\(\int (a+b x^n)^p (c+d x^n)^{-2-\frac {1}{n}-p} \, dx\) [134]

Optimal result

Integrand size = 28, antiderivative size = 164 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-2-\frac {1}{n}-p} \, dx=\frac {x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p}}{a c}+\frac {(a d n (1+p)-b c (1+n+n p)) x^{1+n} \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \operatorname {Hypergeometric2F1}\left (1+\frac {1}{n},-p,2+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a c^2 (1+n)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {907, 905}

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.

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

\(\Big \downarrow \) 907

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

\(\Big \downarrow \) 905

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

Input:

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

Output:

-((b*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(-1 - n^(-1) - p))/(a*(b*c - a*d)*n 
*(1 + p))) + ((1 + (b*c)/((b*c - a*d)*n*(1 + p)))*x*(a + b*x^n)^(1 + p)*(( 
c*(a + b*x^n))/(a*(c + d*x^n)))^(-1 - p)*(c + d*x^n)^(-1 - n^(-1) - p)*Hyp 
ergeometric2F1[n^(-1), -1 - p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x 
^n)))])/(a*c)
 

Defintions of rubi rules used

rule 905

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[x*((a + b*x^n)^p/(c*(c*((a + b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n) 
^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c 
+ d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && 
EqQ[n*(p + q + 1) + 1, 0]
 

rule 907

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) 
  Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} 
, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  ! 
LtQ[q, -1]) && NeQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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