\(\int \frac {x^m}{(a+b x^n)^{10}} \, dx\) [579]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (10,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^{10} (1+m)} \] Output:

x^(1+m)*hypergeom([10, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a^10/(1+m)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (10,\frac {1+m}{n},1+\frac {1+m}{n},-\frac {b x^n}{a}\right )}{a^{10} (1+m)} \] Input:

Integrate[x^m/(a + b*x^n)^10,x]
 

Output:

(x^(1 + m)*Hypergeometric2F1[10, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/ 
(a^10*(1 + m))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {888}

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.

\(\displaystyle \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (10,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a^{10} (m+1)}\)

Input:

Int[x^m/(a + b*x^n)^10,x]
 

Output:

(x^(1 + m)*Hypergeometric2F1[10, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/ 
(a^10*(1 + m))
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {x^{m}}{\left (a +b \,x^{n}\right )^{10}}d x\]

Input:

int(x^m/(a+b*x^n)^10,x)
 

Output:

int(x^m/(a+b*x^n)^10,x)
 

Fricas [F]

\[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {x^{m}}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:

integrate(x^m/(a+b*x^n)^10,x, algorithm="fricas")
 

Output:

integral(x^m/(b^10*x^(10*n) + 10*a*b^9*x^(9*n) + 45*a^2*b^8*x^(8*n) + 120* 
a^3*b^7*x^(7*n) + 210*a^4*b^6*x^(6*n) + 252*a^5*b^5*x^(5*n) + 210*a^6*b^4* 
x^(4*n) + 120*a^7*b^3*x^(3*n) + 45*a^8*b^2*x^(2*n) + 10*a^9*b*x^n + a^10), 
 x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\text {Timed out} \] Input:

integrate(x**m/(a+b*x**n)**10,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {x^{m}}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:

integrate(x^m/(a+b*x^n)^10,x, algorithm="maxima")
 

Output:

-(m^9 - 9*m^8*(5*n - 1) - 362880*n^9 + 6*(145*n^2 - 60*n + 6)*m^7 + 102657 
6*n^8 - 42*(225*n^3 - 145*n^2 + 30*n - 2)*m^6 - 1172700*n^7 + 21*(3013*n^4 
 - 2700*n^3 + 870*n^2 - 120*n + 6)*m^5 + 723680*n^6 - 21*(12825*n^5 - 1506 
5*n^4 + 6750*n^3 - 1450*n^2 + 150*n - 6)*m^4 - 269325*n^5 + 2*(361840*n^6 
- 538650*n^5 + 316365*n^4 - 94500*n^3 + 15225*n^2 - 1260*n + 42)*m^3 + 632 
73*n^4 - 6*(195450*n^7 - 361840*n^6 + 269325*n^5 - 105455*n^4 + 23625*n^3 
- 3045*n^2 + 210*n - 6)*m^2 - 9450*n^3 + 3*(342192*n^8 - 781800*n^7 + 7236 
80*n^6 - 359100*n^5 + 105455*n^4 - 18900*n^3 + 2030*n^2 - 120*n + 3)*m + 8 
70*n^2 - 45*n + 1)*integrate(1/362880*x^m/(a^9*b*n^9*x^n + a^10*n^9), x) + 
 1/362880*((m^8 - m^7*(45*n - 8) + 1026576*n^8 + (870*n^2 - 315*n + 28)*m^ 
6 - 1172700*n^7 - (9450*n^3 - 5220*n^2 + 945*n - 56)*m^5 + 723680*n^6 + (6 
3273*n^4 - 47250*n^3 + 13050*n^2 - 1575*n + 70)*m^4 - 269325*n^5 - (269325 
*n^5 - 253092*n^4 + 94500*n^3 - 17400*n^2 + 1575*n - 56)*m^3 + 63273*n^4 + 
 (723680*n^6 - 807975*n^5 + 379638*n^4 - 94500*n^3 + 13050*n^2 - 945*n + 2 
8)*m^2 - 9450*n^3 - (1172700*n^7 - 1447360*n^6 + 807975*n^5 - 253092*n^4 + 
 47250*n^3 - 5220*n^2 + 315*n - 8)*m + 870*n^2 - 45*n + 1)*a^8*x*x^m + (m^ 
8 - 4*m^7*(11*n - 2) + 362880*n^8 + 14*(59*n^2 - 22*n + 2)*m^6 - 663696*n^ 
7 - 28*(308*n^3 - 177*n^2 + 33*n - 2)*m^5 + 509004*n^6 + 7*(7807*n^4 - 616 
0*n^3 + 1770*n^2 - 220*n + 10)*m^4 - 214676*n^5 - 28*(7667*n^5 - 7807*n^4 
+ 3080*n^3 - 590*n^2 + 55*n - 2)*m^3 + 54649*n^4 + 2*(254502*n^6 - 3220...
 

Giac [F]

\[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {x^{m}}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:

integrate(x^m/(a+b*x^n)^10,x, algorithm="giac")
 

Output:

integrate(x^m/(b*x^n + a)^10, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\int \frac {x^m}{{\left (a+b\,x^n\right )}^{10}} \,d x \] Input:

int(x^m/(a + b*x^n)^10,x)
 

Output:

int(x^m/(a + b*x^n)^10, x)
 

Reduce [F]

\[ \int \frac {x^m}{\left (a+b x^n\right )^{10}} \, dx=\int \frac {x^{m}}{x^{10 n} b^{10}+10 x^{9 n} a \,b^{9}+45 x^{8 n} a^{2} b^{8}+120 x^{7 n} a^{3} b^{7}+210 x^{6 n} a^{4} b^{6}+252 x^{5 n} a^{5} b^{5}+210 x^{4 n} a^{6} b^{4}+120 x^{3 n} a^{7} b^{3}+45 x^{2 n} a^{8} b^{2}+10 x^{n} a^{9} b +a^{10}}d x \] Input:

int(x^m/(a+b*x^n)^10,x)
 

Output:

int(x**m/(x**(10*n)*b**10 + 10*x**(9*n)*a*b**9 + 45*x**(8*n)*a**2*b**8 + 1 
20*x**(7*n)*a**3*b**7 + 210*x**(6*n)*a**4*b**6 + 252*x**(5*n)*a**5*b**5 + 
210*x**(4*n)*a**6*b**4 + 120*x**(3*n)*a**7*b**3 + 45*x**(2*n)*a**8*b**2 + 
10*x**n*a**9*b + a**10),x)