\(\int \frac {x^{-1-n (-2+p)} (a+b x^n)^p}{(c+d x^n)^3} \, dx\) [551]

Optimal result

Integrand size = 29, antiderivative size = 73 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {a^2 x^{n (2-p)} \left (a+b x^n\right )^{-2+p} \operatorname {Hypergeometric2F1}\left (3,2-p,3-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{c^3 n (2-p)} \] Output:

a^2*x^(n*(-p+2))*(a+b*x^n)^(-2+p)*hypergeom([3, -p+2],[3-p],(-a*d+b*c)*x^n 
/c/(a+b*x^n))/c^3/n/(-p+2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=-\frac {x^{-n (-2+p)} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^{-2+p} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,\frac {-\frac {b x^n}{a}+\frac {d x^n}{c}}{1+\frac {d x^n}{c}}\right )}{c^3 n (-2+p)} \] Input:

Integrate[(x^(-1 - n*(-2 + p))*(a + b*x^n)^p)/(c + d*x^n)^3,x]
 

Output:

-(((a + b*x^n)^p*(1 + (d*x^n)/c)^(-2 + p)*Hypergeometric2F1[2 - p, -p, 3 - 
 p, (-((b*x^n)/a) + (d*x^n)/c)/(1 + (d*x^n)/c)])/(c^3*n*(-2 + p)*x^(n*(-2 
+ p))*(1 + (b*x^n)/a)^p))
 

Rubi [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45, 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.

\(\displaystyle \int \frac {x^{-n (p-2)-1} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {x^{n (2-p)-1} \left (\frac {b x^n}{a}+1\right )^p}{\left (d x^n+c\right )^3}dx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{n (2-p)} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (\frac {d x^n}{c}+1\right )^{p-2} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {\frac {b x^n}{a}-\frac {d x^n}{c}}{\frac {d x^n}{c}+1}\right )}{c^3 n (2-p)}\)

Input:

Int[(x^(-1 - n*(-2 + p))*(a + b*x^n)^p)/(c + d*x^n)^3,x]
 

Output:

(x^(n*(2 - p))*(a + b*x^n)^p*(1 + (d*x^n)/c)^(-2 + p)*Hypergeometric2F1[2 
- p, -p, 3 - p, -(((b*x^n)/a - (d*x^n)/c)/(1 + (d*x^n)/c))])/(c^3*n*(2 - 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]

\[\int \frac {x^{-1-n \left (-2+p \right )} \left (a +b \,x^{n}\right )^{p}}{\left (c +d \,x^{n}\right )^{3}}d x\]

Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x)
 

Output:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x)
 

Fricas [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^p*x^(-n*p + 2*n - 1)/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 
 3*c^2*d*x^n + c^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(-1-n*(-2+p))*(a+b*x**n)**p/(c+d*x**n)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 2) - 1)/(d*x^n + c)^3, x)
 

Giac [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 2) - 1)/(d*x^n + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,\left (p-2\right )+1}\,{\left (c+d\,x^n\right )}^3} \,d x \] Input:

int((a + b*x^n)^p/(x^(n*(p - 2) + 1)*(c + d*x^n)^3),x)
 

Output:

int((a + b*x^n)^p/(x^(n*(p - 2) + 1)*(c + d*x^n)^3), x)
 

Reduce [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\text {too large to display} \] Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^3,x)
 

Output:

( - x**(2*n)*(x**n*b + a)**p*a**2*b*d**2*p**2 - 3*x**(2*n)*(x**n*b + a)**p 
*a**2*b*d**2*p - 2*x**(2*n)*(x**n*b + a)**p*a**2*b*d**2 + 4*x**(2*n)*(x**n 
*b + a)**p*a*b**2*c*d*p + 5*x**(2*n)*(x**n*b + a)**p*a*b**2*c*d - 3*x**(2* 
n)*(x**n*b + a)**p*b**3*c**2 - x**n*(x**n*b + a)**p*a**3*d**2*p**2 - 3*x** 
n*(x**n*b + a)**p*a**3*d**2*p - 2*x**n*(x**n*b + a)**p*a**3*d**2 + x**n*(x 
**n*b + a)**p*a**2*b*c*d*p**2 + 6*x**n*(x**n*b + a)**p*a**2*b*c*d*p + 2*x* 
*n*(x**n*b + a)**p*a**2*b*c*d - 3*x**n*(x**n*b + a)**p*a*b**2*c**2*p + 2*x 
**(n*p + 2*n)*int((x**(3*n)*(x**n*b + a)**p)/(x**(n*p + 4*n)*a**3*b*d**6*p 
**3*x + 4*x**(n*p + 4*n)*a**3*b*d**6*p**2*x + 5*x**(n*p + 4*n)*a**3*b*d**6 
*p*x + 2*x**(n*p + 4*n)*a**3*b*d**6*x - 7*x**(n*p + 4*n)*a**2*b**2*c*d**5* 
p**2*x - 15*x**(n*p + 4*n)*a**2*b**2*c*d**5*p*x - 8*x**(n*p + 4*n)*a**2*b* 
*2*c*d**5*x + 12*x**(n*p + 4*n)*a*b**3*c**2*d**4*p*x + 12*x**(n*p + 4*n)*a 
*b**3*c**2*d**4*x - 6*x**(n*p + 4*n)*b**4*c**3*d**3*x + x**(n*p + 3*n)*a** 
4*d**6*p**3*x + 4*x**(n*p + 3*n)*a**4*d**6*p**2*x + 5*x**(n*p + 3*n)*a**4* 
d**6*p*x + 2*x**(n*p + 3*n)*a**4*d**6*x + 3*x**(n*p + 3*n)*a**3*b*c*d**5*p 
**3*x + 5*x**(n*p + 3*n)*a**3*b*c*d**5*p**2*x - 2*x**(n*p + 3*n)*a**3*b*c* 
d**5*x - 21*x**(n*p + 3*n)*a**2*b**2*c**2*d**4*p**2*x - 33*x**(n*p + 3*n)* 
a**2*b**2*c**2*d**4*p*x - 12*x**(n*p + 3*n)*a**2*b**2*c**2*d**4*x + 36*x** 
(n*p + 3*n)*a*b**3*c**3*d**3*p*x + 30*x**(n*p + 3*n)*a*b**3*c**3*d**3*x - 
18*x**(n*p + 3*n)*b**4*c**4*d**2*x + 3*x**(n*p + 2*n)*a**4*c*d**5*p**3*...