\(\int \frac {x^{-1-n (-1+p)} (a+b x^n)^p}{(c+d x^n)^2} \, dx\) [543]

Optimal result

Integrand size = 29, antiderivative size = 71 \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\frac {a x^{n (1-p)} \left (a+b x^n\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{c^2 n (1-p)} \] Output:

a*x^(n*(1-p))*(a+b*x^n)^(-1+p)*hypergeom([2, 1-p],[-p+2],(-a*d+b*c)*x^n/c/ 
(a+b*x^n))/c^2/n/(1-p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.45 \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=-\frac {x^{n-n p} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{c n (-1+p) \left (c+d x^n\right )} \] Input:

Integrate[(x^(-1 - n*(-1 + p))*(a + b*x^n)^p)/(c + d*x^n)^2,x]
 

Output:

-((x^(n - n*p)*(a + b*x^n)^p*(1 + (d*x^n)/c)^p*Hypergeometric2F1[1 - p, -p 
, 2 - p, ((-(b*c) + a*d)*x^n)/(a*(c + d*x^n))])/(c*n*(-1 + p)*(1 + (b*x^n) 
/a)^p*(c + d*x^n)))
 

Rubi [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1013, 1012}

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.

\(\displaystyle \int \frac {x^{-n (p-1)-1} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {x^{-p n+n-1} \left (\frac {b x^n}{a}+1\right )^p}{\left (d x^n+c\right )^2}dx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{n-n p} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (\frac {d x^n}{c}+1\right )^{p-1} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {\frac {b x^n}{a}-\frac {d x^n}{c}}{\frac {d x^n}{c}+1}\right )}{c^2 n (1-p)}\)

Input:

Int[(x^(-1 - n*(-1 + p))*(a + b*x^n)^p)/(c + d*x^n)^2,x]
 

Output:

(x^(n - n*p)*(a + b*x^n)^p*(1 + (d*x^n)/c)^(-1 + p)*Hypergeometric2F1[1 - 
p, -p, 2 - p, -(((b*x^n)/a - (d*x^n)/c)/(1 + (d*x^n)/c))])/(c^2*n*(1 - p)* 
(1 + (b*x^n)/a)^p)
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ 
n/a))^FracPart[p])   Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {x^{-1-n \left (p -1\right )} \left (a +b \,x^{n}\right )^{p}}{\left (c +d \,x^{n}\right )^{2}}d x\]

Input:

int(x^(-1-n*(p-1))*(a+b*x^n)^p/(c+d*x^n)^2,x)
 

Output:

int(x^(-1-n*(p-1))*(a+b*x^n)^p/(c+d*x^n)^2,x)
 

Fricas [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n)^2,x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^p*x^(-n*p + n - 1)/(d^2*x^(2*n) + 2*c*d*x^n + c^2), x 
)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(-1-n*(-1+p))*(a+b*x**n)**p/(c+d*x**n)**2,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 1) - 1)/(d*x^n + c)^2, x)
 

Giac [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 1\right )} - 1}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \] Input:

integrate(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n)^2,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 1) - 1)/(d*x^n + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,\left (p-1\right )+1}\,{\left (c+d\,x^n\right )}^2} \,d x \] Input:

int((a + b*x^n)^p/(x^(n*(p - 1) + 1)*(c + d*x^n)^2),x)
 

Output:

int((a + b*x^n)^p/(x^(n*(p - 1) + 1)*(c + d*x^n)^2), x)
 

Reduce [F]

\[ \int \frac {x^{-1-n (-1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^2} \, dx=\text {too large to display} \] Input:

int(x^(-1-n*(-1+p))*(a+b*x^n)^p/(c+d*x^n)^2,x)
 

Output:

( - x**n*(x**n*b + a)**p*b + x**(n*p + n)*int((x**n*(x**n*b + a)**p)/(x**( 
n*p + 3*n)*a*b*d**3*p*x - x**(n*p + 3*n)*b**2*c*d**2*x + x**(n*p + 2*n)*a* 
*2*d**3*p*x + 2*x**(n*p + 2*n)*a*b*c*d**2*p*x - x**(n*p + 2*n)*a*b*c*d**2* 
x - 2*x**(n*p + 2*n)*b**2*c**2*d*x + 2*x**(n*p + n)*a**2*c*d**2*p*x + x**( 
n*p + n)*a*b*c**2*d*p*x - 2*x**(n*p + n)*a*b*c**2*d*x - x**(n*p + n)*b**2* 
c**3*x + x**(n*p)*a**2*c**2*d*p*x - x**(n*p)*a*b*c**3*x),x)*a**3*d**3*n*p* 
*2 - x**(n*p + n)*int((x**n*(x**n*b + a)**p)/(x**(n*p + 3*n)*a*b*d**3*p*x 
- x**(n*p + 3*n)*b**2*c*d**2*x + x**(n*p + 2*n)*a**2*d**3*p*x + 2*x**(n*p 
+ 2*n)*a*b*c*d**2*p*x - x**(n*p + 2*n)*a*b*c*d**2*x - 2*x**(n*p + 2*n)*b** 
2*c**2*d*x + 2*x**(n*p + n)*a**2*c*d**2*p*x + x**(n*p + n)*a*b*c**2*d*p*x 
- 2*x**(n*p + n)*a*b*c**2*d*x - x**(n*p + n)*b**2*c**3*x + x**(n*p)*a**2*c 
**2*d*p*x - x**(n*p)*a*b*c**3*x),x)*a**2*b*c*d**2*n*p**2 - x**(n*p + n)*in 
t((x**n*(x**n*b + a)**p)/(x**(n*p + 3*n)*a*b*d**3*p*x - x**(n*p + 3*n)*b** 
2*c*d**2*x + x**(n*p + 2*n)*a**2*d**3*p*x + 2*x**(n*p + 2*n)*a*b*c*d**2*p* 
x - x**(n*p + 2*n)*a*b*c*d**2*x - 2*x**(n*p + 2*n)*b**2*c**2*d*x + 2*x**(n 
*p + n)*a**2*c*d**2*p*x + x**(n*p + n)*a*b*c**2*d*p*x - 2*x**(n*p + n)*a*b 
*c**2*d*x - x**(n*p + n)*b**2*c**3*x + x**(n*p)*a**2*c**2*d*p*x - x**(n*p) 
*a*b*c**3*x),x)*a**2*b*c*d**2*n*p + x**(n*p + n)*int((x**n*(x**n*b + a)**p 
)/(x**(n*p + 3*n)*a*b*d**3*p*x - x**(n*p + 3*n)*b**2*c*d**2*x + x**(n*p + 
2*n)*a**2*d**3*p*x + 2*x**(n*p + 2*n)*a*b*c*d**2*p*x - x**(n*p + 2*n)*a...