Integrand size = 9, antiderivative size = 24 \[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (10,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^{10}} \] Output:
x*hypergeom([10, 1/n],[1+1/n],-b*x^n/a)/a^10
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (10,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^{10}} \] Input:
Integrate[(a + b*x^n)^(-10),x]
Output:
(x*Hypergeometric2F1[10, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^10
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {778}
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.
\(\displaystyle \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {x \operatorname {Hypergeometric2F1}\left (10,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^{10}}\) |
Input:
Int[(a + b*x^n)^(-10),x]
Output:
(x*Hypergeometric2F1[10, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^10
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
\[\int \frac {1}{\left (a +b \,x^{n}\right )^{10}}d x\]
Input:
int(1/(a+b*x^n)^10,x)
Output:
int(1/(a+b*x^n)^10,x)
\[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:
integrate(1/(a+b*x^n)^10,x, algorithm="fricas")
Output:
integral(1/(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 )
Timed out. \[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*x**n)**10,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:
integrate(1/(a+b*x^n)^10,x, algorithm="maxima")
Output:
(362880*n^9 - 1026576*n^8 + 1172700*n^7 - 723680*n^6 + 269325*n^5 - 63273* n^4 + 9450*n^3 - 870*n^2 + 45*n - 1)*integrate(1/362880/(a^9*b*n^9*x^n + a ^10*n^9), x) + 1/362880*((362880*n^8 - 663696*n^7 + 509004*n^6 - 214676*n^ 5 + 54649*n^4 - 8624*n^3 + 826*n^2 - 44*n + 1)*b^8*x*x^(8*n) + (3084480*n^ 8 - 5550696*n^7 + 4205970*n^6 - 1757777*n^5 + 444332*n^4 - 69734*n^3 + 665 0*n^2 - 353*n + 8)*a*b^7*x*x^(7*n) + (11551680*n^8 - 20391816*n^7 + 152388 26*n^6 - 6304041*n^5 + 1581402*n^4 - 246744*n^3 + 23424*n^2 - 1239*n + 28) *a^2*b^6*x*x^(6*n) + (24948000*n^8 - 43020900*n^7 + 31631760*n^6 - 1293569 5*n^5 + 3217994*n^4 - 499000*n^3 + 47150*n^2 - 2485*n + 56)*a^3*b^5*x*x^(5 *n) + (34092576*n^8 - 57082500*n^7 + 41163120*n^6 - 16613455*n^5 + 4095154 *n^4 - 630850*n^3 + 59320*n^2 - 3115*n + 70)*a^4*b^4*x*x^(4*n) + (30348864 *n^8 - 48872856*n^7 + 34409554*n^6 - 13677471*n^5 + 3337440*n^4 - 510534*n ^3 + 47766*n^2 - 2499*n + 56)*a^5*b^3*x*x^(3*n) + (17361216*n^8 - 26448696 *n^7 + 18059130*n^6 - 7050407*n^5 + 1701106*n^4 - 258284*n^3 + 24040*n^2 - 1253*n + 28)*a^6*b^2*x*x^(2*n) + (5973264*n^8 - 8314956*n^7 + 5446476*n^6 - 2080961*n^5 + 495818*n^4 - 74684*n^3 + 6914*n^2 - 359*n + 8)*a^7*b*x*x^ n + (1026576*n^8 - 1172700*n^7 + 723680*n^6 - 269325*n^5 + 63273*n^4 - 945 0*n^3 + 870*n^2 - 45*n + 1)*a^8*x)/(a^9*b^9*n^9*x^(9*n) + 9*a^10*b^8*n^9*x ^(8*n) + 36*a^11*b^7*n^9*x^(7*n) + 84*a^12*b^6*n^9*x^(6*n) + 126*a^13*b^5* n^9*x^(5*n) + 126*a^14*b^4*n^9*x^(4*n) + 84*a^15*b^3*n^9*x^(3*n) + 36*a...
\[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{10}} \,d x } \] Input:
integrate(1/(a+b*x^n)^10,x, algorithm="giac")
Output:
integrate((b*x^n + a)^(-10), x)
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\frac {x\,{{}}_2{\mathrm {F}}_1\left (10,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^{10}} \] Input:
int(1/(a + b*x^n)^10,x)
Output:
(x*hypergeom([10, 1/n], 1/n + 1, -(b*x^n)/a))/a^10
\[ \int \frac {1}{\left (a+b x^n\right )^{10}} \, dx=\int \frac {1}{x^{10 n} b^{10}+10 x^{9 n} a \,b^{9}+45 x^{8 n} a^{2} b^{8}+120 x^{7 n} a^{3} b^{7}+210 x^{6 n} a^{4} b^{6}+252 x^{5 n} a^{5} b^{5}+210 x^{4 n} a^{6} b^{4}+120 x^{3 n} a^{7} b^{3}+45 x^{2 n} a^{8} b^{2}+10 x^{n} a^{9} b +a^{10}}d x \] Input:
int(1/(a+b*x^n)^10,x)
Output:
int(1/(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),x)