m} + 445 \, b^{5} m^{3} x^{16} x^{m} + 5 \, a b^{4} m^{5} x^{13} x^{m} + 2485 \, b^{5} m^{2} x^{16} x^{m} + 190 \, a b^{4} m^{4} x^{13} x^{m} + 5714 \, b^{5} m x^{16} x^{m} + 2555 \, a b^{4} m^{3} x^{13} x^{m} + 3640 \, b^{5} x^{16} x^{m} + 10 \, a^{2} b^{3} m^{5} x^{10} x^{m} + 14810 \, a b^{4} m^{2} x^{13} x^{m} + 410 \, a^{2} b^{3} m^{4} x^{10} x^{m} + 34840 \, a b^{4} m x^{13} x^{m} + 5950 \, a^{2} b^{3} m^{3} x^{10} x^{m} + 22400 \, a b^{4} x^{13} x^{m} + 10 \, a^{3} b^{2} m^{5} x^{7} x^{m} + 36550 \, a^{2} b^{3} m^{2} x^{10} x^{m} + 440 \, a^{3} b^{2} m^{4} x^{7} x^{m} + 89240 \, a^{2} b^{3} m x^{10} x^{m} + 6970 \, a^{3} b^{2} m^{3} x^{7} x^{m} + 58240 \, a^{2} b^{3} x^{10} x^{m} + 5 \, a^{4} b m^{5} x^{4} x^{m} + 47260 \, a^{3} b^{2} m^{2} x^{7} x^{m} + 235 \, a^{4} b m^{4} x^{4} x^{m} + 123920 \, a^{3} b^{2} m x^{7} x^{m} + 4085 \, a^{4} b m^{3} x^{4} x^{m} + 83200 \, a^{3} b^{2} x^{7} x^{m} + a^{5} m^{5} x x^{m} + 31685 \, a^{4} b m^{2} x^{4} x^{m} + 50 \, a^{5} m^{4} x x^{m} + 100630 \, a^{4} b m x^{4} x^{m} + 955 \, a^{5} m^{3} x x^{m} + 72800 \, a^{4} b x^{4} x^{m} + 8650 \, a^{5} m^{2} x x^{m} + 36824 \, a^{5} m x x^{m} + 58240 \, a^{5} x x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \]
(b^5*m^5*x^16*x^m + 35*b^5*m^4*x^16*x^m + 445*b^5*m^3*x^16*x^m + 5*a*b^4*m ^5*x^13*x^m + 2485*b^5*m^2*x^16*x^m + 190*a*b^4*m^4*x^13*x^m + 5714*b^5*m* x^16*x^m + 2555*a*b^4*m^3*x^13*x^m + 3640*b^5*x^16*x^m + 10*a^2*b^3*m^5*x^ 10*x^m + 14810*a*b^4*m^2*x^13*x^m + 410*a^2*b^3*m^4*x^10*x^m + 34840*a*b^4 *m*x^13*x^m + 5950*a^2*b^3*m^3*x^10*x^m + 22400*a*b^4*x^13*x^m + 10*a^3*b^ 2*m^5*x^7*x^m + 36550*a^2*b^3*m^2*x^10*x^m + 440*a^3*b^2*m^4*x^7*x^m + 892 40*a^2*b^3*m*x^10*x^m + 6970*a^3*b^2*m^3*x^7*x^m + 58240*a^2*b^3*x^10*x^m + 5*a^4*b*m^5*x^4*x^m + 47260*a^3*b^2*m^2*x^7*x^m + 235*a^4*b*m^4*x^4*x^m + 123920*a^3*b^2*m*x^7*x^m + 4085*a^4*b*m^3*x^4*x^m + 83200*a^3*b^2*x^7*x^ m + a^5*m^5*x*x^m + 31685*a^4*b*m^2*x^4*x^m + 50*a^5*m^4*x*x^m + 100630*a^ 4*b*m*x^4*x^m + 955*a^5*m^3*x*x^m + 72800*a^4*b*x^4*x^m + 8650*a^5*m^2*x*x ^m + 36824*a^5*m*x*x^m + 58240*a^5*x*x^m)/(m^6 + 51*m^5 + 1005*m^4 + 9605* m^3 + 45474*m^2 + 95064*m + 58240)
Time = 6.24 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.01 \[ \int x^m \left (a+b x^3\right )^5 \, dx=\frac {a^5\,x\,x^m\,\left (m^5+50\,m^4+955\,m^3+8650\,m^2+36824\,m+58240\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {b^5\,x^m\,x^{16}\,\left (m^5+35\,m^4+445\,m^3+2485\,m^2+5714\,m+3640\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {5\,a\,b^4\,x^m\,x^{13}\,\left (m^5+38\,m^4+511\,m^3+2962\,m^2+6968\,m+4480\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {5\,a^4\,b\,x^m\,x^4\,\left (m^5+47\,m^4+817\,m^3+6337\,m^2+20126\,m+14560\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {10\,a^2\,b^3\,x^m\,x^{10}\,\left (m^5+41\,m^4+595\,m^3+3655\,m^2+8924\,m+5824\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240}+\frac {10\,a^3\,b^2\,x^m\,x^7\,\left (m^5+44\,m^4+697\,m^3+4726\,m^2+12392\,m+8320\right )}{m^6+51\,m^5+1005\,m^4+9605\,m^3+45474\,m^2+95064\,m+58240} \]
(a^5*x*x^m*(36824*m + 8650*m^2 + 955*m^3 + 50*m^4 + m^5 + 58240))/(95064*m + 45474*m^2 + 9605*m^3 + 1005*m^4 + 51*m^5 + m^6 + 58240) + (b^5*x^m*x^16 *(5714*m + 2485*m^2 + 445*m^3 + 35*m^4 + m^5 + 3640))/(95064*m + 45474*m^2 + 9605*m^3 + 1005*m^4 + 51*m^5 + m^6 + 58240) + (5*a*b^4*x^m*x^13*(6968*m + 2962*m^2 + 511*m^3 + 38*m^4 + m^5 + 4480))/(95064*m + 45474*m^2 + 9605* m^3 + 1005*m^4 + 51*m^5 + m^6 + 58240) + (5*a^4*b*x^m*x^4*(20126*m + 6337* m^2 + 817*m^3 + 47*m^4 + m^5 + 14560))/(95064*m + 45474*m^2 + 9605*m^3 + 1 005*m^4 + 51*m^5 + m^6 + 58240) + (10*a^2*b^3*x^m*x^10*(8924*m + 3655*m^2 + 595*m^3 + 41*m^4 + m^5 + 5824))/(95064*m + 45474*m^2 + 9605*m^3 + 1005*m ^4 + 51*m^5 + m^6 + 58240) + (10*a^3*b^2*x^m*x^7*(12392*m + 4726*m^2 + 697 *m^3 + 44*m^4 + m^5 + 8320))/(95064*m + 45474*m^2 + 9605*m^3 + 1005*m^4 + 51*m^5 + m^6 + 58240)